| {"message": {"transcript": [{"chunks": [{"end": 1.76, "start": 0.0, "text": "Hello,"}, {"end": 2.28, "start": 1.76, "text": "my"}, {"end": 2.64, "start": 2.28, "text": "name"}, {"end": 2.84, "start": 2.64, "text": "is"}, {"end": 2.96, "start": 2.84, "text": "Jesper"}, {"end": 3.32, "start": 2.96, "text": "Nedeloff"}, {"end": 3.56, "start": 3.32, "text": "and"}, {"end": 3.68, "start": 3.56, "text": "I'm"}, {"end": 3.76, "start": 3.68, "text": "going"}, {"end": 3.84, "start": 3.76, "text": "to"}, {"end": 4.16, "start": 3.84, "text": "talk"}, {"end": 4.28, "start": 4.16, "text": "about"}, {"end": 4.44, "start": 4.28, "text": "my"}, {"end": 4.96, "start": 4.44, "text": "paper"}, {"end": 5.2, "start": 4.96, "text": "about"}, {"end": 5.52, "start": 5.2, "text": "the"}, {"end": 5.68, "start": 5.52, "text": "bipartite"}, {"end": 6.04, "start": 5.68, "text": "traveling"}, {"end": 6.4, "start": 6.04, "text": "salesman"}, {"end": 7.92, "start": 6.4, "text": "problem."}, {"end": 8.12, "start": 7.92, "text": "In"}, {"end": 8.32, "start": 8.12, "text": "the"}, {"end": 8.72, "start": 8.32, "text": "traveling"}, {"end": 9.08, "start": 8.72, "text": "salesman"}, {"end": 9.4, "start": 9.08, "text": "problem"}, {"end": 9.6, "start": 9.4, "text": "we"}, {"end": 9.64, "start": 9.6, "text": "are"}, {"end": 10.04, "start": 9.64, "text": "given"}, {"end": 10.28, "start": 10.04, "text": "n"}, {"end": 11.16, "start": 10.28, "text": "vertices"}, {"end": 11.36, "start": 11.16, "text": "and"}, {"end": 11.64, "start": 11.36, "text": "between"}, {"end": 11.84, "start": 11.64, "text": "each"}, {"end": 12.04, "start": 11.84, "text": "pair"}, {"end": 12.2, "start": 12.04, "text": "of"}, {"end": 13.12, "start": 12.2, "text": "vertices"}, {"end": 13.28, "start": 13.12, "text": "we"}, {"end": 13.32, "start": 13.28, "text": "are"}, {"end": 13.64, "start": 13.32, "text": "given"}, {"end": 13.8, "start": 13.64, "text": "a"}, {"end": 14.88, "start": 13.8, "text": "distance."}, {"end": 15.04, "start": 14.88, "text": "In"}, {"end": 15.4, "start": 15.04, "text": "this"}, {"end": 16.16, "start": 15.4, "text": "talk"}, {"end": 16.36, "start": 16.16, "text": "this"}, {"end": 16.88, "start": 16.36, "text": "distance"}, {"end": 17.04, "start": 16.88, "text": "will"}, {"end": 17.24, "start": 17.04, "text": "be"}, {"end": 17.24, "start": 17.24, "text": "an"}, {"end": 17.52, "start": 17.24, "text": "integer"}, {"end": 17.88, "start": 17.52, "text": "between"}, {"end": 18.0, "start": 17.88, "text": "1"}, {"end": 18.2, "start": 18.0, "text": "and"}, {"end": 18.52, "start": 18.2, "text": "capital"}, {"end": 19.08, "start": 18.52, "text": "W"}, {"end": 19.24, "start": 19.08, "text": "or"}, {"end": 20.4, "start": 19.24, "text": "infinity."}, {"end": 20.52, "start": 20.4, "text": "The"}, {"end": 20.84, "start": 20.52, "text": "goal"}, {"end": 21.08, "start": 20.84, "text": "is"}, {"end": 21.12, "start": 21.08, "text": "to"}, {"end": 21.4, "start": 21.12, "text": "find"}, {"end": 21.48, "start": 21.4, "text": "the"}, {"end": 21.92, "start": 21.48, "text": "shortest"}, {"end": 22.08, "start": 21.92, "text": "round"}, {"end": 22.24, "start": 22.08, "text": "trip"}, {"end": 22.4, "start": 22.24, "text": "that"}, {"end": 22.8, "start": 22.4, "text": "visits"}, {"end": 22.88, "start": 22.8, "text": "all"}, {"end": 23.08, "start": 22.88, "text": "the"}, {"end": 23.44, "start": 23.08, "text": "vertices"}, {"end": 24.12, "start": 23.44, "text": "once."}, {"end": 24.12, "start": 24.12, "text": "In"}, {"end": 24.24, "start": 24.12, "text": "the"}, {"end": 24.8, "start": 24.24, "text": "example"}, {"end": 24.92, "start": 24.8, "text": "we"}, {"end": 24.92, "start": 24.92, "text": "can"}, {"end": 25.08, "start": 24.92, "text": "see"}, {"end": 25.4, "start": 25.08, "text": "here"}, {"end": 25.72, "start": 25.4, "text": "on"}, {"end": 26.28, "start": 25.72, "text": "the"}, {"end": 26.68, "start": 26.28, "text": "right"}, {"end": 27.0, "start": 26.68, "text": "hand"}, {"end": 27.36, "start": 27.0, "text": "side"}, {"end": 27.68, "start": 27.36, "text": "I"}, {"end": 28.0, "start": 27.68, "text": "just"}, {"end": 28.56, "start": 28.0, "text": "indicated"}, {"end": 28.76, "start": 28.56, "text": "all"}, {"end": 28.76, "start": 28.76, "text": "the"}, {"end": 29.2, "start": 28.76, "text": "finite"}, {"end": 29.84, "start": 29.2, "text": "distances"}, {"end": 29.96, "start": 29.84, "text": "on"}], "text": " Hello, my name is Jesper Nedeloff and I'm going to talk about my paper about the bipartite traveling salesman problem. In the traveling salesman problem we are given n vertices and between each pair of vertices we are given a distance. In this talk this distance will be an integer between 1 and capital W or infinity. The goal is to find the shortest round trip that visits all the vertices once. In the example we can see here on the right hand side I just indicated all the finite distances on"}, {"chunks": [{"end": 30.68, "start": 30.0, "text": "The"}, {"end": 31.84, "start": 30.68, "text": "clean"}, {"end": 32.44, "start": 31.84, "text": "formulation"}, {"end": 34.12, "start": 32.44, "text": "and"}, {"end": 34.36, "start": 34.12, "text": "the"}, {"end": 34.56, "start": 34.36, "text": "clear"}, {"end": 35.64, "start": 34.56, "text": "applicability"}, {"end": 35.88, "start": 35.64, "text": "of"}, {"end": 36.2, "start": 35.88, "text": "the"}, {"end": 37.12, "start": 36.2, "text": "problem"}, {"end": 37.68, "start": 37.12, "text": "make"}, {"end": 38.2, "start": 37.68, "text": "it"}, {"end": 38.76, "start": 38.2, "text": "a"}, {"end": 39.16, "start": 38.76, "text": "favorite"}, {"end": 39.4, "start": 39.16, "text": "one"}, {"end": 39.68, "start": 39.4, "text": "in"}, {"end": 40.16, "start": 39.68, "text": "theoretical"}, {"end": 40.480000000000004, "start": 40.16, "text": "computer"}, {"end": 42.28, "start": 40.480000000000004, "text": "science."}, {"end": 42.56, "start": 42.28, "text": "In"}, {"end": 42.88, "start": 42.56, "text": "this"}, {"end": 43.12, "start": 42.88, "text": "talk"}, {"end": 43.24, "start": 43.12, "text": "we"}, {"end": 43.480000000000004, "start": 43.24, "text": "are"}, {"end": 43.8, "start": 43.480000000000004, "text": "going"}, {"end": 43.88, "start": 43.8, "text": "to"}, {"end": 44.28, "start": 43.88, "text": "focus"}, {"end": 44.32, "start": 44.28, "text": "on"}, {"end": 44.519999999999996, "start": 44.32, "text": "the"}, {"end": 44.72, "start": 44.519999999999996, "text": "worst"}, {"end": 44.96, "start": 44.72, "text": "case"}, {"end": 45.72, "start": 44.96, "text": "complexity"}, {"end": 45.84, "start": 45.72, "text": "of"}, {"end": 45.96, "start": 45.84, "text": "the"}, {"end": 47.68, "start": 45.96, "text": "problem."}, {"end": 48.28, "start": 47.68, "text": "Unfortunately"}, {"end": 48.56, "start": 48.28, "text": "the"}, {"end": 49.08, "start": 48.56, "text": "problem"}, {"end": 49.879999999999995, "start": 49.08, "text": "is"}, {"end": 50.56, "start": 49.879999999999995, "text": "NP-complete"}, {"end": 50.56, "start": 50.56, "text": "and"}, {"end": 50.96, "start": 50.56, "text": "therefore"}, {"end": 51.08, "start": 50.96, "text": "we"}, {"end": 51.36, "start": 51.08, "text": "expect"}, {"end": 51.64, "start": 51.36, "text": "to"}, {"end": 51.68, "start": 51.64, "text": "have"}, {"end": 52.0, "start": 51.68, "text": "some"}, {"end": 52.64, "start": 52.0, "text": "wall"}, {"end": 52.96, "start": 52.64, "text": "of"}, {"end": 53.480000000000004, "start": 52.96, "text": "improvements"}, {"end": 53.64, "start": 53.480000000000004, "text": "when"}, {"end": 53.68, "start": 53.64, "text": "we"}, {"end": 53.879999999999995, "start": 53.68, "text": "try"}, {"end": 54.04, "start": 53.879999999999995, "text": "to"}, {"end": 54.239999999999995, "start": 54.04, "text": "keep"}, {"end": 54.36, "start": 54.239999999999995, "text": "and"}, {"end": 54.56, "start": 54.36, "text": "keep"}, {"end": 54.92, "start": 54.56, "text": "improving."}, {"end": 55.120000000000005, "start": 54.92, "text": "If"}, {"end": 56.519999999999996, "start": 55.120000000000005, "text": "we"}, {"end": 56.92, "start": 56.519999999999996, "text": "assume"}, {"end": 57.0, "start": 56.92, "text": "that"}, {"end": 57.44, "start": 57.0, "text": "P"}, {"end": 58.120000000000005, "start": 57.44, "text": "is"}, {"end": 58.84, "start": 58.120000000000005, "text": "not"}, {"end": 59.4, "start": 58.84, "text": "equal"}, {"end": 59.72, "start": 59.4, "text": "to"}, {"end": 59.96, "start": 59.72, "text": "NP,"}], "text": " The clean formulation and the clear applicability of the problem make it a favorite one in theoretical computer science. In this talk we are going to focus on the worst case complexity of the problem. Unfortunately the problem is NP-complete and therefore we expect to have some wall of improvements when we try to keep and keep improving. If we assume that P is not equal to NP,"}, {"chunks": [{"end": 60.32, "start": 60.0, "text": "Then"}, {"end": 60.52, "start": 60.32, "text": "we"}, {"end": 61.0, "start": 60.52, "text": "know"}, {"end": 61.0, "start": 61.0, "text": "that"}, {"end": 61.4, "start": 61.0, "text": "TSP"}, {"end": 61.56, "start": 61.4, "text": "cannot"}, {"end": 61.96, "start": 61.56, "text": "be"}, {"end": 62.4, "start": 61.96, "text": "solved"}, {"end": 62.56, "start": 62.4, "text": "in"}, {"end": 62.88, "start": 62.56, "text": "polynomial"}, {"end": 63.52, "start": 62.88, "text": "time,"}, {"end": 63.68, "start": 63.52, "text": "but"}, {"end": 64.56, "start": 63.68, "text": "nevertheless"}, {"end": 64.64, "start": 64.56, "text": "it"}, {"end": 64.8, "start": 64.64, "text": "could"}, {"end": 65.0, "start": 64.8, "text": "still"}, {"end": 65.12, "start": 65.0, "text": "be"}, {"end": 65.36, "start": 65.12, "text": "solved"}, {"end": 65.44, "start": 65.36, "text": "in"}, {"end": 66.28, "start": 65.44, "text": "quasi-polynomial"}, {"end": 66.72, "start": 66.28, "text": "time"}, {"end": 67.04, "start": 66.72, "text": "or"}, {"end": 67.72, "start": 67.04, "text": "sub-exponential"}, {"end": 68.6, "start": 67.72, "text": "time"}, {"end": 68.68, "start": 68.6, "text": "or"}, {"end": 69.84, "start": 68.68, "text": "even"}, {"end": 70.44, "start": 69.84, "text": "moderately"}, {"end": 71.0, "start": 70.44, "text": "exponential"}, {"end": 73.03999999999999, "start": 71.0, "text": "time."}, {"end": 73.36, "start": 73.03999999999999, "text": "And"}, {"end": 73.52, "start": 73.36, "text": "when"}, {"end": 73.76, "start": 73.52, "text": "trying"}, {"end": 73.88, "start": 73.76, "text": "to"}, {"end": 74.03999999999999, "start": 73.88, "text": "get"}, {"end": 74.36, "start": 74.03999999999999, "text": "more"}, {"end": 74.64, "start": 74.36, "text": "and"}, {"end": 74.84, "start": 74.64, "text": "more"}, {"end": 75.64, "start": 74.84, "text": "improvements,"}, {"end": 76.32, "start": 75.64, "text": "we"}, {"end": 76.72, "start": 76.32, "text": "expect"}, {"end": 77.0, "start": 76.72, "text": "to"}, {"end": 77.28, "start": 77.0, "text": "hit"}, {"end": 77.48, "start": 77.28, "text": "some"}, {"end": 77.88, "start": 77.48, "text": "wall"}, {"end": 78.16, "start": 77.88, "text": "and"}, {"end": 78.96000000000001, "start": 78.16, "text": "maybe"}, {"end": 79.32, "start": 78.96000000000001, "text": "at"}, {"end": 79.76, "start": 79.32, "text": "that"}, {"end": 81.24, "start": 79.76, "text": "moment"}, {"end": 81.64, "start": 81.24, "text": "we"}, {"end": 84.0, "start": 81.64, "text": "get"}, {"end": 84.4, "start": 84.0, "text": "some"}, {"end": 84.76, "start": 84.4, "text": "more"}, {"end": 86.4, "start": 84.76, "text": "refined"}, {"end": 87.96000000000001, "start": 86.4, "text": "understanding"}, {"end": 88.28, "start": 87.96000000000001, "text": "of"}, {"end": 88.64, "start": 88.28, "text": "the"}, {"end": 89.96000000000001, "start": 88.64, "text": "impicompleteness."}], "text": " Then we know that TSP cannot be solved in polynomial time, but nevertheless it could still be solved in quasi-polynomial time or sub-exponential time or even moderately exponential time. And when trying to get more and more improvements, we expect to hit some wall and maybe at that moment we get some more refined understanding of the impicompleteness."}, {"chunks": [{"end": 90.84, "start": 90.0, "text": "problems,"}, {"end": 90.96, "start": 90.84, "text": "in"}, {"end": 91.44, "start": 90.96, "text": "particular"}, {"end": 91.52, "start": 91.44, "text": "for"}, {"end": 91.68, "start": 91.52, "text": "the"}, {"end": 92.4, "start": 91.68, "text": "K-CNF"}, {"end": 92.92, "start": 92.4, "text": "satisfied"}, {"end": 93.28, "start": 92.92, "text": "reality"}, {"end": 94.48, "start": 93.28, "text": "problem."}, {"end": 94.96, "start": 94.48, "text": "The"}, {"end": 95.2, "start": 94.96, "text": "strong"}, {"end": 95.68, "start": 95.2, "text": "exponential"}, {"end": 96.04, "start": 95.68, "text": "time"}, {"end": 96.88, "start": 96.04, "text": "hypothesis,"}, {"end": 97.0, "start": 96.88, "text": "which"}, {"end": 97.52, "start": 97.0, "text": "says"}, {"end": 98.03999999999999, "start": 97.52, "text": "that"}, {"end": 98.24, "start": 98.03999999999999, "text": "brute"}, {"end": 98.76, "start": 98.24, "text": "force"}, {"end": 99.0, "start": 98.76, "text": "for"}, {"end": 99.08, "start": 99.0, "text": "that"}, {"end": 99.36, "start": 99.08, "text": "problem"}, {"end": 99.88, "start": 99.36, "text": "cannot"}, {"end": 99.96000000000001, "start": 99.88, "text": "be"}, {"end": 100.2, "start": 99.96000000000001, "text": "solved"}, {"end": 100.84, "start": 100.2, "text": "significantly,"}, {"end": 101.12, "start": 100.84, "text": "led"}, {"end": 102.32, "start": 101.12, "text": "to"}, {"end": 102.6, "start": 102.32, "text": "a"}, {"end": 102.88, "start": 102.6, "text": "great"}, {"end": 103.08, "start": 102.88, "text": "amount"}, {"end": 103.24, "start": 103.08, "text": "of"}, {"end": 103.92, "start": 103.24, "text": "work"}, {"end": 104.4, "start": 103.92, "text": "on"}, {"end": 104.88, "start": 104.4, "text": "fine-grained"}, {"end": 105.08, "start": 104.88, "text": "low"}, {"end": 106.92, "start": 105.08, "text": "bounds"}, {"end": 107.03999999999999, "start": 106.92, "text": "on"}, {"end": 107.68, "start": 107.03999999999999, "text": "the"}, {"end": 108.48, "start": 107.68, "text": "complexity"}, {"end": 108.6, "start": 108.48, "text": "of"}, {"end": 108.84, "start": 108.6, "text": "problems"}, {"end": 108.88, "start": 108.84, "text": "in"}, {"end": 108.96000000000001, "start": 108.88, "text": "the"}, {"end": 109.44, "start": 108.96000000000001, "text": "complexity"}, {"end": 109.72, "start": 109.44, "text": "class"}, {"end": 111.88, "start": 109.72, "text": "P."}, {"end": 112.12, "start": 111.88, "text": "So"}, {"end": 112.72, "start": 112.12, "text": "now"}, {"end": 112.96000000000001, "start": 112.72, "text": "I"}, {"end": 113.16, "start": 112.96000000000001, "text": "want"}, {"end": 113.72, "start": 113.16, "text": "to"}, {"end": 114.36, "start": 113.72, "text": "survey"}, {"end": 114.52, "start": 114.36, "text": "a"}, {"end": 115.0, "start": 114.52, "text": "bit"}, {"end": 115.68, "start": 115.0, "text": "of"}, {"end": 115.8, "start": 115.68, "text": "the"}, {"end": 116.28, "start": 115.8, "text": "worst-case"}, {"end": 116.6, "start": 116.28, "text": "complexity"}, {"end": 116.8, "start": 116.6, "text": "of"}, {"end": 116.84, "start": 116.8, "text": "the"}, {"end": 118.12, "start": 116.84, "text": "TSP."}, {"end": 118.24, "start": 118.12, "text": "So"}, {"end": 118.48, "start": 118.24, "text": "already"}, {"end": 118.64, "start": 118.48, "text": "in"}, {"end": 118.84, "start": 118.64, "text": "the"}, {"end": 119.96000000000001, "start": 118.84, "text": "1930s,"}], "text": " problems, in particular for the K-CNF satisfied reality problem. The strong exponential time hypothesis, which says that brute force for that problem cannot be solved significantly, led to a great amount of work on fine-grained low bounds on the complexity of problems in the complexity class P. So now I want to survey a bit of the worst-case complexity of the TSP. So already in the 1930s,"}, {"chunks": [{"end": 120.36, "start": 120.0, "text": "Menger"}, {"end": 120.92, "start": 120.36, "text": "observed"}, {"end": 120.92, "start": 120.92, "text": "that"}, {"end": 121.08, "start": 120.92, "text": "there"}, {"end": 121.32, "start": 121.08, "text": "is"}, {"end": 121.4, "start": 121.32, "text": "an"}, {"end": 121.88, "start": 121.4, "text": "easy"}, {"end": 122.52, "start": 121.88, "text": "n-victorial"}, {"end": 123.04, "start": 122.52, "text": "algorithm,"}, {"end": 123.28, "start": 123.04, "text": "and"}, {"end": 123.36, "start": 123.28, "text": "he"}, {"end": 123.8, "start": 123.36, "text": "wondered"}, {"end": 124.0, "start": 123.8, "text": "when"}, {"end": 124.12, "start": 124.0, "text": "you"}, {"end": 124.16, "start": 124.12, "text": "can"}, {"end": 124.52, "start": 124.16, "text": "do"}, {"end": 125.56, "start": 124.52, "text": "faster."}, {"end": 125.68, "start": 125.56, "text": "30"}, {"end": 126.08, "start": 125.68, "text": "years"}, {"end": 126.72, "start": 126.08, "text": "later,"}, {"end": 126.96, "start": 126.72, "text": "this"}, {"end": 127.16, "start": 126.96, "text": "was"}, {"end": 127.68, "start": 127.16, "text": "solved"}, {"end": 128.16, "start": 127.68, "text": "affirmatively"}, {"end": 128.36, "start": 128.16, "text": "by"}, {"end": 128.76, "start": 128.36, "text": "Bellman"}, {"end": 129.12, "start": 128.76, "text": "and"}, {"end": 129.72, "start": 129.12, "text": "independently"}, {"end": 130.08, "start": 129.72, "text": "held"}, {"end": 130.16, "start": 130.08, "text": "in"}, {"end": 130.6, "start": 130.16, "text": "CARP,"}, {"end": 130.84, "start": 130.6, "text": "who"}, {"end": 131.0, "start": 130.84, "text": "gave"}, {"end": 131.36, "start": 131.0, "text": "a"}, {"end": 131.56, "start": 131.36, "text": "2"}, {"end": 131.76, "start": 131.56, "text": "to"}, {"end": 131.96, "start": 131.76, "text": "the"}, {"end": 132.44, "start": 131.96, "text": "n"}, {"end": 132.92, "start": 132.44, "text": "to"}, {"end": 132.92, "start": 132.92, "text": "the"}, {"end": 133.36, "start": 132.92, "text": "power"}, {"end": 134.12, "start": 133.36, "text": "n-square"}, {"end": 135.08, "start": 134.12, "text": "algorithm."}, {"end": 135.16, "start": 135.08, "text": "And"}, {"end": 135.32, "start": 135.16, "text": "this"}, {"end": 135.72, "start": 135.32, "text": "dynamic"}, {"end": 136.32, "start": 135.72, "text": "programming"}, {"end": 137.12, "start": 136.32, "text": "algorithm"}, {"end": 137.32, "start": 137.12, "text": "is"}, {"end": 137.88, "start": 137.32, "text": "now"}, {"end": 138.16, "start": 137.88, "text": "a"}, {"end": 138.76, "start": 138.16, "text": "household"}, {"end": 139.36, "start": 138.76, "text": "algorithm"}, {"end": 140.16, "start": 139.36, "text": "in"}, {"end": 140.72, "start": 140.16, "text": "many"}, {"end": 140.84, "start": 140.72, "text": "undergraduate"}, {"end": 141.24, "start": 140.84, "text": "algorithms"}, {"end": 142.32, "start": 141.24, "text": "classes."}, {"end": 144.72, "start": 142.32, "text": "20"}, {"end": 145.0, "start": 144.72, "text": "years"}, {"end": 145.44, "start": 145.0, "text": "later,"}, {"end": 146.12, "start": 145.44, "text": "CARP"}, {"end": 146.56, "start": 146.12, "text": "gave"}, {"end": 146.56, "start": 146.56, "text": "a"}, {"end": 146.68, "start": 146.56, "text": "2"}, {"end": 146.96, "start": 146.68, "text": "to"}, {"end": 147.16, "start": 146.96, "text": "the"}, {"end": 148.12, "start": 147.16, "text": "power"}, {"end": 148.48, "start": 148.12, "text": "n,"}, {"end": 149.16, "start": 148.48, "text": "n-square"}, {"end": 149.48, "start": 149.16, "text": "times"}, {"end": 149.84, "start": 149.48, "text": "capital"}, {"end": 149.96, "start": 149.84, "text": "W."}], "text": " Menger observed that there is an easy n-victorial algorithm, and he wondered when you can do faster. 30 years later, this was solved affirmatively by Bellman and independently held in CARP, who gave a 2 to the n to the power n-square algorithm. And this dynamic programming algorithm is now a household algorithm in many undergraduate algorithms classes. 20 years later, CARP gave a 2 to the power n, n-square times capital W."}, {"chunks": [{"end": 150.4, "start": 150.0, "text": "So"}, {"end": 151.68, "start": 150.4, "text": "this"}, {"end": 152.44, "start": 151.68, "text": "algorithm"}, {"end": 152.84, "start": 152.44, "text": "is"}, {"end": 153.16, "start": 152.84, "text": "a"}, {"end": 153.28, "start": 153.16, "text": "bit"}, {"end": 153.76, "start": 153.28, "text": "slower"}, {"end": 154.16, "start": 153.76, "text": "because"}, {"end": 154.24, "start": 154.16, "text": "there"}, {"end": 154.4, "start": 154.24, "text": "is"}, {"end": 154.76, "start": 154.4, "text": "this"}, {"end": 154.92, "start": 154.76, "text": "factor"}, {"end": 155.8, "start": 154.92, "text": "W,"}, {"end": 156.2, "start": 155.8, "text": "but"}, {"end": 156.4, "start": 156.2, "text": "it"}, {"end": 156.68, "start": 156.4, "text": "can"}, {"end": 156.92, "start": 156.68, "text": "also"}, {"end": 157.2, "start": 156.92, "text": "use"}, {"end": 157.48, "start": 157.2, "text": "quite"}, {"end": 157.6, "start": 157.48, "text": "a"}, {"end": 158.12, "start": 157.6, "text": "bit"}, {"end": 158.4, "start": 158.12, "text": "less"}, {"end": 159.2, "start": 158.4, "text": "space."}, {"end": 159.56, "start": 159.2, "text": "So"}, {"end": 159.76, "start": 159.56, "text": "the"}, {"end": 160.12, "start": 159.76, "text": "space"}, {"end": 160.84, "start": 160.12, "text": "usage"}, {"end": 161.0, "start": 160.84, "text": "is"}, {"end": 161.24, "start": 161.0, "text": "only"}, {"end": 162.28, "start": 161.24, "text": "order"}, {"end": 162.56, "start": 162.28, "text": "W."}, {"end": 162.72, "start": 162.56, "text": "And"}, {"end": 163.04, "start": 162.72, "text": "whereas"}, {"end": 163.04, "start": 163.04, "text": "the"}, {"end": 163.32, "start": 163.04, "text": "dynamic"}, {"end": 163.76, "start": 163.32, "text": "program"}, {"end": 164.28, "start": 163.76, "text": "algorithm"}, {"end": 164.6, "start": 164.28, "text": "is"}, {"end": 164.88, "start": 164.6, "text": "as"}, {"end": 165.16, "start": 164.88, "text": "large"}, {"end": 165.52, "start": 165.16, "text": "as"}, {"end": 165.64, "start": 165.52, "text": "2"}, {"end": 165.64, "start": 165.64, "text": "to"}, {"end": 165.68, "start": 165.64, "text": "the"}, {"end": 165.76, "start": 165.68, "text": "n,"}, {"end": 166.07999999999998, "start": 165.76, "text": "the"}, {"end": 166.6, "start": 166.07999999999998, "text": "space"}, {"end": 167.0, "start": 166.6, "text": "usage,"}, {"end": 169.76, "start": 167.0, "text": "that"}, {"end": 170.36, "start": 169.76, "text": "is."}, {"end": 170.68, "start": 170.36, "text": "Then"}, {"end": 170.88, "start": 170.68, "text": "in"}, {"end": 171.56, "start": 170.88, "text": "2010,"}, {"end": 171.68, "start": 171.56, "text": "there"}, {"end": 171.92000000000002, "start": 171.68, "text": "was"}, {"end": 172.32, "start": 171.92000000000002, "text": "a"}, {"end": 172.8, "start": 172.32, "text": "significant"}, {"end": 173.32, "start": 172.8, "text": "breakthrough"}, {"end": 174.56, "start": 173.32, "text": "by"}, {"end": 175.36, "start": 174.56, "text": "Bj\u00f6rkloed."}, {"end": 175.76, "start": 175.36, "text": "So"}, {"end": 175.84, "start": 175.76, "text": "he"}, {"end": 176.24, "start": 175.84, "text": "showed"}, {"end": 176.32, "start": 176.24, "text": "that"}, {"end": 176.56, "start": 176.32, "text": "the"}, {"end": 176.8, "start": 176.56, "text": "problem"}, {"end": 176.96, "start": 176.8, "text": "can"}, {"end": 177.24, "start": 176.96, "text": "also"}, {"end": 177.28, "start": 177.24, "text": "be"}, {"end": 177.56, "start": 177.28, "text": "solved"}, {"end": 177.64, "start": 177.56, "text": "in"}, {"end": 178.48, "start": 177.64, "text": "1.66"}, {"end": 178.68, "start": 178.48, "text": "to"}, {"end": 178.76, "start": 178.68, "text": "the"}, {"end": 179.0, "start": 178.76, "text": "power"}, {"end": 179.32, "start": 179.0, "text": "n"}, {"end": 179.56, "start": 179.32, "text": "times"}, {"end": 179.96, "start": 179.56, "text": "capital"}], "text": " So this algorithm is a bit slower because there is this factor W, but it can also use quite a bit less space. So the space usage is only order W. And whereas the dynamic program algorithm is as large as 2 to the n, the space usage, that is. Then in 2010, there was a significant breakthrough by Bj\u00f6rkloed. So he showed that the problem can also be solved in 1.66 to the power n times capital"}, {"chunks": [{"end": 180.76, "start": 180.0, "text": "In"}, {"end": 181.0, "start": 180.76, "text": "the"}, {"end": 181.96, "start": 181.0, "text": "case"}, {"end": 182.24, "start": 181.96, "text": "that"}, {"end": 182.4, "start": 182.24, "text": "the"}, {"end": 182.92, "start": 182.4, "text": "distances"}, {"end": 183.08, "start": 182.92, "text": "are"}, {"end": 183.88, "start": 183.08, "text": "symmetric,"}, {"end": 184.12, "start": 183.88, "text": "so"}, {"end": 184.16, "start": 184.12, "text": "the"}, {"end": 184.52, "start": 184.16, "text": "distance"}, {"end": 184.68, "start": 184.52, "text": "from"}, {"end": 184.92, "start": 184.68, "text": "i"}, {"end": 185.04, "start": 184.92, "text": "to"}, {"end": 185.48, "start": 185.04, "text": "j"}, {"end": 185.92, "start": 185.48, "text": "equals"}, {"end": 185.96, "start": 185.92, "text": "the"}, {"end": 186.16, "start": 185.96, "text": "distance"}, {"end": 186.4, "start": 186.16, "text": "from"}, {"end": 186.56, "start": 186.4, "text": "j"}, {"end": 186.76, "start": 186.56, "text": "to"}, {"end": 187.88, "start": 186.76, "text": "i."}, {"end": 188.08, "start": 187.88, "text": "And"}, {"end": 188.4, "start": 188.08, "text": "as"}, {"end": 188.56, "start": 188.4, "text": "a"}, {"end": 189.08, "start": 188.56, "text": "stepping"}, {"end": 189.36, "start": 189.08, "text": "stone"}, {"end": 189.72, "start": 189.36, "text": "towards"}, {"end": 189.88, "start": 189.72, "text": "this"}, {"end": 190.36, "start": 189.88, "text": "result,"}, {"end": 191.04, "start": 190.36, "text": "he"}, {"end": 191.52, "start": 191.04, "text": "showed"}, {"end": 192.04, "start": 191.52, "text": "that"}, {"end": 192.28, "start": 192.04, "text": "if"}, {"end": 193.88, "start": 192.28, "text": "the"}, {"end": 194.48, "start": 193.88, "text": "underlying"}, {"end": 194.88, "start": 194.48, "text": "graph"}, {"end": 194.96, "start": 194.88, "text": "of"}, {"end": 195.32, "start": 194.96, "text": "finite"}, {"end": 195.92, "start": 195.32, "text": "distances"}, {"end": 196.07999999999998, "start": 195.92, "text": "is"}, {"end": 196.8, "start": 196.07999999999998, "text": "bipartite,"}, {"end": 197.24, "start": 196.8, "text": "then"}, {"end": 197.64, "start": 197.24, "text": "the"}, {"end": 197.8, "start": 197.64, "text": "problem"}, {"end": 198.16, "start": 197.8, "text": "can"}, {"end": 198.28, "start": 198.16, "text": "even"}, {"end": 198.72, "start": 198.28, "text": "be"}, {"end": 199.2, "start": 198.72, "text": "solved"}, {"end": 199.24, "start": 199.2, "text": "in"}, {"end": 199.64, "start": 199.24, "text": "square"}, {"end": 199.68, "start": 199.64, "text": "root"}, {"end": 199.88, "start": 199.68, "text": "2"}, {"end": 200.12, "start": 199.88, "text": "to"}, {"end": 200.48, "start": 200.12, "text": "the"}, {"end": 201.36, "start": 200.48, "text": "power"}, {"end": 201.4, "start": 201.36, "text": "n."}, {"end": 201.52, "start": 201.4, "text": "So"}, {"end": 201.6, "start": 201.52, "text": "the"}, {"end": 202.16, "start": 201.6, "text": "second"}, {"end": 202.44, "start": 202.16, "text": "result"}, {"end": 202.84, "start": 202.44, "text": "was"}, {"end": 203.0, "start": 202.84, "text": "a"}, {"end": 203.48, "start": 203.0, "text": "stepping"}, {"end": 203.64, "start": 203.48, "text": "stone"}, {"end": 204.07999999999998, "start": 203.64, "text": "to"}, {"end": 204.8, "start": 204.07999999999998, "text": "the"}, {"end": 205.12, "start": 204.8, "text": "first"}, {"end": 205.4, "start": 205.12, "text": "result."}, {"end": 205.72, "start": 205.4, "text": "And"}, {"end": 205.96, "start": 205.72, "text": "bipartite"}, {"end": 206.44, "start": 205.96, "text": "here"}, {"end": 206.6, "start": 206.44, "text": "again"}, {"end": 206.88, "start": 206.6, "text": "refers"}, {"end": 207.32, "start": 206.88, "text": "to"}, {"end": 207.72, "start": 207.32, "text": "the"}, {"end": 208.24, "start": 207.72, "text": "graph"}, {"end": 208.68, "start": 208.24, "text": "that"}, {"end": 208.8, "start": 208.68, "text": "you"}, {"end": 209.04, "start": 208.8, "text": "get"}, {"end": 209.12, "start": 209.04, "text": "with"}, {"end": 209.48, "start": 209.12, "text": "finite"}, {"end": 209.96, "start": 209.48, "text": "distances."}], "text": " In the case that the distances are symmetric, so the distance from i to j equals the distance from j to i. And as a stepping stone towards this result, he showed that if the underlying graph of finite distances is bipartite, then the problem can even be solved in square root 2 to the power n. So the second result was a stepping stone to the first result. And bipartite here again refers to the graph that you get with finite distances."}, {"chunks": [{"end": 210.12, "start": 210.0, "text": "So"}, {"end": 210.24, "start": 210.12, "text": "for"}, {"end": 210.6, "start": 210.24, "text": "example"}, {"end": 210.68, "start": 210.6, "text": "in"}, {"end": 211.0, "start": 210.68, "text": "this"}, {"end": 211.64, "start": 211.0, "text": "instance"}, {"end": 211.92, "start": 211.64, "text": "is"}, {"end": 212.28, "start": 211.92, "text": "just"}, {"end": 212.32, "start": 212.28, "text": "a"}, {"end": 212.32, "start": 212.32, "text": "graph"}, {"end": 212.72, "start": 212.32, "text": "that"}, {"end": 212.84, "start": 212.72, "text": "you"}, {"end": 213.0, "start": 212.84, "text": "see"}, {"end": 215.08, "start": 213.0, "text": "there."}, {"end": 215.48, "start": 215.08, "text": "In"}, {"end": 216.8, "start": 215.48, "text": "2013,"}, {"end": 217.2, "start": 216.8, "text": "Sigan,"}, {"end": 217.52, "start": 217.2, "text": "Krajci"}, {"end": 217.56, "start": 217.52, "text": "and"}, {"end": 219.0, "start": 217.56, "text": "myself"}, {"end": 219.44, "start": 219.0, "text": "relaxed"}, {"end": 220.04, "start": 219.44, "text": "this"}, {"end": 220.64, "start": 220.04, "text": "condition"}, {"end": 221.24, "start": 220.64, "text": "to"}, {"end": 222.16, "start": 221.24, "text": "also"}, {"end": 222.28, "start": 222.16, "text": "get"}, {"end": 222.4, "start": 222.28, "text": "to"}, {"end": 222.56, "start": 222.4, "text": "the"}, {"end": 223.04, "start": 222.56, "text": "asymmetric"}, {"end": 223.48, "start": 223.04, "text": "case"}, {"end": 223.52, "start": 223.48, "text": "and"}, {"end": 223.6, "start": 223.52, "text": "we"}, {"end": 223.88, "start": 223.6, "text": "gave"}, {"end": 224.56, "start": 223.88, "text": "this"}, {"end": 225.48, "start": 224.56, "text": "1.89"}, {"end": 225.68, "start": 225.48, "text": "to"}, {"end": 225.68, "start": 225.68, "text": "the"}, {"end": 226.0, "start": 225.68, "text": "power"}, {"end": 226.12, "start": 226.0, "text": "n"}, {"end": 226.4, "start": 226.12, "text": "running"}, {"end": 227.07999999999998, "start": 226.4, "text": "time,"}, {"end": 227.52, "start": 227.07999999999998, "text": "which"}, {"end": 228.24, "start": 227.52, "text": "was"}, {"end": 229.4, "start": 228.24, "text": "subsequently"}, {"end": 229.76, "start": 229.4, "text": "improved"}, {"end": 229.92000000000002, "start": 229.76, "text": "by"}, {"end": 230.28, "start": 229.92000000000002, "text": "Bj\u00f6rklund,"}, {"end": 230.84, "start": 230.28, "text": "Karski"}, {"end": 231.32, "start": 230.84, "text": "and"}, {"end": 233.64, "start": 231.32, "text": "Kutis."}, {"end": 234.04, "start": 233.64, "text": "Given"}, {"end": 234.16, "start": 234.04, "text": "all"}, {"end": 234.4, "start": 234.16, "text": "these"}, {"end": 235.28, "start": 234.4, "text": "results,"}, {"end": 235.92000000000002, "start": 235.28, "text": "one"}, {"end": 236.4, "start": 235.92000000000002, "text": "natural"}, {"end": 237.24, "start": 236.4, "text": "question"}, {"end": 238.24, "start": 237.24, "text": "is"}, {"end": 238.72, "start": 238.24, "text": "if"}, {"end": 238.84, "start": 238.72, "text": "you"}, {"end": 239.04, "start": 238.84, "text": "want"}, {"end": 239.28, "start": 239.04, "text": "to"}, {"end": 239.96, "start": 239.28, "text": "get"}], "text": " So for example in this instance is just a graph that you see there. In 2013, Sigan, Krajci and myself relaxed this condition to also get to the asymmetric case and we gave this 1.89 to the power n running time, which was subsequently improved by Bj\u00f6rklund, Karski and Kutis. Given all these results, one natural question is if you want to get"}, {"chunks": [{"end": 240.28, "start": 240.0, "text": "One"}, {"end": 240.68, "start": 240.28, "text": "natural"}, {"end": 240.96, "start": 240.68, "text": "question"}, {"end": 241.12, "start": 240.96, "text": "is"}, {"end": 241.28, "start": 241.12, "text": "can"}, {"end": 241.44, "start": 241.28, "text": "we"}, {"end": 242.0, "start": 241.44, "text": "actually"}, {"end": 242.6, "start": 242.0, "text": "improve"}, {"end": 242.88, "start": 242.6, "text": "the"}, {"end": 243.12, "start": 242.88, "text": "algorithm"}, {"end": 243.32, "start": 243.12, "text": "by"}, {"end": 243.36, "start": 243.32, "text": "Bellman,"}, {"end": 243.56, "start": 243.36, "text": "Held"}, {"end": 243.76, "start": 243.56, "text": "and"}, {"end": 244.12, "start": 243.76, "text": "Krab?"}, {"end": 244.52, "start": 244.12, "text": "Strictly"}, {"end": 244.92, "start": 244.52, "text": "speaking,"}, {"end": 245.36, "start": 244.92, "text": "all"}, {"end": 246.52, "start": 245.36, "text": "these"}, {"end": 247.08, "start": 246.52, "text": "algorithms"}, {"end": 247.4, "start": 247.08, "text": "we"}, {"end": 247.6, "start": 247.4, "text": "see"}, {"end": 247.76, "start": 247.6, "text": "here,"}, {"end": 247.88, "start": 247.76, "text": "they"}, {"end": 247.92, "start": 247.88, "text": "are"}, {"end": 248.12, "start": 247.92, "text": "not"}, {"end": 248.2, "start": 248.12, "text": "an"}, {"end": 249.04, "start": 248.2, "text": "improvement"}, {"end": 249.32, "start": 249.04, "text": "because"}, {"end": 249.64, "start": 249.32, "text": "there"}, {"end": 250.52, "start": 249.64, "text": "is"}, {"end": 250.84, "start": 250.52, "text": "this"}, {"end": 251.52, "start": 250.84, "text": "factor"}, {"end": 251.88, "start": 251.52, "text": "W,"}, {"end": 252.32, "start": 251.88, "text": "capital"}, {"end": 252.6, "start": 252.32, "text": "W"}, {"end": 252.76, "start": 252.6, "text": "in"}, {"end": 253.2, "start": 252.76, "text": "there."}, {"end": 253.76, "start": 253.2, "text": "And"}, {"end": 254.12, "start": 253.76, "text": "recall"}, {"end": 254.16, "start": 254.12, "text": "that"}, {"end": 254.44, "start": 254.16, "text": "capital"}, {"end": 254.6, "start": 254.44, "text": "W"}, {"end": 255.24, "start": 254.6, "text": "can"}, {"end": 255.6, "start": 255.24, "text": "very"}, {"end": 256.36, "start": 255.6, "text": "well"}, {"end": 256.52, "start": 256.36, "text": "be"}, {"end": 257.36, "start": 256.52, "text": "exponential"}, {"end": 257.48, "start": 257.36, "text": "in"}, {"end": 257.92, "start": 257.48, "text": "the"}, {"end": 258.28, "start": 257.92, "text": "input"}, {"end": 259.08, "start": 258.28, "text": "size"}, {"end": 259.44, "start": 259.08, "text": "because"}, {"end": 259.68, "start": 259.44, "text": "it's"}, {"end": 259.92, "start": 259.68, "text": "only"}, {"end": 260.36, "start": 259.92, "text": "represented"}, {"end": 260.48, "start": 260.36, "text": "in"}, {"end": 260.88, "start": 260.48, "text": "binary."}, {"end": 266.44, "start": 260.88, "text": "So"}, {"end": 266.88, "start": 266.44, "text": "the"}, {"end": 267.28, "start": 266.88, "text": "natural"}, {"end": 267.76, "start": 267.28, "text": "question,"}, {"end": 268.04, "start": 267.76, "text": "which"}, {"end": 268.08, "start": 268.04, "text": "we"}, {"end": 268.32, "start": 268.08, "text": "try"}, {"end": 269.0, "start": 268.32, "text": "to"}, {"end": 269.96, "start": 269.0, "text": "address"}], "text": " One natural question is can we actually improve the algorithm by Bellman, Held and Krab? Strictly speaking, all these algorithms we see here, they are not an improvement because there is this factor W, capital W in there. And recall that capital W can very well be exponential in the input size because it's only represented in binary. So the natural question, which we try to address"}, {"chunks": [{"end": 270.32, "start": 270.0, "text": "in"}, {"end": 271.04, "start": 270.32, "text": "this"}, {"end": 271.72, "start": 271.04, "text": "work"}, {"end": 271.88, "start": 271.72, "text": "is"}, {"end": 272.04, "start": 271.88, "text": "can"}, {"end": 272.32, "start": 272.04, "text": "we"}, {"end": 273.12, "start": 272.32, "text": "solve"}, {"end": 273.28, "start": 273.12, "text": "the"}, {"end": 273.52, "start": 273.28, "text": "TSP"}, {"end": 273.76, "start": 273.52, "text": "problem"}, {"end": 273.96, "start": 273.76, "text": "or"}, {"end": 274.24, "start": 273.96, "text": "the"}, {"end": 274.6, "start": 274.24, "text": "symmetric"}, {"end": 275.04, "start": 274.6, "text": "TSP"}, {"end": 275.72, "start": 275.04, "text": "problem"}, {"end": 275.8, "start": 275.72, "text": "a"}, {"end": 276.0, "start": 275.8, "text": "bit"}, {"end": 276.4, "start": 276.0, "text": "faster"}, {"end": 276.48, "start": 276.4, "text": "than"}, {"end": 276.64, "start": 276.48, "text": "2"}, {"end": 276.76, "start": 276.64, "text": "to"}, {"end": 276.88, "start": 276.76, "text": "the"}, {"end": 276.96, "start": 276.88, "text": "n?"}, {"end": 277.16, "start": 276.96, "text": "Can"}, {"end": 277.36, "start": 277.16, "text": "we"}, {"end": 278.16, "start": 277.36, "text": "significantly"}, {"end": 278.96, "start": 278.16, "text": "improve"}, {"end": 279.8, "start": 278.96, "text": "this"}, {"end": 280.48, "start": 279.8, "text": "Bellman-Helth"}, {"end": 280.68, "start": 280.48, "text": "and"}, {"end": 282.04, "start": 280.68, "text": "Krahp"}, {"end": 282.72, "start": 282.04, "text": "algorithm?"}, {"end": 283.08, "start": 282.72, "text": "And"}, {"end": 283.32, "start": 283.08, "text": "this"}, {"end": 283.6, "start": 283.32, "text": "doesn't"}, {"end": 283.84, "start": 283.6, "text": "seem"}, {"end": 283.96, "start": 283.84, "text": "to"}, {"end": 284.08, "start": 283.96, "text": "be"}, {"end": 284.4, "start": 284.08, "text": "easy"}, {"end": 284.52, "start": 284.4, "text": "at"}, {"end": 284.96, "start": 284.52, "text": "all"}, {"end": 285.6, "start": 284.96, "text": "because"}, {"end": 285.76, "start": 285.6, "text": "all"}, {"end": 286.16, "start": 285.76, "text": "these"}, {"end": 286.6, "start": 286.16, "text": "improved"}, {"end": 287.04, "start": 286.6, "text": "algorithms"}, {"end": 287.04, "start": 287.04, "text": "that"}, {"end": 287.04, "start": 287.04, "text": "we"}, {"end": 287.24, "start": 287.04, "text": "can"}, {"end": 287.48, "start": 287.24, "text": "see"}, {"end": 288.04, "start": 287.48, "text": "here"}, {"end": 288.12, "start": 288.04, "text": "with"}, {"end": 288.32, "start": 288.12, "text": "this"}, {"end": 288.68, "start": 288.32, "text": "capital"}, {"end": 288.96, "start": 288.68, "text": "W"}, {"end": 290.32, "start": 288.96, "text": "factor,"}, {"end": 290.68, "start": 290.32, "text": "they"}, {"end": 290.88, "start": 290.68, "text": "are"}, {"end": 291.64, "start": 290.88, "text": "algebraic"}, {"end": 292.6, "start": 291.64, "text": "algorithms,"}, {"end": 293.12, "start": 292.6, "text": "which"}, {"end": 293.68, "start": 293.12, "text": "means"}, {"end": 293.92, "start": 293.68, "text": "that"}, {"end": 294.16, "start": 293.92, "text": "they"}, {"end": 294.16, "start": 294.16, "text": "are"}, {"end": 294.32, "start": 294.16, "text": "counting"}, {"end": 294.64, "start": 294.32, "text": "certain"}, {"end": 295.24, "start": 294.64, "text": "objects"}, {"end": 295.52, "start": 295.24, "text": "and"}, {"end": 295.76, "start": 295.52, "text": "they"}, {"end": 296.4, "start": 295.76, "text": "inherently"}, {"end": 296.8, "start": 296.4, "text": "rely"}, {"end": 296.88, "start": 296.8, "text": "on"}, {"end": 297.0, "start": 296.88, "text": "the"}, {"end": 297.4, "start": 297.0, "text": "power"}, {"end": 297.6, "start": 297.4, "text": "of"}, {"end": 298.4, "start": 297.6, "text": "cancellation"}, {"end": 298.6, "start": 298.4, "text": "in"}, {"end": 298.88, "start": 298.6, "text": "some"}, {"end": 299.36, "start": 298.88, "text": "algebraic"}, {"end": 299.96, "start": 299.36, "text": "setting."}], "text": " in this work is can we solve the TSP problem or the symmetric TSP problem a bit faster than 2 to the n? Can we significantly improve this Bellman-Helth and Krahp algorithm? And this doesn't seem to be easy at all because all these improved algorithms that we can see here with this capital W factor, they are algebraic algorithms, which means that they are counting certain objects and they inherently rely on the power of cancellation in some algebraic setting."}, {"chunks": [{"end": 301.2, "start": 300.0, "text": "And"}, {"end": 301.68, "start": 301.2, "text": "the"}, {"end": 303.92, "start": 301.68, "text": "objects"}, {"end": 304.32, "start": 303.92, "text": "they"}, {"end": 304.44, "start": 304.32, "text": "are"}, {"end": 304.8, "start": 304.44, "text": "counting,"}, {"end": 305.16, "start": 304.8, "text": "for"}, {"end": 305.68, "start": 305.16, "text": "example,"}, {"end": 306.52, "start": 305.68, "text": "Karp,"}, {"end": 307.0, "start": 306.52, "text": "what"}, {"end": 307.16, "start": 307.0, "text": "he"}, {"end": 307.64, "start": 307.16, "text": "counted"}, {"end": 307.96, "start": 307.64, "text": "was"}, {"end": 308.44, "start": 307.96, "text": "walks"}, {"end": 309.28, "start": 308.44, "text": "of"}, {"end": 309.88, "start": 309.28, "text": "a"}, {"end": 310.32, "start": 309.88, "text": "certain"}, {"end": 311.04, "start": 310.32, "text": "distance."}, {"end": 311.48, "start": 311.04, "text": "And"}, {"end": 311.72, "start": 311.48, "text": "if"}, {"end": 311.76, "start": 311.72, "text": "the"}, {"end": 312.4, "start": 311.76, "text": "distances"}, {"end": 312.68, "start": 312.4, "text": "are"}, {"end": 313.68, "start": 312.68, "text": "large,"}, {"end": 314.08, "start": 313.68, "text": "it's"}, {"end": 314.24, "start": 314.08, "text": "not"}, {"end": 314.52, "start": 314.24, "text": "easy"}, {"end": 314.92, "start": 314.52, "text": "to"}, {"end": 315.36, "start": 314.92, "text": "see,"}, {"end": 315.52, "start": 315.36, "text": "or"}, {"end": 316.32, "start": 315.52, "text": "it's"}, {"end": 316.76, "start": 316.32, "text": "not"}, {"end": 316.96, "start": 316.76, "text": "clear"}, {"end": 317.2, "start": 316.96, "text": "that"}, {"end": 317.56, "start": 317.2, "text": "we"}, {"end": 317.68, "start": 317.56, "text": "can"}, {"end": 318.24, "start": 317.68, "text": "actually"}, {"end": 318.6, "start": 318.24, "text": "still"}, {"end": 318.72, "start": 318.6, "text": "get"}, {"end": 318.84, "start": 318.72, "text": "a"}, {"end": 319.2, "start": 318.84, "text": "fast"}, {"end": 320.36, "start": 319.2, "text": "algorithm."}, {"end": 321.28, "start": 320.36, "text": "For"}, {"end": 321.76, "start": 321.28, "text": "example,"}, {"end": 321.96, "start": 321.76, "text": "if"}, {"end": 322.04, "start": 321.96, "text": "you"}, {"end": 322.44, "start": 322.04, "text": "want"}, {"end": 322.56, "start": 322.44, "text": "to"}, {"end": 323.04, "start": 322.56, "text": "count"}, {"end": 323.44, "start": 323.04, "text": "walks"}, {"end": 323.44, "start": 323.44, "text": "of"}, {"end": 323.52, "start": 323.44, "text": "a"}, {"end": 323.88, "start": 323.52, "text": "certain"}, {"end": 324.16, "start": 323.88, "text": "distance,"}, {"end": 325.2, "start": 324.16, "text": "this"}, {"end": 325.76, "start": 325.2, "text": "is"}, {"end": 326.04, "start": 325.76, "text": "a"}, {"end": 326.4, "start": 326.04, "text": "Sharpie"}, {"end": 327.16, "start": 326.4, "text": "complete"}, {"end": 328.68, "start": 327.16, "text": "problem."}, {"end": 328.84, "start": 328.68, "text": "It"}, {"end": 329.04, "start": 328.84, "text": "can"}, {"end": 329.24, "start": 329.04, "text": "be"}, {"end": 329.56, "start": 329.24, "text": "solved"}, {"end": 329.64, "start": 329.56, "text": "in"}, {"end": 329.96, "start": 329.64, "text": "pseudopods."}], "text": " And the objects they are counting, for example, Karp, what he counted was walks of a certain distance. And if the distances are large, it's not easy to see, or it's not clear that we can actually still get a fast algorithm. For example, if you want to count walks of a certain distance, this is a Sharpie complete problem. It can be solved in pseudopods."}, {"chunks": [{"end": 330.4, "start": 330.0, "text": "polynomial"}, {"end": 331.2, "start": 330.4, "text": "time,"}, {"end": 331.44, "start": 331.2, "text": "but"}, {"end": 331.64, "start": 331.44, "text": "then"}, {"end": 331.76, "start": 331.64, "text": "we"}, {"end": 331.88, "start": 331.76, "text": "get"}, {"end": 332.48, "start": 331.88, "text": "this"}, {"end": 333.32, "start": 332.48, "text": "capital"}, {"end": 333.6, "start": 333.32, "text": "W"}, {"end": 333.88, "start": 333.6, "text": "factor,"}, {"end": 336.76, "start": 333.88, "text": "unfortunately."}, {"end": 337.2, "start": 336.76, "text": "OK,"}, {"end": 337.32, "start": 337.2, "text": "and"}, {"end": 337.64, "start": 337.32, "text": "so"}, {"end": 337.72, "start": 337.64, "text": "in"}, {"end": 338.36, "start": 337.72, "text": "this"}, {"end": 338.64, "start": 338.36, "text": "talk,"}, {"end": 338.76, "start": 338.64, "text": "we"}, {"end": 339.28, "start": 338.76, "text": "do"}, {"end": 339.4, "start": 339.28, "text": "get"}, {"end": 339.6, "start": 339.4, "text": "some"}, {"end": 340.72, "start": 339.6, "text": "progress,"}, {"end": 341.08, "start": 340.72, "text": "but"}, {"end": 341.08, "start": 341.08, "text": "there"}, {"end": 341.08, "start": 341.08, "text": "are"}, {"end": 341.08, "start": 341.08, "text": "some"}, {"end": 341.6, "start": 341.08, "text": "caveats,"}, {"end": 342.68, "start": 341.6, "text": "unfortunately."}, {"end": 343.08, "start": 342.68, "text": "So"}, {"end": 343.32, "start": 343.08, "text": "the"}, {"end": 343.72, "start": 343.32, "text": "result"}, {"end": 343.92, "start": 343.72, "text": "is"}, {"end": 344.0, "start": 343.92, "text": "that"}, {"end": 344.44, "start": 344.0, "text": "symmetric"}, {"end": 344.84, "start": 344.44, "text": "biopatite"}, {"end": 345.08, "start": 344.84, "text": "TSP"}, {"end": 345.6, "start": 345.08, "text": "can"}, {"end": 345.76, "start": 345.6, "text": "be"}, {"end": 346.04, "start": 345.76, "text": "solved"}, {"end": 346.2, "start": 346.04, "text": "in"}, {"end": 347.32, "start": 346.2, "text": "1.9999"}, {"end": 347.56, "start": 347.32, "text": "to"}, {"end": 348.0, "start": 347.56, "text": "the"}, {"end": 349.08, "start": 348.0, "text": "power"}, {"end": 349.4, "start": 349.08, "text": "n"}, {"end": 349.76, "start": 349.4, "text": "time,"}, {"end": 350.32, "start": 349.76, "text": "assuming"}, {"end": 350.52, "start": 350.32, "text": "omega"}, {"end": 350.96, "start": 350.52, "text": "is"}, {"end": 351.76, "start": 350.96, "text": "2."}, {"end": 352.12, "start": 351.76, "text": "So"}, {"end": 352.44, "start": 352.12, "text": "let"}, {"end": 352.72, "start": 352.44, "text": "me"}, {"end": 353.6, "start": 352.72, "text": "elaborate"}, {"end": 353.68, "start": 353.6, "text": "a"}, {"end": 354.0, "start": 353.68, "text": "bit"}, {"end": 354.8, "start": 354.0, "text": "on"}, {"end": 355.48, "start": 354.8, "text": "this."}, {"end": 355.72, "start": 355.48, "text": "So"}, {"end": 356.0, "start": 355.72, "text": "one"}, {"end": 356.44, "start": 356.0, "text": "hope"}, {"end": 356.6, "start": 356.44, "text": "would"}, {"end": 356.72, "start": 356.6, "text": "be"}, {"end": 356.96, "start": 356.72, "text": "to"}, {"end": 357.44, "start": 356.96, "text": "actually"}, {"end": 357.84, "start": 357.44, "text": "get"}, {"end": 357.96, "start": 357.84, "text": "a"}, {"end": 358.16, "start": 357.96, "text": "faster"}, {"end": 358.64, "start": 358.16, "text": "algorithm"}, {"end": 358.84, "start": 358.64, "text": "for"}, {"end": 359.2, "start": 358.84, "text": "general"}, {"end": 359.96, "start": 359.2, "text": "TSP."}], "text": " polynomial time, but then we get this capital W factor, unfortunately. OK, and so in this talk, we do get some progress, but there are some caveats, unfortunately. So the result is that symmetric biopatite TSP can be solved in 1.9999 to the power n time, assuming omega is 2. So let me elaborate a bit on this. So one hope would be to actually get a faster algorithm for general TSP."}, {"chunks": [{"end": 360.6, "start": 360.0, "text": "But"}, {"end": 360.76, "start": 360.6, "text": "one"}, {"end": 361.16, "start": 360.76, "text": "should"}, {"end": 361.64, "start": 361.16, "text": "realize"}, {"end": 361.84, "start": 361.64, "text": "here"}, {"end": 362.6, "start": 361.84, "text": "that"}, {"end": 362.76, "start": 362.6, "text": "to"}, {"end": 362.92, "start": 362.76, "text": "do"}, {"end": 363.72, "start": 362.92, "text": "that,"}, {"end": 364.16, "start": 363.72, "text": "even"}, {"end": 364.24, "start": 364.16, "text": "if"}, {"end": 364.36, "start": 364.24, "text": "the"}, {"end": 364.76, "start": 364.36, "text": "weights"}, {"end": 364.8, "start": 364.76, "text": "are"}, {"end": 364.92, "start": 364.8, "text": "all"}, {"end": 365.2, "start": 364.92, "text": "zero"}, {"end": 365.44, "start": 365.2, "text": "and"}, {"end": 365.92, "start": 365.44, "text": "one,"}, {"end": 366.04, "start": 365.92, "text": "so"}, {"end": 366.24, "start": 366.04, "text": "then"}, {"end": 366.52, "start": 366.24, "text": "we"}, {"end": 366.6, "start": 366.52, "text": "have"}, {"end": 366.84, "start": 366.6, "text": "the"}, {"end": 367.52, "start": 366.84, "text": "problem"}, {"end": 367.64, "start": 367.52, "text": "of"}, {"end": 368.08, "start": 367.64, "text": "finding"}, {"end": 368.56, "start": 368.08, "text": "Hamiltonian"}, {"end": 369.04, "start": 368.56, "text": "cycles"}, {"end": 369.12, "start": 369.04, "text": "in"}, {"end": 369.52, "start": 369.12, "text": "directed"}, {"end": 370.44, "start": 369.52, "text": "graphs,"}, {"end": 370.68, "start": 370.44, "text": "even"}, {"end": 370.92, "start": 370.68, "text": "that"}, {"end": 371.24, "start": 370.92, "text": "special"}, {"end": 371.6, "start": 371.24, "text": "case"}, {"end": 371.6, "start": 371.6, "text": "at"}, {"end": 371.8, "start": 371.6, "text": "the"}, {"end": 372.24, "start": 371.8, "text": "moment"}, {"end": 372.44, "start": 372.24, "text": "is"}, {"end": 372.64, "start": 372.44, "text": "not"}, {"end": 373.04, "start": 372.64, "text": "known"}, {"end": 374.6, "start": 373.04, "text": "to"}, {"end": 374.96, "start": 374.6, "text": "be"}, {"end": 375.4, "start": 374.96, "text": "solvable"}, {"end": 375.84, "start": 375.4, "text": "faster"}, {"end": 375.92, "start": 375.84, "text": "than"}, {"end": 376.08, "start": 375.92, "text": "two"}, {"end": 376.16, "start": 376.08, "text": "to"}, {"end": 376.2, "start": 376.16, "text": "the"}, {"end": 376.28, "start": 376.2, "text": "n."}, {"end": 376.48, "start": 376.28, "text": "So"}, {"end": 377.0, "start": 376.48, "text": "there"}, {"end": 377.32, "start": 377.0, "text": "was"}, {"end": 377.36, "start": 377.32, "text": "some"}, {"end": 377.64, "start": 377.36, "text": "very"}, {"end": 377.84, "start": 377.64, "text": "nice"}, {"end": 378.32, "start": 377.84, "text": "progress"}, {"end": 378.68, "start": 378.32, "text": "by"}, {"end": 379.32, "start": 378.68, "text": "Bj\u00f6rklund,"}, {"end": 380.92, "start": 379.32, "text": "Karski"}, {"end": 381.12, "start": 380.92, "text": "and"}, {"end": 383.24, "start": 381.12, "text": "Kutis,"}, {"end": 383.92, "start": 383.24, "text": "which"}, {"end": 384.32, "start": 383.92, "text": "seems"}, {"end": 384.32, "start": 384.32, "text": "that"}, {"end": 384.32, "start": 384.32, "text": "we"}, {"end": 384.32, "start": 384.32, "text": "are"}, {"end": 384.96, "start": 384.32, "text": "close"}, {"end": 385.2, "start": 384.96, "text": "to"}, {"end": 385.36, "start": 385.2, "text": "that"}, {"end": 386.28, "start": 385.36, "text": "question,"}, {"end": 386.72, "start": 386.28, "text": "but"}, {"end": 386.92, "start": 386.72, "text": "so"}, {"end": 387.24, "start": 386.92, "text": "far"}, {"end": 387.76, "start": 387.24, "text": "this"}, {"end": 388.64, "start": 387.76, "text": "is"}, {"end": 389.08, "start": 388.64, "text": "still"}, {"end": 389.96, "start": 389.08, "text": "unresolved."}], "text": " But one should realize here that to do that, even if the weights are all zero and one, so then we have the problem of finding Hamiltonian cycles in directed graphs, even that special case at the moment is not known to be solvable faster than two to the n. So there was some very nice progress by Bj\u00f6rklund, Karski and Kutis, which seems that we are close to that question, but so far this is still unresolved."}, {"chunks": [{"end": 390.44, "start": 390.0, "text": "TSP"}, {"end": 390.96, "start": 390.44, "text": "faster"}, {"end": 391.0, "start": 390.96, "text": "than"}, {"end": 391.16, "start": 391.0, "text": "2"}, {"end": 391.28, "start": 391.16, "text": "to"}, {"end": 392.08, "start": 391.28, "text": "the"}, {"end": 392.2, "start": 392.08, "text": "n,"}, {"end": 393.04, "start": 392.2, "text": "one"}, {"end": 393.32, "start": 393.04, "text": "first"}, {"end": 393.52, "start": 393.32, "text": "needs"}, {"end": 394.0, "start": 393.52, "text": "to"}, {"end": 394.36, "start": 394.0, "text": "overcome"}, {"end": 394.64, "start": 394.36, "text": "that"}, {"end": 394.88, "start": 394.64, "text": "obstacle."}, {"end": 395.08, "start": 394.88, "text": "So"}, {"end": 395.8, "start": 395.08, "text": "the"}, {"end": 396.04, "start": 395.8, "text": "bipartite"}, {"end": 396.68, "start": 396.04, "text": "case,"}, {"end": 396.92, "start": 396.68, "text": "so"}, {"end": 397.16, "start": 396.92, "text": "we"}, {"end": 397.24, "start": 397.16, "text": "only"}, {"end": 398.0, "start": 397.24, "text": "consider"}, {"end": 398.28, "start": 398.0, "text": "bipartite"}, {"end": 398.64, "start": 398.28, "text": "graphs,"}, {"end": 399.08, "start": 398.64, "text": "but"}, {"end": 399.48, "start": 399.08, "text": "as"}, {"end": 399.84, "start": 399.48, "text": "we"}, {"end": 400.08, "start": 399.84, "text": "already"}, {"end": 400.48, "start": 400.08, "text": "saw"}, {"end": 400.56, "start": 400.48, "text": "in"}, {"end": 400.92, "start": 400.56, "text": "the"}, {"end": 401.32, "start": 400.92, "text": "previous"}, {"end": 401.76, "start": 401.32, "text": "slide,"}, {"end": 402.44, "start": 401.76, "text": "also"}, {"end": 402.72, "start": 402.44, "text": "in"}, {"end": 402.8, "start": 402.72, "text": "the"}, {"end": 403.76, "start": 402.8, "text": "past,"}, {"end": 404.04, "start": 403.76, "text": "this"}, {"end": 404.28, "start": 404.04, "text": "turned"}, {"end": 404.48, "start": 404.28, "text": "out"}, {"end": 404.48, "start": 404.48, "text": "to"}, {"end": 404.8, "start": 404.48, "text": "be"}, {"end": 405.0, "start": 404.8, "text": "a"}, {"end": 405.04, "start": 405.0, "text": "very"}, {"end": 405.32, "start": 405.04, "text": "good"}, {"end": 405.64, "start": 405.32, "text": "stepping"}, {"end": 405.96, "start": 405.64, "text": "stone"}, {"end": 406.28, "start": 405.96, "text": "towards"}, {"end": 406.6, "start": 406.28, "text": "solving"}, {"end": 406.72, "start": 406.6, "text": "the"}, {"end": 407.08, "start": 406.72, "text": "general"}, {"end": 407.56, "start": 407.08, "text": "case."}, {"end": 409.32, "start": 407.56, "text": "So"}, {"end": 409.48, "start": 409.32, "text": "in"}, {"end": 409.64, "start": 409.48, "text": "that"}, {"end": 410.36, "start": 409.64, "text": "sense,"}, {"end": 410.48, "start": 410.36, "text": "our"}, {"end": 410.8, "start": 410.48, "text": "theorem"}, {"end": 410.92, "start": 410.8, "text": "is"}, {"end": 411.24, "start": 410.92, "text": "also"}, {"end": 411.64, "start": 411.24, "text": "pretty"}, {"end": 412.52, "start": 411.64, "text": "valuable."}, {"end": 412.88, "start": 412.52, "text": "And"}, {"end": 413.0, "start": 412.88, "text": "then"}, {"end": 413.2, "start": 413.0, "text": "there"}, {"end": 413.52, "start": 413.2, "text": "is"}, {"end": 414.0, "start": 413.52, "text": "this"}, {"end": 414.36, "start": 414.0, "text": "last"}, {"end": 414.84, "start": 414.36, "text": "assumption,"}, {"end": 415.28, "start": 414.84, "text": "omega"}, {"end": 415.48, "start": 415.28, "text": "is"}, {"end": 416.32, "start": 415.48, "text": "2."}, {"end": 416.8, "start": 416.32, "text": "So"}, {"end": 417.12, "start": 416.8, "text": "omega"}, {"end": 417.44, "start": 417.12, "text": "is"}, {"end": 417.72, "start": 417.44, "text": "the"}, {"end": 418.16, "start": 417.72, "text": "smallest"}, {"end": 418.48, "start": 418.16, "text": "constant,"}, {"end": 418.6, "start": 418.48, "text": "so"}, {"end": 418.68, "start": 418.6, "text": "it's"}, {"end": 418.72, "start": 418.68, "text": "at"}, {"end": 418.88, "start": 418.72, "text": "S"}, {"end": 419.16, "start": 418.88, "text": "times"}, {"end": 419.4, "start": 419.16, "text": "S"}, {"end": 419.96, "start": 419.4, "text": "matrices,"}], "text": " TSP faster than 2 to the n, one first needs to overcome that obstacle. So the bipartite case, so we only consider bipartite graphs, but as we already saw in the previous slide, also in the past, this turned out to be a very good stepping stone towards solving the general case. So in that sense, our theorem is also pretty valuable. And then there is this last assumption, omega is 2. So omega is the smallest constant, so it's at S times S matrices,"}, {"chunks": [{"end": 420.44, "start": 420.0, "text": "can"}, {"end": 422.04, "start": 420.44, "text": "be"}, {"end": 422.64, "start": 422.04, "text": "multiplied"}, {"end": 422.8, "start": 422.64, "text": "in"}, {"end": 423.24, "start": 422.8, "text": "time"}, {"end": 423.4, "start": 423.24, "text": "s"}, {"end": 423.6, "start": 423.4, "text": "to"}, {"end": 423.6, "start": 423.6, "text": "the"}, {"end": 424.04, "start": 423.6, "text": "power"}, {"end": 424.28, "start": 424.04, "text": "omega"}, {"end": 424.76, "start": 424.28, "text": "plus"}, {"end": 425.12, "start": 424.76, "text": "little"}, {"end": 425.2, "start": 425.12, "text": "o"}, {"end": 425.36, "start": 425.2, "text": "of"}, {"end": 425.68, "start": 425.36, "text": "1."}, {"end": 426.48, "start": 425.68, "text": "So"}, {"end": 426.56, "start": 426.48, "text": "here"}, {"end": 426.64, "start": 426.56, "text": "you"}, {"end": 427.04, "start": 426.64, "text": "can"}, {"end": 427.2, "start": 427.04, "text": "see"}, {"end": 427.52, "start": 427.2, "text": "some"}, {"end": 428.56, "start": 427.52, "text": "progress"}, {"end": 429.96, "start": 428.56, "text": "on"}, {"end": 430.68, "start": 429.96, "text": "omega."}, {"end": 431.4, "start": 430.68, "text": "So"}, {"end": 431.64, "start": 431.4, "text": "this"}, {"end": 432.56, "start": 431.64, "text": "assumption"}, {"end": 432.84, "start": 432.56, "text": "omega"}, {"end": 433.2, "start": 432.84, "text": "is"}, {"end": 433.52, "start": 433.2, "text": "2,"}, {"end": 433.96, "start": 433.52, "text": "that's"}, {"end": 434.12, "start": 433.96, "text": "a"}, {"end": 434.52, "start": 434.12, "text": "highly"}, {"end": 435.04, "start": 434.52, "text": "non-trivial"}, {"end": 435.72, "start": 435.04, "text": "assumption."}, {"end": 436.88, "start": 435.72, "text": "Sometimes"}, {"end": 437.24, "start": 436.88, "text": "this"}, {"end": 437.76, "start": 437.24, "text": "conjecture"}, {"end": 437.88, "start": 437.76, "text": "is"}, {"end": 438.24, "start": 437.88, "text": "mentioned"}, {"end": 438.24, "start": 438.24, "text": "that"}, {"end": 438.64, "start": 438.24, "text": "this"}, {"end": 438.64, "start": 438.64, "text": "is"}, {"end": 438.92, "start": 438.64, "text": "true,"}, {"end": 439.28, "start": 438.92, "text": "but"}, {"end": 439.52, "start": 439.28, "text": "it's"}, {"end": 439.68, "start": 439.52, "text": "not"}, {"end": 439.96, "start": 439.68, "text": "clear"}, {"end": 440.04, "start": 439.96, "text": "at"}, {"end": 440.2, "start": 440.04, "text": "all,"}, {"end": 440.32, "start": 440.2, "text": "and"}, {"end": 440.48, "start": 440.32, "text": "I"}, {"end": 440.68, "start": 440.48, "text": "don't"}, {"end": 440.92, "start": 440.68, "text": "even"}, {"end": 441.12, "start": 440.92, "text": "dare"}, {"end": 441.52, "start": 441.12, "text": "to"}, {"end": 442.2, "start": 441.52, "text": "say"}, {"end": 442.44, "start": 442.2, "text": "what"}, {"end": 442.76, "start": 442.44, "text": "the"}, {"end": 443.2, "start": 442.76, "text": "common"}, {"end": 444.52, "start": 443.2, "text": "consensus"}, {"end": 444.88, "start": 444.52, "text": "is"}, {"end": 444.92, "start": 444.88, "text": "at"}, {"end": 445.32, "start": 444.92, "text": "the"}, {"end": 445.68, "start": 445.32, "text": "moment"}, {"end": 445.68, "start": 445.68, "text": "of"}, {"end": 447.68, "start": 445.68, "text": "the"}, {"end": 449.96, "start": 447.68, "text": "experts."}], "text": " can be multiplied in time s to the power omega plus little o of 1. So here you can see some progress on omega. So this assumption omega is 2, that's a highly non-trivial assumption. Sometimes this conjecture is mentioned that this is true, but it's not clear at all, and I don't even dare to say what the common consensus is at the moment of the experts."}, {"chunks": [{"end": 451.52, "start": 450.0, "text": "Okay,"}, {"end": 452.84, "start": 451.52, "text": "now"}, {"end": 453.04, "start": 452.84, "text": "in"}, {"end": 453.24, "start": 453.04, "text": "the"}, {"end": 453.36, "start": 453.24, "text": "remainder"}, {"end": 453.36, "start": 453.36, "text": "of"}, {"end": 453.84, "start": 453.36, "text": "the"}, {"end": 454.32, "start": 453.84, "text": "talk"}, {"end": 454.64, "start": 454.32, "text": "I"}, {"end": 454.8, "start": 454.64, "text": "will"}, {"end": 454.96, "start": 454.8, "text": "tell"}, {"end": 455.0, "start": 454.96, "text": "you"}, {"end": 455.2, "start": 455.0, "text": "a"}, {"end": 455.4, "start": 455.2, "text": "bit"}, {"end": 455.76, "start": 455.4, "text": "about"}, {"end": 455.92, "start": 455.76, "text": "the"}, {"end": 456.16, "start": 455.92, "text": "approach"}, {"end": 456.44, "start": 456.16, "text": "taken"}, {"end": 456.6, "start": 456.44, "text": "in"}, {"end": 457.0, "start": 456.6, "text": "this"}, {"end": 458.28, "start": 457.0, "text": "paper."}, {"end": 458.48, "start": 458.28, "text": "So"}, {"end": 458.76, "start": 458.48, "text": "there"}, {"end": 458.84, "start": 458.76, "text": "are"}, {"end": 459.44, "start": 458.84, "text": "three"}, {"end": 460.08, "start": 459.44, "text": "steps,"}, {"end": 460.6, "start": 460.08, "text": "and"}, {"end": 460.92, "start": 460.6, "text": "the"}, {"end": 461.24, "start": 460.92, "text": "first"}, {"end": 461.36, "start": 461.24, "text": "step"}, {"end": 461.6, "start": 461.36, "text": "is"}, {"end": 461.84, "start": 461.6, "text": "pretty"}, {"end": 462.52, "start": 461.84, "text": "easy."}, {"end": 462.64, "start": 462.52, "text": "We"}, {"end": 462.96, "start": 462.64, "text": "assume"}, {"end": 463.16, "start": 462.96, "text": "that"}, {"end": 463.16, "start": 463.16, "text": "the"}, {"end": 463.36, "start": 463.16, "text": "number"}, {"end": 463.56, "start": 463.36, "text": "of"}, {"end": 464.24, "start": 463.56, "text": "vertices"}, {"end": 464.44, "start": 464.24, "text": "is"}, {"end": 465.56, "start": 464.44, "text": "even."}, {"end": 465.8, "start": 465.56, "text": "And"}, {"end": 466.04, "start": 465.8, "text": "this"}, {"end": 466.28, "start": 466.04, "text": "can"}, {"end": 466.6, "start": 466.28, "text": "be"}, {"end": 467.0, "start": 466.6, "text": "easily"}, {"end": 467.2, "start": 467.0, "text": "done"}, {"end": 467.32, "start": 467.2, "text": "by"}, {"end": 467.56, "start": 467.32, "text": "just"}, {"end": 467.84, "start": 467.56, "text": "adding"}, {"end": 468.04, "start": 467.84, "text": "an"}, {"end": 468.72, "start": 468.04, "text": "artificial"}, {"end": 469.44, "start": 468.72, "text": "vertex."}, {"end": 469.84, "start": 469.44, "text": "So"}, {"end": 470.24, "start": 469.84, "text": "what"}, {"end": 470.52, "start": 470.24, "text": "this"}, {"end": 471.52, "start": 470.52, "text": "achieves"}, {"end": 471.8, "start": 471.52, "text": "is"}, {"end": 471.84, "start": 471.8, "text": "that"}, {"end": 472.12, "start": 471.84, "text": "now"}, {"end": 472.28, "start": 472.12, "text": "we"}, {"end": 472.28, "start": 472.28, "text": "can"}, {"end": 472.52, "start": 472.28, "text": "think"}, {"end": 472.64, "start": 472.52, "text": "of"}, {"end": 472.76, "start": 472.64, "text": "the"}, {"end": 473.56, "start": 472.76, "text": "problem"}, {"end": 474.04, "start": 473.56, "text": "as"}, {"end": 474.4, "start": 474.04, "text": "trying"}, {"end": 474.6, "start": 474.4, "text": "to"}, {"end": 474.92, "start": 474.6, "text": "find"}, {"end": 475.2, "start": 474.92, "text": "two"}, {"end": 475.52, "start": 475.2, "text": "perfect"}, {"end": 476.16, "start": 475.52, "text": "matchings,"}, {"end": 476.44, "start": 476.16, "text": "A"}, {"end": 476.84, "start": 476.44, "text": "and"}, {"end": 477.12, "start": 476.84, "text": "B,"}, {"end": 477.24, "start": 477.12, "text": "that"}, {"end": 477.96, "start": 477.24, "text": "form"}, {"end": 478.04, "start": 477.96, "text": "a"}, {"end": 478.48, "start": 478.04, "text": "Hamiltonian"}, {"end": 479.36, "start": 478.48, "text": "cycle"}, {"end": 479.64, "start": 479.36, "text": "and"}, {"end": 479.96, "start": 479.64, "text": "minimize"}], "text": " Okay, now in the remainder of the talk I will tell you a bit about the approach taken in this paper. So there are three steps, and the first step is pretty easy. We assume that the number of vertices is even. And this can be easily done by just adding an artificial vertex. So what this achieves is that now we can think of the problem as trying to find two perfect matchings, A and B, that form a Hamiltonian cycle and minimize"}, {"chunks": [{"end": 480.92, "start": 480.0, "text": "the"}, {"end": 481.28, "start": 480.92, "text": "sum"}, {"end": 481.44, "start": 481.28, "text": "of"}, {"end": 481.52, "start": 481.44, "text": "the"}, {"end": 481.92, "start": 481.52, "text": "total"}, {"end": 482.48, "start": 481.92, "text": "distance"}, {"end": 482.52, "start": 482.48, "text": "of"}, {"end": 482.56, "start": 482.52, "text": "the"}, {"end": 482.72, "start": 482.56, "text": "two"}, {"end": 483.32, "start": 482.72, "text": "matchings."}, {"end": 484.48, "start": 483.32, "text": "And"}, {"end": 484.8, "start": 484.48, "text": "the"}, {"end": 485.16, "start": 484.8, "text": "distance"}, {"end": 485.88, "start": 485.16, "text": "here"}, {"end": 486.64, "start": 485.88, "text": "denotes"}, {"end": 486.88, "start": 486.64, "text": "the"}, {"end": 487.4, "start": 486.88, "text": "sum"}, {"end": 487.76, "start": 487.4, "text": "of"}, {"end": 488.08, "start": 487.76, "text": "all"}, {"end": 488.12, "start": 488.08, "text": "the"}, {"end": 488.64, "start": 488.12, "text": "distances"}, {"end": 488.76, "start": 488.64, "text": "of"}, {"end": 488.84, "start": 488.76, "text": "the"}, {"end": 489.4, "start": 488.84, "text": "edges"}, {"end": 489.48, "start": 489.4, "text": "in"}, {"end": 489.96, "start": 489.48, "text": "the"}, {"end": 491.52, "start": 489.96, "text": "matching."}, {"end": 491.76, "start": 491.52, "text": "So"}, {"end": 492.08, "start": 491.76, "text": "again,"}, {"end": 492.44, "start": 492.08, "text": "if"}, {"end": 492.56, "start": 492.44, "text": "we"}, {"end": 492.68, "start": 492.56, "text": "look"}, {"end": 493.08, "start": 492.68, "text": "at"}, {"end": 493.4, "start": 493.08, "text": "our"}, {"end": 493.68, "start": 493.4, "text": "example,"}, {"end": 494.96, "start": 493.68, "text": "conveniently,"}, {"end": 495.32, "start": 494.96, "text": "six"}, {"end": 495.56, "start": 495.32, "text": "is"}, {"end": 496.96, "start": 495.56, "text": "even."}, {"end": 497.36, "start": 496.96, "text": "So"}, {"end": 497.52, "start": 497.36, "text": "if"}, {"end": 497.88, "start": 497.52, "text": "we"}, {"end": 498.28, "start": 497.88, "text": "make"}, {"end": 498.56, "start": 498.28, "text": "a"}, {"end": 499.0, "start": 498.56, "text": "tour,"}, {"end": 499.52, "start": 499.0, "text": "then"}, {"end": 500.12, "start": 499.52, "text": "the"}, {"end": 500.44, "start": 500.12, "text": "traveling"}, {"end": 501.56, "start": 500.44, "text": "salesman"}, {"end": 502.48, "start": 501.56, "text": "actually"}, {"end": 502.96, "start": 502.48, "text": "alternates"}, {"end": 503.24, "start": 502.96, "text": "between"}, {"end": 503.56, "start": 503.24, "text": "using"}, {"end": 503.72, "start": 503.56, "text": "a"}, {"end": 503.96, "start": 503.72, "text": "red"}, {"end": 504.8, "start": 503.96, "text": "edge"}, {"end": 505.12, "start": 504.8, "text": "and"}, {"end": 505.28, "start": 505.12, "text": "a"}, {"end": 505.56, "start": 505.28, "text": "yellow"}, {"end": 506.28, "start": 505.56, "text": "edge."}, {"end": 506.44, "start": 506.28, "text": "So"}, {"end": 506.56, "start": 506.44, "text": "the"}, {"end": 506.84, "start": 506.56, "text": "tour"}, {"end": 507.24, "start": 506.84, "text": "splits"}, {"end": 507.44, "start": 507.24, "text": "into"}, {"end": 507.72, "start": 507.44, "text": "perfect"}, {"end": 508.28, "start": 507.72, "text": "matchings,"}, {"end": 509.04, "start": 508.28, "text": "the"}, {"end": 509.4, "start": 509.04, "text": "red"}, {"end": 509.4, "start": 509.4, "text": "and"}, {"end": 509.44, "start": 509.4, "text": "the"}, {"end": 509.76, "start": 509.44, "text": "yellow"}, {"end": 509.96, "start": 509.76, "text": "one."}], "text": " the sum of the total distance of the two matchings. And the distance here denotes the sum of all the distances of the edges in the matching. So again, if we look at our example, conveniently, six is even. So if we make a tour, then the traveling salesman actually alternates between using a red edge and a yellow edge. So the tour splits into perfect matchings, the red and the yellow one."}, {"chunks": [{"end": 510.68, "start": 510.0, "text": "So"}, {"end": 511.12, "start": 510.68, "text": "we"}, {"end": 511.48, "start": 511.12, "text": "could"}, {"end": 511.72, "start": 511.48, "text": "look"}, {"end": 512.16, "start": 511.72, "text": "for"}, {"end": 512.6, "start": 512.16, "text": "the"}, {"end": 513.0, "start": 512.6, "text": "minimum"}, {"end": 513.28, "start": 513.0, "text": "tool"}, {"end": 513.88, "start": 513.28, "text": "by"}, {"end": 514.24, "start": 513.88, "text": "listing"}, {"end": 514.48, "start": 514.24, "text": "all"}, {"end": 514.76, "start": 514.48, "text": "the"}, {"end": 514.88, "start": 514.76, "text": "perfect"}, {"end": 515.24, "start": 514.88, "text": "matchings"}, {"end": 515.28, "start": 515.24, "text": "in"}, {"end": 515.32, "start": 515.28, "text": "the"}, {"end": 515.76, "start": 515.32, "text": "graph"}, {"end": 516.28, "start": 515.76, "text": "and"}, {"end": 516.52, "start": 516.28, "text": "then"}, {"end": 516.84, "start": 516.52, "text": "looking"}, {"end": 517.12, "start": 516.84, "text": "for"}, {"end": 517.12, "start": 517.12, "text": "the"}, {"end": 517.6, "start": 517.12, "text": "pair"}, {"end": 517.84, "start": 517.6, "text": "that"}, {"end": 518.16, "start": 517.84, "text": "forms"}, {"end": 518.36, "start": 518.16, "text": "a"}, {"end": 518.64, "start": 518.36, "text": "Hamiltonian"}, {"end": 519.48, "start": 518.64, "text": "cycle"}, {"end": 519.72, "start": 519.48, "text": "and"}, {"end": 520.04, "start": 519.72, "text": "has"}, {"end": 520.56, "start": 520.04, "text": "minimum"}, {"end": 520.92, "start": 520.56, "text": "total"}, {"end": 522.76, "start": 520.92, "text": "distance."}, {"end": 522.88, "start": 522.76, "text": "But"}, {"end": 523.48, "start": 522.88, "text": "unfortunately,"}, {"end": 523.6, "start": 523.48, "text": "there"}, {"end": 524.12, "start": 523.6, "text": "can"}, {"end": 524.4, "start": 524.12, "text": "be"}, {"end": 524.72, "start": 524.4, "text": "as"}, {"end": 524.88, "start": 524.72, "text": "many"}, {"end": 524.92, "start": 524.88, "text": "as"}, {"end": 525.16, "start": 524.92, "text": "2"}, {"end": 525.44, "start": 525.16, "text": "to"}, {"end": 526.0, "start": 525.44, "text": "the"}, {"end": 526.64, "start": 526.0, "text": "power"}, {"end": 527.52, "start": 526.64, "text": "capital"}, {"end": 528.4, "start": 527.52, "text": "omega"}, {"end": 528.64, "start": 528.4, "text": "of"}, {"end": 529.32, "start": 528.64, "text": "n,"}, {"end": 530.08, "start": 529.32, "text": "sorry,"}, {"end": 530.16, "start": 530.08, "text": "there"}, {"end": 530.68, "start": 530.16, "text": "can"}, {"end": 531.0, "start": 530.68, "text": "be"}, {"end": 531.16, "start": 531.0, "text": "as"}, {"end": 531.68, "start": 531.16, "text": "many"}, {"end": 532.76, "start": 531.68, "text": "as"}, {"end": 533.44, "start": 532.76, "text": "n"}, {"end": 533.76, "start": 533.44, "text": "to"}, {"end": 534.12, "start": 533.76, "text": "the"}, {"end": 534.68, "start": 534.12, "text": "power"}, {"end": 536.76, "start": 534.68, "text": "capital"}, {"end": 537.24, "start": 536.76, "text": "omega"}, {"end": 538.36, "start": 537.24, "text": "of"}, {"end": 538.44, "start": 538.36, "text": "n"}, {"end": 538.72, "start": 538.44, "text": "perfect"}, {"end": 539.96, "start": 538.72, "text": "matchings."}], "text": " So we could look for the minimum tool by listing all the perfect matchings in the graph and then looking for the pair that forms a Hamiltonian cycle and has minimum total distance. But unfortunately, there can be as many as 2 to the power capital omega of n, sorry, there can be as many as n to the power capital omega of n perfect matchings."}, {"chunks": [{"end": 540.52, "start": 540.0, "text": "This"}, {"end": 540.72, "start": 540.52, "text": "is"}, {"end": 542.0, "start": 540.72, "text": "way"}, {"end": 543.24, "start": 542.0, "text": "too"}, {"end": 544.0, "start": 543.24, "text": "slow."}, {"end": 544.08, "start": 544.0, "text": "But"}, {"end": 544.24, "start": 544.08, "text": "to"}, {"end": 544.32, "start": 544.24, "text": "do"}, {"end": 544.56, "start": 544.32, "text": "this"}, {"end": 545.16, "start": 544.56, "text": "faster,"}, {"end": 545.48, "start": 545.16, "text": "what"}, {"end": 545.88, "start": 545.48, "text": "we"}, {"end": 545.88, "start": 545.88, "text": "do"}, {"end": 545.88, "start": 545.88, "text": "in"}, {"end": 546.24, "start": 545.88, "text": "the"}, {"end": 546.56, "start": 546.24, "text": "first"}, {"end": 547.68, "start": 546.56, "text": "step"}, {"end": 547.88, "start": 547.68, "text": "is"}, {"end": 547.96, "start": 547.88, "text": "the"}, {"end": 549.0, "start": 547.96, "text": "following."}, {"end": 549.12, "start": 549.0, "text": "So"}, {"end": 549.32, "start": 549.12, "text": "we"}, {"end": 549.68, "start": 549.32, "text": "find"}, {"end": 549.88, "start": 549.68, "text": "a"}, {"end": 550.8, "start": 549.88, "text": "family"}, {"end": 551.4, "start": 550.8, "text": "calligraphic"}, {"end": 552.36, "start": 551.4, "text": "A"}, {"end": 552.88, "start": 552.36, "text": "of"}, {"end": 553.6, "start": 552.88, "text": "2"}, {"end": 554.08, "start": 553.6, "text": "to"}, {"end": 554.4, "start": 554.08, "text": "the"}, {"end": 554.64, "start": 554.4, "text": "n"}, {"end": 554.92, "start": 554.64, "text": "over"}, {"end": 555.2, "start": 554.92, "text": "two"}, {"end": 555.44, "start": 555.2, "text": "perfect"}, {"end": 557.04, "start": 555.44, "text": "matchings."}, {"end": 557.16, "start": 557.04, "text": "And"}, {"end": 557.32, "start": 557.16, "text": "this"}, {"end": 558.24, "start": 557.32, "text": "family"}, {"end": 558.64, "start": 558.24, "text": "will"}, {"end": 558.8, "start": 558.64, "text": "be"}, {"end": 560.12, "start": 558.8, "text": "representative"}, {"end": 560.44, "start": 560.12, "text": "for"}, {"end": 560.68, "start": 560.44, "text": "all"}, {"end": 561.0, "start": 560.68, "text": "perfect"}, {"end": 562.2, "start": 561.0, "text": "matchings"}, {"end": 562.28, "start": 562.2, "text": "in"}, {"end": 562.76, "start": 562.28, "text": "the"}, {"end": 562.96, "start": 562.76, "text": "input"}, {"end": 563.0, "start": 562.96, "text": "graph."}, {"end": 563.0, "start": 563.0, "text": "And"}, {"end": 563.16, "start": 563.0, "text": "we"}, {"end": 564.08, "start": 563.16, "text": "are"}, {"end": 564.44, "start": 564.08, "text": "going"}, {"end": 564.96, "start": 564.44, "text": "to"}, {"end": 565.24, "start": 564.96, "text": "find"}, {"end": 565.64, "start": 565.24, "text": "this"}, {"end": 566.0, "start": 565.64, "text": "set"}, {"end": 566.28, "start": 566.0, "text": "in"}, {"end": 566.64, "start": 566.28, "text": "fast"}, {"end": 566.96, "start": 566.64, "text": "enough"}, {"end": 567.12, "start": 566.96, "text": "in"}, {"end": 567.36, "start": 567.12, "text": "claimed"}, {"end": 567.56, "start": 567.36, "text": "running"}, {"end": 568.16, "start": 567.56, "text": "time."}, {"end": 568.52, "start": 568.16, "text": "And"}, {"end": 568.68, "start": 568.52, "text": "here"}, {"end": 569.0, "start": 568.68, "text": "we"}, {"end": 569.2, "start": 569.0, "text": "are"}, {"end": 569.52, "start": 569.2, "text": "using"}, {"end": 569.72, "start": 569.52, "text": "this"}, {"end": 569.96, "start": 569.72, "text": "assumption"}], "text": " This is way too slow. But to do this faster, what we do in the first step is the following. So we find a family calligraphic A of 2 to the n over two perfect matchings. And this family will be representative for all perfect matchings in the input graph. And we are going to find this set in fast enough in claimed running time. And here we are using this assumption"}, {"chunks": [{"end": 570.36, "start": 570.0, "text": "that"}, {"end": 570.52, "start": 570.36, "text": "we"}, {"end": 570.84, "start": 570.52, "text": "can"}, {"end": 571.92, "start": 570.84, "text": "multiply"}, {"end": 572.72, "start": 571.92, "text": "matrices"}, {"end": 572.92, "start": 572.72, "text": "in"}, {"end": 573.32, "start": 572.92, "text": "input"}, {"end": 575.24, "start": 573.32, "text": "linear"}, {"end": 575.6, "start": 575.24, "text": "time."}, {"end": 575.88, "start": 575.6, "text": "So"}, {"end": 576.56, "start": 575.88, "text": "representative"}, {"end": 576.84, "start": 576.56, "text": "here"}, {"end": 577.08, "start": 576.84, "text": "means"}, {"end": 577.16, "start": 577.08, "text": "the"}, {"end": 578.6, "start": 577.16, "text": "following."}, {"end": 578.96, "start": 578.6, "text": "So"}, {"end": 579.16, "start": 578.96, "text": "if"}, {"end": 579.16, "start": 579.16, "text": "you"}, {"end": 579.2, "start": 579.16, "text": "have"}, {"end": 579.4, "start": 579.2, "text": "two"}, {"end": 579.96, "start": 579.4, "text": "families"}, {"end": 580.12, "start": 579.96, "text": "of"}, {"end": 580.44, "start": 580.12, "text": "perfect"}, {"end": 581.16, "start": 580.44, "text": "matchings,"}, {"end": 582.0, "start": 581.16, "text": "calligraphic"}, {"end": 582.32, "start": 582.0, "text": "A"}, {"end": 582.4, "start": 582.32, "text": "and"}, {"end": 582.88, "start": 582.4, "text": "calligraphic"}, {"end": 583.08, "start": 582.88, "text": "A"}, {"end": 584.0, "start": 583.08, "text": "prime,"}, {"end": 584.48, "start": 584.0, "text": "then"}, {"end": 584.76, "start": 584.48, "text": "we"}, {"end": 585.04, "start": 584.76, "text": "say"}, {"end": 585.52, "start": 585.04, "text": "calligraphic"}, {"end": 585.88, "start": 585.52, "text": "A"}, {"end": 586.08, "start": 585.88, "text": "is"}, {"end": 586.8, "start": 586.08, "text": "representative"}, {"end": 587.16, "start": 586.8, "text": "for"}, {"end": 587.32, "start": 587.16, "text": "calligraphic"}, {"end": 587.48, "start": 587.32, "text": "A"}, {"end": 588.44, "start": 587.48, "text": "prime."}, {"end": 588.52, "start": 588.44, "text": "If"}, {"end": 588.84, "start": 588.52, "text": "for"}, {"end": 589.28, "start": 588.84, "text": "each"}, {"end": 589.92, "start": 589.28, "text": "perfect"}, {"end": 590.32, "start": 589.92, "text": "matching"}, {"end": 591.0, "start": 590.32, "text": "B,"}, {"end": 591.12, "start": 591.0, "text": "so"}, {"end": 591.28, "start": 591.12, "text": "this"}, {"end": 591.48, "start": 591.28, "text": "can"}, {"end": 591.64, "start": 591.48, "text": "be"}, {"end": 591.88, "start": 591.64, "text": "any"}, {"end": 592.16, "start": 591.88, "text": "perfect"}, {"end": 592.76, "start": 592.16, "text": "matching,"}, {"end": 593.12, "start": 592.76, "text": "so"}, {"end": 593.28, "start": 593.12, "text": "it"}, {"end": 593.64, "start": 593.28, "text": "doesn't"}, {"end": 593.72, "start": 593.64, "text": "even"}, {"end": 594.0, "start": 593.72, "text": "need"}, {"end": 594.16, "start": 594.0, "text": "to"}, {"end": 594.28, "start": 594.16, "text": "be"}, {"end": 594.4, "start": 594.28, "text": "in"}, {"end": 595.12, "start": 594.4, "text": "this"}, {"end": 595.4, "start": 595.12, "text": "input"}, {"end": 597.28, "start": 595.4, "text": "graph."}, {"end": 597.48, "start": 597.28, "text": "If"}, {"end": 597.68, "start": 597.48, "text": "for"}, {"end": 598.16, "start": 597.68, "text": "each"}, {"end": 598.48, "start": 598.16, "text": "perfect"}, {"end": 598.8, "start": 598.48, "text": "matching"}, {"end": 598.96, "start": 598.8, "text": "B,"}, {"end": 599.0, "start": 598.96, "text": "the"}, {"end": 599.36, "start": 599.0, "text": "following"}, {"end": 599.96, "start": 599.36, "text": "holds."}], "text": " that we can multiply matrices in input linear time. So representative here means the following. So if you have two families of perfect matchings, calligraphic A and calligraphic A prime, then we say calligraphic A is representative for calligraphic A prime. If for each perfect matching B, so this can be any perfect matching, so it doesn't even need to be in this input graph. If for each perfect matching B, the following holds."}, {"chunks": [{"end": 601.84, "start": 600.0, "text": "So"}, {"end": 602.52, "start": 601.84, "text": "the"}, {"end": 603.84, "start": 602.52, "text": "minimum"}, {"end": 604.68, "start": 603.84, "text": "distance"}, {"end": 604.8, "start": 604.68, "text": "of"}, {"end": 605.2, "start": 604.8, "text": "a"}, {"end": 605.44, "start": 605.2, "text": "perfect"}, {"end": 605.96, "start": 605.44, "text": "matching"}, {"end": 606.36, "start": 605.96, "text": "A"}, {"end": 606.48, "start": 606.36, "text": "in"}, {"end": 606.84, "start": 606.48, "text": "calligraphic"}, {"end": 607.16, "start": 606.84, "text": "A"}, {"end": 607.56, "start": 607.16, "text": "that"}, {"end": 607.84, "start": 607.56, "text": "forms"}, {"end": 608.08, "start": 607.84, "text": "a"}, {"end": 608.32, "start": 608.08, "text": "Hamiltonian"}, {"end": 608.64, "start": 608.32, "text": "cycle"}, {"end": 608.88, "start": 608.64, "text": "with"}, {"end": 609.76, "start": 608.88, "text": "B"}, {"end": 609.96, "start": 609.76, "text": "is"}, {"end": 610.4, "start": 609.96, "text": "always"}, {"end": 610.64, "start": 610.4, "text": "at"}, {"end": 611.16, "start": 610.64, "text": "most"}, {"end": 611.64, "start": 611.16, "text": "the"}, {"end": 612.04, "start": 611.64, "text": "minimum"}, {"end": 612.8, "start": 612.04, "text": "distance"}, {"end": 613.04, "start": 612.8, "text": "of"}, {"end": 613.2, "start": 613.04, "text": "a"}, {"end": 613.44, "start": 613.2, "text": "perfect"}, {"end": 613.8, "start": 613.44, "text": "matching"}, {"end": 613.96, "start": 613.8, "text": "A"}, {"end": 614.96, "start": 613.96, "text": "prime"}, {"end": 615.12, "start": 614.96, "text": "in"}, {"end": 615.64, "start": 615.12, "text": "calligraphic"}, {"end": 615.8, "start": 615.64, "text": "A"}, {"end": 616.12, "start": 615.8, "text": "prime"}, {"end": 616.28, "start": 616.12, "text": "that"}, {"end": 616.72, "start": 616.28, "text": "forms"}, {"end": 616.92, "start": 616.72, "text": "a"}, {"end": 617.16, "start": 616.92, "text": "Hamiltonian"}, {"end": 618.16, "start": 617.16, "text": "cycle"}, {"end": 619.56, "start": 618.16, "text": "with"}, {"end": 619.72, "start": 619.56, "text": "B."}, {"end": 619.84, "start": 619.72, "text": "So"}, {"end": 620.08, "start": 619.84, "text": "the"}, {"end": 620.44, "start": 620.08, "text": "intuition"}, {"end": 620.8, "start": 620.44, "text": "behind"}, {"end": 621.08, "start": 620.8, "text": "this"}, {"end": 622.0, "start": 621.08, "text": "definition"}, {"end": 622.2, "start": 622.0, "text": "is"}, {"end": 622.28, "start": 622.2, "text": "that"}, {"end": 622.56, "start": 622.28, "text": "we"}, {"end": 623.52, "start": 622.56, "text": "can,"}, {"end": 624.36, "start": 623.52, "text": "when"}, {"end": 625.16, "start": 624.36, "text": "we"}, {"end": 625.2, "start": 625.16, "text": "have"}, {"end": 625.32, "start": 625.2, "text": "a"}, {"end": 625.76, "start": 625.32, "text": "set"}, {"end": 625.96, "start": 625.76, "text": "A"}, {"end": 626.36, "start": 625.96, "text": "prime,"}, {"end": 626.84, "start": 626.36, "text": "calligraphic"}, {"end": 627.0, "start": 626.84, "text": "A"}, {"end": 627.4, "start": 627.0, "text": "prime"}, {"end": 627.4, "start": 627.4, "text": "of"}, {"end": 627.64, "start": 627.4, "text": "perfect"}, {"end": 628.28, "start": 627.64, "text": "matchings,"}, {"end": 628.52, "start": 628.28, "text": "we"}, {"end": 628.76, "start": 628.52, "text": "can"}, {"end": 629.32, "start": 628.76, "text": "always"}, {"end": 629.96, "start": 629.32, "text": "focus,"}], "text": " So the minimum distance of a perfect matching A in calligraphic A that forms a Hamiltonian cycle with B is always at most the minimum distance of a perfect matching A prime in calligraphic A prime that forms a Hamiltonian cycle with B. So the intuition behind this definition is that we can, when we have a set A prime, calligraphic A prime of perfect matchings, we can always focus,"}, {"chunks": [{"end": 630.04, "start": 630.0, "text": "we"}, {"end": 630.64, "start": 630.04, "text": "can"}, {"end": 631.92, "start": 630.64, "text": "throw"}, {"end": 632.96, "start": 631.92, "text": "away"}, {"end": 633.52, "start": 632.96, "text": "matchings"}, {"end": 633.72, "start": 633.52, "text": "in"}, {"end": 633.96, "start": 633.72, "text": "A'"}, {"end": 634.08, "start": 633.96, "text": "to"}, {"end": 634.4, "start": 634.08, "text": "get"}, {"end": 635.0, "start": 634.4, "text": "calligraphic"}, {"end": 635.2, "start": 635.0, "text": "A,"}, {"end": 635.44, "start": 635.2, "text": "and"}, {"end": 636.24, "start": 635.44, "text": "we"}, {"end": 636.68, "start": 636.24, "text": "will"}, {"end": 636.76, "start": 636.68, "text": "not"}, {"end": 637.12, "start": 636.76, "text": "lose"}, {"end": 637.4, "start": 637.12, "text": "an"}, {"end": 637.76, "start": 637.4, "text": "optimal"}, {"end": 638.32, "start": 637.76, "text": "solution"}, {"end": 638.64, "start": 638.32, "text": "with"}, {"end": 639.04, "start": 638.64, "text": "this,"}, {"end": 639.48, "start": 639.04, "text": "because"}, {"end": 640.08, "start": 639.48, "text": "whenever"}, {"end": 640.24, "start": 640.08, "text": "an"}, {"end": 640.6, "start": 640.24, "text": "optimal"}, {"end": 641.04, "start": 640.6, "text": "solution"}, {"end": 641.24, "start": 641.04, "text": "was"}, {"end": 641.48, "start": 641.24, "text": "formed"}, {"end": 641.6, "start": 641.48, "text": "with"}, {"end": 641.88, "start": 641.6, "text": "some"}, {"end": 642.16, "start": 641.88, "text": "A'"}, {"end": 642.48, "start": 642.16, "text": "and"}, {"end": 642.8, "start": 642.48, "text": "some"}, {"end": 643.44, "start": 642.8, "text": "B,"}, {"end": 643.84, "start": 643.44, "text": "then"}, {"end": 644.04, "start": 643.84, "text": "we"}, {"end": 644.36, "start": 644.04, "text": "can"}, {"end": 644.64, "start": 644.36, "text": "replace"}, {"end": 645.04, "start": 644.64, "text": "this"}, {"end": 645.48, "start": 645.04, "text": "A'"}, {"end": 645.64, "start": 645.48, "text": "with"}, {"end": 645.84, "start": 645.64, "text": "some"}, {"end": 646.16, "start": 645.84, "text": "other"}, {"end": 646.48, "start": 646.16, "text": "A"}, {"end": 646.76, "start": 646.48, "text": "that"}, {"end": 647.36, "start": 646.76, "text": "actually"}, {"end": 647.64, "start": 647.36, "text": "lives"}, {"end": 647.72, "start": 647.64, "text": "in"}, {"end": 650.12, "start": 647.72, "text": "calligraphic"}, {"end": 650.28, "start": 650.12, "text": "A."}, {"end": 650.52, "start": 650.28, "text": "And"}, {"end": 650.52, "start": 650.52, "text": "note"}, {"end": 650.56, "start": 650.52, "text": "that"}, {"end": 650.88, "start": 650.56, "text": "actually"}, {"end": 651.2, "start": 650.88, "text": "this"}, {"end": 651.4, "start": 651.2, "text": "will"}, {"end": 651.56, "start": 651.4, "text": "be"}, {"end": 652.12, "start": 651.56, "text": "an"}, {"end": 653.16, "start": 652.12, "text": "equality,"}, {"end": 653.56, "start": 653.16, "text": "since"}, {"end": 653.84, "start": 653.56, "text": "we"}, {"end": 654.16, "start": 653.84, "text": "assumed"}, {"end": 654.36, "start": 654.16, "text": "a"}, {"end": 655.08, "start": 654.36, "text": "subset"}, {"end": 655.2, "start": 655.08, "text": "thing"}, {"end": 655.28, "start": 655.2, "text": "here,"}, {"end": 655.32, "start": 655.28, "text": "but"}, {"end": 655.52, "start": 655.32, "text": "for"}, {"end": 656.96, "start": 655.52, "text": "us"}, {"end": 657.08, "start": 656.96, "text": "that"}, {"end": 659.52, "start": 657.08, "text": "is"}, {"end": 659.88, "start": 659.52, "text": "not"}, {"end": 659.96, "start": 659.88, "text": "important."}], "text": " we can throw away matchings in A' to get calligraphic A, and we will not lose an optimal solution with this, because whenever an optimal solution was formed with some A' and some B, then we can replace this A' with some other A that actually lives in calligraphic A. And note that actually this will be an equality, since we assumed a subset thing here, but for us that is not important."}, {"chunks": [{"end": 660.84, "start": 660.0, "text": "Okay,"}, {"end": 663.28, "start": 660.84, "text": "so"}, {"end": 663.68, "start": 663.28, "text": "that"}, {"end": 664.2, "start": 663.68, "text": "is"}, {"end": 664.64, "start": 664.2, "text": "step"}, {"end": 665.32, "start": 664.64, "text": "one."}, {"end": 665.76, "start": 665.32, "text": "And"}, {"end": 665.96, "start": 665.76, "text": "now"}, {"end": 666.16, "start": 665.96, "text": "step"}, {"end": 666.28, "start": 666.16, "text": "two"}, {"end": 666.56, "start": 666.28, "text": "is"}, {"end": 667.04, "start": 666.56, "text": "already"}, {"end": 667.32, "start": 667.04, "text": "pretty"}, {"end": 668.88, "start": 667.32, "text": "natural."}, {"end": 669.44, "start": 668.88, "text": "So"}, {"end": 669.84, "start": 669.44, "text": "by"}, {"end": 670.28, "start": 669.84, "text": "this"}, {"end": 670.96, "start": 670.28, "text": "property"}, {"end": 670.96, "start": 670.96, "text": "of"}, {"end": 671.8, "start": 670.96, "text": "representative"}, {"end": 672.0, "start": 671.8, "text": "set,"}, {"end": 672.2, "start": 672.0, "text": "now"}, {"end": 672.64, "start": 672.2, "text": "actually"}, {"end": 672.92, "start": 672.64, "text": "we"}, {"end": 672.92, "start": 672.92, "text": "can"}, {"end": 673.2, "start": 672.92, "text": "just"}, {"end": 673.64, "start": 673.2, "text": "restrict"}, {"end": 674.4, "start": 673.64, "text": "attention"}, {"end": 674.52, "start": 674.4, "text": "to"}, {"end": 674.92, "start": 674.52, "text": "finding"}, {"end": 675.32, "start": 674.92, "text": "two"}, {"end": 675.6, "start": 675.32, "text": "perfect"}, {"end": 676.36, "start": 675.6, "text": "matchings"}, {"end": 676.52, "start": 676.36, "text": "in"}, {"end": 676.92, "start": 676.52, "text": "calligraphic"}, {"end": 677.48, "start": 676.92, "text": "A"}, {"end": 677.84, "start": 677.48, "text": "that"}, {"end": 678.16, "start": 677.84, "text": "jointly"}, {"end": 678.44, "start": 678.16, "text": "form"}, {"end": 678.6, "start": 678.44, "text": "a"}, {"end": 679.4, "start": 678.6, "text": "Hamiltonian"}, {"end": 679.68, "start": 679.4, "text": "cycle"}, {"end": 680.04, "start": 679.68, "text": "and"}, {"end": 680.84, "start": 680.04, "text": "minimize"}, {"end": 681.28, "start": 680.84, "text": "the"}, {"end": 681.72, "start": 681.28, "text": "total"}, {"end": 682.2, "start": 681.72, "text": "distance,"}, {"end": 682.48, "start": 682.2, "text": "the"}, {"end": 682.6, "start": 682.48, "text": "sum"}, {"end": 682.6, "start": 682.6, "text": "of"}, {"end": 682.64, "start": 682.6, "text": "the"}, {"end": 683.04, "start": 682.64, "text": "distances."}, {"end": 683.04, "start": 683.04, "text": "And"}, {"end": 683.44, "start": 683.04, "text": "this"}, {"end": 683.68, "start": 683.44, "text": "is"}, {"end": 684.36, "start": 683.68, "text": "exactly"}, {"end": 684.64, "start": 684.36, "text": "what"}, {"end": 684.72, "start": 684.64, "text": "we"}, {"end": 684.84, "start": 684.72, "text": "will"}, {"end": 684.88, "start": 684.84, "text": "do"}, {"end": 685.08, "start": 684.88, "text": "in"}, {"end": 685.32, "start": 685.08, "text": "step"}, {"end": 685.84, "start": 685.32, "text": "two"}, {"end": 686.24, "start": 685.84, "text": "in"}, {"end": 686.52, "start": 686.24, "text": "this"}, {"end": 686.96, "start": 686.52, "text": "particular"}, {"end": 687.16, "start": 686.96, "text": "running"}, {"end": 687.16, "start": 687.16, "text": "time."}, {"end": 687.56, "start": 687.16, "text": "So"}, {"end": 687.76, "start": 687.56, "text": "note"}, {"end": 688.12, "start": 687.76, "text": "this"}, {"end": 688.2, "start": 688.12, "text": "is"}, {"end": 688.76, "start": 688.2, "text": "fast"}, {"end": 689.36, "start": 688.76, "text": "enough"}, {"end": 689.96, "start": 689.36, "text": "because"}], "text": " Okay, so that is step one. And now step two is already pretty natural. So by this property of representative set, now actually we can just restrict attention to finding two perfect matchings in calligraphic A that jointly form a Hamiltonian cycle and minimize the total distance, the sum of the distances. And this is exactly what we will do in step two in this particular running time. So note this is fast enough because"}, {"chunks": [{"end": 690.28, "start": 690.0, "text": "The"}, {"end": 690.6, "start": 690.28, "text": "size"}, {"end": 690.64, "start": 690.6, "text": "of"}, {"end": 691.2, "start": 690.64, "text": "calligraphic"}, {"end": 691.44, "start": 691.2, "text": "A"}, {"end": 691.56, "start": 691.44, "text": "is"}, {"end": 691.88, "start": 691.56, "text": "promised"}, {"end": 692.0, "start": 691.88, "text": "to"}, {"end": 692.16, "start": 692.0, "text": "be"}, {"end": 692.24, "start": 692.16, "text": "at"}, {"end": 692.6, "start": 692.24, "text": "most"}, {"end": 692.96, "start": 692.6, "text": "2"}, {"end": 693.12, "start": 692.96, "text": "to"}, {"end": 693.44, "start": 693.12, "text": "the"}, {"end": 693.44, "start": 693.44, "text": "n"}, {"end": 693.52, "start": 693.44, "text": "over"}, {"end": 693.68, "start": 693.52, "text": "2."}, {"end": 694.8, "start": 693.68, "text": "And"}, {"end": 696.56, "start": 694.8, "text": "now"}, {"end": 696.68, "start": 696.56, "text": "I"}, {"end": 696.72, "start": 696.68, "text": "will"}, {"end": 697.6, "start": 696.72, "text": "focus"}, {"end": 697.68, "start": 697.6, "text": "in"}, {"end": 697.96, "start": 697.68, "text": "a"}, {"end": 698.16, "start": 697.96, "text": "bit"}, {"end": 698.32, "start": 698.16, "text": "more"}, {"end": 699.12, "start": 698.32, "text": "detail"}, {"end": 699.28, "start": 699.12, "text": "on"}, {"end": 699.52, "start": 699.28, "text": "step"}, {"end": 700.88, "start": 699.52, "text": "1."}, {"end": 701.44, "start": 700.88, "text": "And"}, {"end": 701.72, "start": 701.44, "text": "the"}, {"end": 702.28, "start": 701.72, "text": "very"}, {"end": 703.12, "start": 702.28, "text": "important"}, {"end": 703.72, "start": 703.12, "text": "matrix"}, {"end": 703.88, "start": 703.72, "text": "that"}, {"end": 704.68, "start": 703.88, "text": "actually"}, {"end": 705.04, "start": 704.68, "text": "helps"}, {"end": 705.2, "start": 705.04, "text": "in"}, {"end": 705.56, "start": 705.2, "text": "both"}, {"end": 705.6, "start": 705.56, "text": "of"}, {"end": 705.96, "start": 705.6, "text": "these"}, {"end": 706.84, "start": 705.96, "text": "steps"}, {"end": 707.08, "start": 706.84, "text": "is"}, {"end": 707.12, "start": 707.08, "text": "the"}, {"end": 707.44, "start": 707.12, "text": "matching"}, {"end": 708.16, "start": 707.44, "text": "connectivity"}, {"end": 710.04, "start": 708.16, "text": "matrix."}, {"end": 710.4, "start": 710.04, "text": "And"}, {"end": 710.52, "start": 710.4, "text": "it's"}, {"end": 711.2, "start": 710.52, "text": "defined"}, {"end": 711.84, "start": 711.2, "text": "as"}, {"end": 712.56, "start": 711.84, "text": "follows."}, {"end": 712.72, "start": 712.56, "text": "For"}, {"end": 712.84, "start": 712.72, "text": "every"}, {"end": 713.0, "start": 712.84, "text": "even"}, {"end": 713.4, "start": 713.0, "text": "integer"}, {"end": 714.2, "start": 713.4, "text": "n"}, {"end": 714.48, "start": 714.2, "text": "and"}, {"end": 714.72, "start": 714.48, "text": "two"}, {"end": 715.4, "start": 714.72, "text": "perfect"}, {"end": 715.92, "start": 715.4, "text": "matchings"}, {"end": 716.52, "start": 715.92, "text": "a"}, {"end": 716.64, "start": 716.52, "text": "and"}, {"end": 716.96, "start": 716.64, "text": "b"}, {"end": 717.36, "start": 716.96, "text": "of"}, {"end": 717.6, "start": 717.36, "text": "the"}, {"end": 718.68, "start": 717.6, "text": "complete"}, {"end": 718.84, "start": 718.68, "text": "graph"}, {"end": 719.24, "start": 718.84, "text": "on"}, {"end": 719.4, "start": 719.24, "text": "n"}, {"end": 719.96, "start": 719.4, "text": "versus,"}], "text": " The size of calligraphic A is promised to be at most 2 to the n over 2. And now I will focus in a bit more detail on step 1. And the very important matrix that actually helps in both of these steps is the matching connectivity matrix. And it's defined as follows. For every even integer n and two perfect matchings a and b of the complete graph on n versus,"}, {"chunks": [{"end": 720.0, "start": 720.0, "text": "The"}, {"end": 720.32, "start": 720.0, "text": "identity"}, {"end": 720.92, "start": 720.32, "text": "matrix"}, {"end": 721.04, "start": 720.92, "text": "is"}, {"end": 721.28, "start": 721.04, "text": "defined"}, {"end": 721.4, "start": 721.28, "text": "as"}, {"end": 721.88, "start": 721.4, "text": "follows,"}, {"end": 722.04, "start": 721.88, "text": "it's"}, {"end": 722.36, "start": 722.04, "text": "called"}, {"end": 723.16, "start": 722.36, "text": "Hn"}, {"end": 723.68, "start": 723.16, "text": "and"}, {"end": 724.28, "start": 723.68, "text": "if"}, {"end": 724.52, "start": 724.28, "text": "you"}, {"end": 724.56, "start": 724.52, "text": "look"}, {"end": 724.68, "start": 724.56, "text": "at"}, {"end": 725.2, "start": 724.68, "text": "the"}, {"end": 725.72, "start": 725.2, "text": "row"}, {"end": 726.16, "start": 725.72, "text": "indexed"}, {"end": 726.48, "start": 726.16, "text": "by"}, {"end": 726.8, "start": 726.48, "text": "A"}, {"end": 726.88, "start": 726.8, "text": "and"}, {"end": 726.92, "start": 726.88, "text": "the"}, {"end": 727.28, "start": 726.92, "text": "column"}, {"end": 727.56, "start": 727.28, "text": "indexed"}, {"end": 727.84, "start": 727.56, "text": "by"}, {"end": 728.4, "start": 727.84, "text": "B"}, {"end": 728.68, "start": 728.4, "text": "you"}, {"end": 728.96, "start": 728.68, "text": "write"}, {"end": 729.12, "start": 728.96, "text": "the"}, {"end": 729.84, "start": 729.12, "text": "1"}, {"end": 730.04, "start": 729.84, "text": "if"}, {"end": 730.44, "start": 730.04, "text": "A"}, {"end": 730.6, "start": 730.44, "text": "and"}, {"end": 730.64, "start": 730.6, "text": "B"}, {"end": 730.88, "start": 730.64, "text": "together"}, {"end": 731.08, "start": 730.88, "text": "form"}, {"end": 731.2, "start": 731.08, "text": "a"}, {"end": 731.8, "start": 731.2, "text": "single"}, {"end": 732.08, "start": 731.8, "text": "Hamiltonian"}, {"end": 733.08, "start": 732.08, "text": "cycle"}, {"end": 733.24, "start": 733.08, "text": "and"}, {"end": 733.64, "start": 733.24, "text": "0"}, {"end": 735.68, "start": 733.64, "text": "otherwise."}, {"end": 736.04, "start": 735.68, "text": "So"}, {"end": 736.24, "start": 736.04, "text": "here"}, {"end": 736.24, "start": 736.24, "text": "you"}, {"end": 736.4, "start": 736.24, "text": "can"}, {"end": 736.68, "start": 736.4, "text": "see"}, {"end": 737.36, "start": 736.68, "text": "H4,"}, {"end": 737.48, "start": 737.36, "text": "so"}, {"end": 737.6, "start": 737.48, "text": "there"}, {"end": 737.68, "start": 737.6, "text": "are"}, {"end": 738.2, "start": 737.68, "text": "3"}, {"end": 738.48, "start": 738.2, "text": "perfect"}, {"end": 738.92, "start": 738.48, "text": "matchings"}, {"end": 739.0, "start": 738.92, "text": "on"}, {"end": 739.24, "start": 739.0, "text": "4"}, {"end": 739.84, "start": 739.24, "text": "elements"}, {"end": 740.16, "start": 739.84, "text": "and"}, {"end": 740.36, "start": 740.16, "text": "you"}, {"end": 740.48, "start": 740.36, "text": "will"}, {"end": 740.76, "start": 740.48, "text": "get"}, {"end": 741.0, "start": 740.76, "text": "the"}, {"end": 741.12, "start": 741.0, "text": "complement"}, {"end": 741.4, "start": 741.12, "text": "of"}, {"end": 741.56, "start": 741.4, "text": "the"}, {"end": 742.16, "start": 741.56, "text": "identity"}, {"end": 743.12, "start": 742.16, "text": "matrix"}, {"end": 743.48, "start": 743.12, "text": "and"}, {"end": 743.64, "start": 743.48, "text": "here"}, {"end": 743.96, "start": 743.64, "text": "you"}, {"end": 744.24, "start": 743.96, "text": "see"}, {"end": 744.84, "start": 744.24, "text": "H6,"}, {"end": 745.16, "start": 744.84, "text": "so"}, {"end": 745.2, "start": 745.16, "text": "it"}, {"end": 745.48, "start": 745.2, "text": "turns"}, {"end": 745.6, "start": 745.48, "text": "out"}, {"end": 745.68, "start": 745.6, "text": "that"}, {"end": 745.88, "start": 745.68, "text": "there"}, {"end": 746.16, "start": 745.88, "text": "are"}, {"end": 746.8, "start": 746.16, "text": "15"}, {"end": 747.28, "start": 746.8, "text": "perfect"}, {"end": 747.64, "start": 747.28, "text": "matchings"}, {"end": 747.76, "start": 747.64, "text": "on"}, {"end": 748.24, "start": 747.76, "text": "6"}, {"end": 748.64, "start": 748.24, "text": "elements"}, {"end": 748.72, "start": 748.64, "text": "and"}, {"end": 749.08, "start": 748.72, "text": "you"}, {"end": 749.28, "start": 749.08, "text": "get"}, {"end": 749.96, "start": 749.28, "text": "this"}], "text": " The identity matrix is defined as follows, it's called Hn and if you look at the row indexed by A and the column indexed by B you write the 1 if A and B together form a single Hamiltonian cycle and 0 otherwise. So here you can see H4, so there are 3 perfect matchings on 4 elements and you will get the complement of the identity matrix and here you see H6, so it turns out that there are 15 perfect matchings on 6 elements and you get this"}, {"chunks": [{"end": 750.44, "start": 750.0, "text": "This"}, {"end": 750.72, "start": 750.44, "text": "kind"}, {"end": 752.44, "start": 750.72, "text": "of"}, {"end": 752.88, "start": 752.44, "text": "structure."}, {"end": 753.76, "start": 752.88, "text": "Okay,"}, {"end": 754.52, "start": 753.76, "text": "so"}, {"end": 754.76, "start": 754.52, "text": "why"}, {"end": 754.8, "start": 754.76, "text": "is"}, {"end": 754.92, "start": 754.8, "text": "this"}, {"end": 755.24, "start": 754.92, "text": "matrix"}, {"end": 755.52, "start": 755.24, "text": "so"}, {"end": 756.44, "start": 755.52, "text": "important?"}, {"end": 756.96, "start": 756.44, "text": "So"}, {"end": 757.2, "start": 756.96, "text": "let's"}, {"end": 757.48, "start": 757.2, "text": "again"}, {"end": 757.92, "start": 757.48, "text": "think"}, {"end": 757.92, "start": 757.92, "text": "of"}, {"end": 757.92, "start": 757.92, "text": "the"}, {"end": 758.2, "start": 757.92, "text": "problem"}, {"end": 758.44, "start": 758.2, "text": "that"}, {"end": 758.56, "start": 758.44, "text": "we"}, {"end": 758.56, "start": 758.56, "text": "are"}, {"end": 758.8, "start": 758.56, "text": "trying"}, {"end": 758.92, "start": 758.8, "text": "to"}, {"end": 759.44, "start": 758.92, "text": "solve"}, {"end": 759.48, "start": 759.44, "text": "here"}, {"end": 759.56, "start": 759.48, "text": "in"}, {"end": 759.88, "start": 759.56, "text": "terms"}, {"end": 760.2, "start": 759.88, "text": "of"}, {"end": 760.68, "start": 760.2, "text": "this"}, {"end": 760.88, "start": 760.68, "text": "matching"}, {"end": 761.36, "start": 760.88, "text": "connectivity"}, {"end": 762.32, "start": 761.36, "text": "matrix."}, {"end": 762.76, "start": 762.32, "text": "It"}, {"end": 763.0, "start": 762.76, "text": "just"}, {"end": 763.32, "start": 763.0, "text": "means"}, {"end": 763.64, "start": 763.32, "text": "that"}, {"end": 764.0, "start": 763.64, "text": "we"}, {"end": 764.0, "start": 764.0, "text": "are"}, {"end": 764.32, "start": 764.0, "text": "given"}, {"end": 764.52, "start": 764.32, "text": "this"}, {"end": 764.84, "start": 764.52, "text": "bunch"}, {"end": 764.92, "start": 764.84, "text": "of"}, {"end": 765.2, "start": 764.92, "text": "perfect"}, {"end": 766.16, "start": 765.2, "text": "matchings"}, {"end": 766.44, "start": 766.16, "text": "so"}, {"end": 766.44, "start": 766.44, "text": "we"}, {"end": 766.84, "start": 766.44, "text": "can"}, {"end": 767.2, "start": 766.84, "text": "annotate"}, {"end": 767.32, "start": 767.2, "text": "all"}, {"end": 767.36, "start": 767.32, "text": "the"}, {"end": 768.0, "start": 767.36, "text": "rows"}, {"end": 768.08, "start": 768.0, "text": "with"}, {"end": 768.16, "start": 768.08, "text": "the"}, {"end": 768.52, "start": 768.16, "text": "weights"}, {"end": 768.64, "start": 768.52, "text": "that"}, {"end": 769.08, "start": 768.64, "text": "we"}, {"end": 769.12, "start": 769.08, "text": "have"}, {"end": 769.16, "start": 769.12, "text": "and"}, {"end": 769.28, "start": 769.16, "text": "we"}, {"end": 769.4, "start": 769.28, "text": "do"}, {"end": 769.6, "start": 769.4, "text": "the"}, {"end": 769.84, "start": 769.6, "text": "same"}, {"end": 769.92, "start": 769.84, "text": "with"}, {"end": 770.0, "start": 769.92, "text": "the"}, {"end": 770.76, "start": 770.0, "text": "columns."}, {"end": 771.16, "start": 770.76, "text": "And"}, {"end": 772.0, "start": 771.16, "text": "now"}, {"end": 772.24, "start": 772.0, "text": "this"}, {"end": 772.76, "start": 772.24, "text": "task"}, {"end": 773.24, "start": 772.76, "text": "that"}, {"end": 773.6, "start": 773.24, "text": "we"}, {"end": 773.64, "start": 773.6, "text": "had"}, {"end": 773.68, "start": 773.64, "text": "of"}, {"end": 774.32, "start": 773.68, "text": "finding"}, {"end": 774.6, "start": 774.32, "text": "two"}, {"end": 775.08, "start": 774.6, "text": "perfect"}, {"end": 775.6, "start": 775.08, "text": "matchings"}, {"end": 775.6, "start": 775.6, "text": "that"}, {"end": 775.8, "start": 775.6, "text": "form"}, {"end": 776.0, "start": 775.8, "text": "a"}, {"end": 776.36, "start": 776.0, "text": "Hamiltonian"}, {"end": 776.84, "start": 776.36, "text": "cycle"}, {"end": 776.84, "start": 776.84, "text": "with"}, {"end": 777.16, "start": 776.84, "text": "minimum"}, {"end": 777.68, "start": 777.16, "text": "weight"}, {"end": 777.92, "start": 777.68, "text": "in"}, {"end": 778.08, "start": 777.92, "text": "terms"}, {"end": 778.24, "start": 778.08, "text": "of"}, {"end": 778.48, "start": 778.24, "text": "the"}, {"end": 779.96, "start": 778.48, "text": "matrix,"}], "text": " This kind of structure. Okay, so why is this matrix so important? So let's again think of the problem that we are trying to solve here in terms of this matching connectivity matrix. It just means that we are given this bunch of perfect matchings so we can annotate all the rows with the weights that we have and we do the same with the columns. And now this task that we had of finding two perfect matchings that form a Hamiltonian cycle with minimum weight in terms of the matrix,"}, {"chunks": [{"end": 780.28, "start": 780.0, "text": "What"}, {"end": 780.28, "start": 780.28, "text": "we"}, {"end": 780.28, "start": 780.28, "text": "have"}, {"end": 780.32, "start": 780.28, "text": "to"}, {"end": 780.52, "start": 780.32, "text": "do"}, {"end": 780.72, "start": 780.52, "text": "is"}, {"end": 781.04, "start": 780.72, "text": "we"}, {"end": 781.24, "start": 781.04, "text": "have"}, {"end": 781.24, "start": 781.24, "text": "to"}, {"end": 781.56, "start": 781.24, "text": "go"}, {"end": 782.08, "start": 781.56, "text": "over"}, {"end": 782.24, "start": 782.08, "text": "all"}, {"end": 782.48, "start": 782.24, "text": "these"}, {"end": 782.8, "start": 782.48, "text": "ones"}, {"end": 782.88, "start": 782.8, "text": "that"}, {"end": 783.08, "start": 782.88, "text": "we"}, {"end": 783.84, "start": 783.08, "text": "see"}, {"end": 783.92, "start": 783.84, "text": "on"}, {"end": 784.16, "start": 783.92, "text": "the"}, {"end": 784.76, "start": 784.16, "text": "intersection"}, {"end": 784.96, "start": 784.76, "text": "of"}, {"end": 785.72, "start": 784.96, "text": "the"}, {"end": 786.12, "start": 785.72, "text": "orange"}, {"end": 786.72, "start": 786.12, "text": "rows"}, {"end": 787.2, "start": 786.72, "text": "and"}, {"end": 787.4, "start": 787.2, "text": "the"}, {"end": 787.6, "start": 787.4, "text": "orange"}, {"end": 789.08, "start": 787.6, "text": "columns."}, {"end": 789.28, "start": 789.08, "text": "And"}, {"end": 789.56, "start": 789.28, "text": "for"}, {"end": 789.8, "start": 789.56, "text": "each"}, {"end": 789.84, "start": 789.8, "text": "of"}, {"end": 790.08, "start": 789.84, "text": "these"}, {"end": 790.92, "start": 790.08, "text": "ones,"}, {"end": 791.08, "start": 790.92, "text": "we're"}, {"end": 791.48, "start": 791.08, "text": "interested"}, {"end": 791.64, "start": 791.48, "text": "in"}, {"end": 792.92, "start": 791.64, "text": "the"}, {"end": 793.6, "start": 792.92, "text": "one"}, {"end": 793.84, "start": 793.6, "text": "where"}, {"end": 795.16, "start": 793.84, "text": "the"}, {"end": 796.16, "start": 795.16, "text": "associated"}, {"end": 796.44, "start": 796.16, "text": "number"}, {"end": 796.92, "start": 796.44, "text": "with"}, {"end": 797.12, "start": 796.92, "text": "the"}, {"end": 797.64, "start": 797.12, "text": "row"}, {"end": 798.04, "start": 797.64, "text": "plus"}, {"end": 798.44, "start": 798.04, "text": "the"}, {"end": 799.04, "start": 798.44, "text": "associated"}, {"end": 799.32, "start": 799.04, "text": "number"}, {"end": 799.4, "start": 799.32, "text": "on"}, {"end": 799.52, "start": 799.4, "text": "the"}, {"end": 800.24, "start": 799.52, "text": "column"}, {"end": 800.4, "start": 800.24, "text": "is"}, {"end": 802.0, "start": 800.4, "text": "minimum."}, {"end": 802.12, "start": 802.0, "text": "So"}, {"end": 802.28, "start": 802.12, "text": "I"}, {"end": 802.72, "start": 802.28, "text": "believe"}, {"end": 803.36, "start": 802.72, "text": "this"}, {"end": 803.6, "start": 803.36, "text": "guy"}, {"end": 804.16, "start": 803.6, "text": "here"}, {"end": 804.44, "start": 804.16, "text": "is"}, {"end": 804.8, "start": 804.44, "text": "achieving"}, {"end": 807.08, "start": 804.8, "text": "the"}, {"end": 807.44, "start": 807.08, "text": "minimum"}, {"end": 807.72, "start": 807.44, "text": "in"}, {"end": 808.2, "start": 807.72, "text": "this"}, {"end": 809.52, "start": 808.2, "text": "particular"}, {"end": 809.96, "start": 809.52, "text": "case."}], "text": " What we have to do is we have to go over all these ones that we see on the intersection of the orange rows and the orange columns. And for each of these ones, we're interested in the one where the associated number with the row plus the associated number on the column is minimum. So I believe this guy here is achieving the minimum in this particular case."}, {"chunks": [{"end": 810.28, "start": 810.0, "text": "do"}, {"end": 810.68, "start": 810.28, "text": "directly"}, {"end": 811.24, "start": 810.68, "text": "translates"}, {"end": 811.44, "start": 811.24, "text": "in"}, {"end": 811.72, "start": 811.44, "text": "this"}, {"end": 812.04, "start": 811.72, "text": "matching"}, {"end": 812.36, "start": 812.04, "text": "connectivity"}, {"end": 815.24, "start": 812.36, "text": "matrix."}, {"end": 815.44, "start": 815.24, "text": "But"}, {"end": 815.88, "start": 815.44, "text": "actually"}, {"end": 816.68, "start": 815.88, "text": "what"}, {"end": 817.24, "start": 816.68, "text": "is"}, {"end": 817.52, "start": 817.24, "text": "here"}, {"end": 817.72, "start": 817.52, "text": "the"}, {"end": 818.08, "start": 817.72, "text": "special"}, {"end": 818.72, "start": 818.08, "text": "property"}, {"end": 818.8, "start": 818.72, "text": "in"}, {"end": 819.24, "start": 818.8, "text": "this"}, {"end": 820.72, "start": 819.24, "text": "example,"}, {"end": 820.88, "start": 820.72, "text": "if"}, {"end": 820.96, "start": 820.88, "text": "you"}, {"end": 821.16, "start": 820.96, "text": "look"}, {"end": 821.48, "start": 821.16, "text": "for"}, {"end": 821.64, "start": 821.48, "text": "a"}, {"end": 821.96, "start": 821.64, "text": "while,"}, {"end": 822.24, "start": 821.96, "text": "you"}, {"end": 822.84, "start": 822.24, "text": "will"}, {"end": 823.72, "start": 822.84, "text": "actually"}, {"end": 824.08, "start": 823.72, "text": "see"}, {"end": 824.4, "start": 824.08, "text": "that"}, {"end": 824.72, "start": 824.4, "text": "in"}, {"end": 824.96, "start": 824.72, "text": "every"}, {"end": 826.08, "start": 824.96, "text": "column,"}, {"end": 826.32, "start": 826.08, "text": "the"}, {"end": 826.68, "start": 826.32, "text": "number"}, {"end": 827.0, "start": 826.68, "text": "of"}, {"end": 827.68, "start": 827.0, "text": "ones"}, {"end": 827.8, "start": 827.68, "text": "in"}, {"end": 828.0, "start": 827.8, "text": "an"}, {"end": 828.32, "start": 828.0, "text": "orange"}, {"end": 828.72, "start": 828.32, "text": "row"}, {"end": 829.0, "start": 828.72, "text": "is"}, {"end": 829.48, "start": 829.0, "text": "always"}, {"end": 829.92, "start": 829.48, "text": "even"}, {"end": 830.0, "start": 829.92, "text": "in"}, {"end": 830.6, "start": 830.0, "text": "this"}, {"end": 830.88, "start": 830.6, "text": "setting."}, {"end": 830.88, "start": 830.88, "text": "For"}, {"end": 831.16, "start": 830.88, "text": "example,"}, {"end": 831.28, "start": 831.16, "text": "in"}, {"end": 831.6, "start": 831.28, "text": "the"}, {"end": 831.92, "start": 831.6, "text": "last"}, {"end": 832.16, "start": 831.92, "text": "column,"}, {"end": 832.32, "start": 832.16, "text": "we"}, {"end": 832.52, "start": 832.32, "text": "see"}, {"end": 833.2, "start": 832.52, "text": "one,"}, {"end": 833.6, "start": 833.2, "text": "two,"}, {"end": 834.2, "start": 833.6, "text": "three,"}, {"end": 834.92, "start": 834.2, "text": "four"}, {"end": 835.28, "start": 834.92, "text": "number"}, {"end": 835.48, "start": 835.28, "text": "of"}, {"end": 835.84, "start": 835.48, "text": "ones,"}, {"end": 835.92, "start": 835.84, "text": "and"}, {"end": 836.12, "start": 835.92, "text": "four"}, {"end": 836.32, "start": 836.12, "text": "is"}, {"end": 836.52, "start": 836.32, "text": "even,"}, {"end": 836.56, "start": 836.52, "text": "of"}, {"end": 837.04, "start": 836.56, "text": "course."}, {"end": 837.84, "start": 837.04, "text": "So"}, {"end": 837.92, "start": 837.84, "text": "what"}, {"end": 838.4, "start": 837.92, "text": "we"}, {"end": 839.0, "start": 838.4, "text": "could"}, {"end": 839.28, "start": 839.0, "text": "also"}, {"end": 839.64, "start": 839.28, "text": "say,"}, {"end": 839.8, "start": 839.64, "text": "since"}, {"end": 839.96, "start": 839.8, "text": "this"}], "text": " do directly translates in this matching connectivity matrix. But actually what is here the special property in this example, if you look for a while, you will actually see that in every column, the number of ones in an orange row is always even in this setting. For example, in the last column, we see one, two, three, four number of ones, and four is even, of course. So what we could also say, since this"}, {"chunks": [{"end": 840.24, "start": 840.0, "text": "It"}, {"end": 840.24, "start": 840.24, "text": "holds"}, {"end": 840.28, "start": 840.24, "text": "in"}, {"end": 840.52, "start": 840.28, "text": "every"}, {"end": 840.96, "start": 840.52, "text": "column,"}, {"end": 841.28, "start": 840.96, "text": "we"}, {"end": 841.52, "start": 841.28, "text": "could"}, {"end": 841.84, "start": 841.52, "text": "say"}, {"end": 842.28, "start": 841.84, "text": "that"}, {"end": 842.68, "start": 842.28, "text": "the"}, {"end": 843.04, "start": 842.68, "text": "five"}, {"end": 843.6, "start": 843.04, "text": "rows"}, {"end": 843.96, "start": 843.6, "text": "that"}, {"end": 844.32, "start": 843.96, "text": "we"}, {"end": 844.4, "start": 844.32, "text": "have,"}, {"end": 844.72, "start": 844.4, "text": "they"}, {"end": 844.92, "start": 844.72, "text": "are"}, {"end": 845.8, "start": 844.92, "text": "linearly"}, {"end": 846.36, "start": 845.8, "text": "dependent"}, {"end": 846.64, "start": 846.36, "text": "over"}, {"end": 848.52, "start": 846.64, "text": "GF2."}, {"end": 848.84, "start": 848.52, "text": "And"}, {"end": 849.28, "start": 848.84, "text": "actually"}, {"end": 849.4, "start": 849.28, "text": "in"}, {"end": 849.72, "start": 849.4, "text": "general"}, {"end": 849.84, "start": 849.72, "text": "it"}, {"end": 850.32, "start": 849.84, "text": "holds"}, {"end": 850.48, "start": 850.32, "text": "when"}, {"end": 850.48, "start": 850.48, "text": "you"}, {"end": 850.76, "start": 850.48, "text": "have"}, {"end": 851.04, "start": 850.76, "text": "some"}, {"end": 851.48, "start": 851.04, "text": "rows"}, {"end": 851.8, "start": 851.48, "text": "that"}, {"end": 852.2, "start": 851.8, "text": "are"}, {"end": 853.32, "start": 852.2, "text": "linearly"}, {"end": 854.0, "start": 853.32, "text": "dependent,"}, {"end": 854.12, "start": 854.0, "text": "you"}, {"end": 854.52, "start": 854.12, "text": "can"}, {"end": 855.24, "start": 854.52, "text": "always"}, {"end": 857.12, "start": 855.24, "text": "remove"}, {"end": 858.0, "start": 857.12, "text": "the"}, {"end": 858.72, "start": 858.0, "text": "row"}, {"end": 859.36, "start": 858.72, "text": "that"}, {"end": 859.72, "start": 859.36, "text": "has"}, {"end": 860.08, "start": 859.72, "text": "the"}, {"end": 860.56, "start": 860.08, "text": "largest"}, {"end": 860.96, "start": 860.56, "text": "weight"}, {"end": 861.0, "start": 860.96, "text": "and"}, {"end": 861.08, "start": 861.0, "text": "you"}, {"end": 861.36, "start": 861.08, "text": "will"}, {"end": 862.08, "start": 861.36, "text": "get"}, {"end": 862.28, "start": 862.08, "text": "a"}, {"end": 862.96, "start": 862.28, "text": "representative"}, {"end": 863.76, "start": 862.96, "text": "set."}, {"end": 863.88, "start": 863.76, "text": "And"}, {"end": 864.24, "start": 863.88, "text": "this"}, {"end": 864.36, "start": 864.24, "text": "is"}, {"end": 865.2, "start": 864.36, "text": "because"}, {"end": 865.96, "start": 865.2, "text": "whenever"}, {"end": 866.24, "start": 865.96, "text": "there"}, {"end": 866.56, "start": 866.24, "text": "was"}, {"end": 866.88, "start": 866.56, "text": "some"}, {"end": 867.12, "start": 866.88, "text": "perf"}, {"end": 867.56, "start": 867.12, "text": "matching"}, {"end": 867.92, "start": 867.56, "text": "B,"}, {"end": 869.12, "start": 867.92, "text": "some"}, {"end": 869.96, "start": 869.12, "text": "used,"}], "text": " It holds in every column, we could say that the five rows that we have, they are linearly dependent over GF2. And actually in general it holds when you have some rows that are linearly dependent, you can always remove the row that has the largest weight and you will get a representative set. And this is because whenever there was some perf matching B, some used,"}, {"chunks": [{"end": 870.04, "start": 870.0, "text": "and"}, {"end": 870.6, "start": 870.04, "text": "that"}, {"end": 870.92, "start": 870.6, "text": "works"}, {"end": 871.0, "start": 870.92, "text": "with"}, {"end": 871.12, "start": 871.0, "text": "the"}, {"end": 871.32, "start": 871.12, "text": "row"}, {"end": 871.72, "start": 871.32, "text": "that"}, {"end": 871.8, "start": 871.72, "text": "was"}, {"end": 872.12, "start": 871.8, "text": "thrown"}, {"end": 872.6, "start": 872.12, "text": "away,"}, {"end": 872.92, "start": 872.6, "text": "you"}, {"end": 873.12, "start": 872.92, "text": "can"}, {"end": 873.56, "start": 873.12, "text": "always"}, {"end": 873.84, "start": 873.56, "text": "replace"}, {"end": 874.16, "start": 873.84, "text": "it"}, {"end": 874.52, "start": 874.16, "text": "with"}, {"end": 874.68, "start": 874.52, "text": "an"}, {"end": 874.76, "start": 874.68, "text": "other"}, {"end": 874.96, "start": 874.76, "text": "row"}, {"end": 875.32, "start": 874.96, "text": "that"}, {"end": 875.6, "start": 875.32, "text": "also"}, {"end": 875.8, "start": 875.6, "text": "has"}, {"end": 875.84, "start": 875.8, "text": "a"}, {"end": 876.08, "start": 875.84, "text": "one"}, {"end": 876.56, "start": 876.08, "text": "there,"}, {"end": 876.8, "start": 876.56, "text": "since"}, {"end": 876.92, "start": 876.8, "text": "there"}, {"end": 877.04, "start": 876.92, "text": "is"}, {"end": 877.12, "start": 877.04, "text": "an"}, {"end": 877.36, "start": 877.12, "text": "even"}, {"end": 877.48, "start": 877.36, "text": "number"}, {"end": 877.68, "start": 877.48, "text": "of"}, {"end": 878.08, "start": 877.68, "text": "ones."}, {"end": 879.04, "start": 878.08, "text": "And"}, {"end": 879.16, "start": 879.04, "text": "it"}, {"end": 879.36, "start": 879.16, "text": "will"}, {"end": 879.8, "start": 879.36, "text": "have"}, {"end": 880.36, "start": 879.8, "text": "weight"}, {"end": 881.4, "start": 880.36, "text": "that"}, {"end": 882.68, "start": 881.4, "text": "is"}, {"end": 883.52, "start": 882.68, "text": "not"}, {"end": 883.84, "start": 883.52, "text": "larger."}, {"end": 884.12, "start": 883.84, "text": "So"}, {"end": 884.92, "start": 884.12, "text": "based"}, {"end": 885.16, "start": 884.92, "text": "on"}, {"end": 885.76, "start": 885.16, "text": "this,"}, {"end": 885.88, "start": 885.76, "text": "we"}, {"end": 886.16, "start": 885.88, "text": "are"}, {"end": 886.28, "start": 886.16, "text": "now"}, {"end": 886.36, "start": 886.28, "text": "interested"}, {"end": 886.52, "start": 886.36, "text": "in"}, {"end": 887.28, "start": 886.52, "text": "the"}, {"end": 887.6, "start": 887.28, "text": "linear"}, {"end": 888.08, "start": 887.6, "text": "algebraic"}, {"end": 888.64, "start": 888.08, "text": "properties"}, {"end": 888.76, "start": 888.64, "text": "of"}, {"end": 889.16, "start": 888.76, "text": "the"}, {"end": 889.76, "start": 889.16, "text": "Metzins"}, {"end": 890.08, "start": 889.76, "text": "conductivity"}, {"end": 894.28, "start": 890.08, "text": "matrix."}, {"end": 894.44, "start": 894.28, "text": "And"}, {"end": 894.6, "start": 894.44, "text": "it"}, {"end": 895.0, "start": 894.6, "text": "shows"}, {"end": 895.2, "start": 895.0, "text": "to"}, {"end": 895.2, "start": 895.2, "text": "have"}, {"end": 895.2, "start": 895.2, "text": "some"}, {"end": 895.44, "start": 895.2, "text": "very"}, {"end": 895.76, "start": 895.44, "text": "nice"}, {"end": 896.36, "start": 895.76, "text": "ones."}, {"end": 897.08, "start": 896.36, "text": "So"}, {"end": 897.88, "start": 897.08, "text": "the"}, {"end": 898.36, "start": 897.88, "text": "starry"}, {"end": 898.76, "start": 898.36, "text": "means"}, {"end": 899.08, "start": 898.76, "text": "that"}, {"end": 899.32, "start": 899.08, "text": "I'm"}, {"end": 899.96, "start": 899.32, "text": "oversimplifying"}], "text": " and that works with the row that was thrown away, you can always replace it with an other row that also has a one there, since there is an even number of ones. And it will have weight that is not larger. So based on this, we are now interested in the linear algebraic properties of the Metzins conductivity matrix. And it shows to have some very nice ones. So the starry means that I'm oversimplifying"}, {"chunks": [{"end": 900.4, "start": 900.0, "text": "So"}, {"end": 901.56, "start": 900.4, "text": "what"}, {"end": 902.28, "start": 901.56, "text": "we"}, {"end": 902.72, "start": 902.28, "text": "showed"}, {"end": 903.0, "start": 902.72, "text": "in"}, {"end": 904.36, "start": 903.0, "text": "2013"}, {"end": 904.72, "start": 904.36, "text": "is"}, {"end": 905.0, "start": 904.72, "text": "that"}, {"end": 905.28, "start": 905.0, "text": "the"}, {"end": 905.48, "start": 905.28, "text": "matching"}, {"end": 905.6, "start": 905.48, "text": "conductivity"}, {"end": 906.0, "start": 905.6, "text": "matrix"}, {"end": 906.24, "start": 906.0, "text": "can"}, {"end": 906.32, "start": 906.24, "text": "be"}, {"end": 907.52, "start": 906.32, "text": "factorized"}, {"end": 907.84, "start": 907.52, "text": "in"}, {"end": 908.08, "start": 907.84, "text": "this"}, {"end": 908.84, "start": 908.08, "text": "way."}, {"end": 909.04, "start": 908.84, "text": "So"}, {"end": 909.6, "start": 909.04, "text": "these"}, {"end": 909.6, "start": 909.6, "text": "are"}, {"end": 909.88, "start": 909.6, "text": "just"}, {"end": 909.88, "start": 909.88, "text": "the"}, {"end": 910.52, "start": 909.88, "text": "sub-matrices"}, {"end": 910.56, "start": 910.52, "text": "of"}, {"end": 910.6, "start": 910.56, "text": "the"}, {"end": 910.96, "start": 910.6, "text": "matching"}, {"end": 911.6, "start": 910.96, "text": "conductivity"}, {"end": 912.48, "start": 911.6, "text": "matrix"}, {"end": 913.08, "start": 912.48, "text": "restricted"}, {"end": 913.24, "start": 913.08, "text": "to"}, {"end": 913.48, "start": 913.24, "text": "a"}, {"end": 914.72, "start": 913.48, "text": "special"}, {"end": 915.0, "start": 914.72, "text": "set"}, {"end": 915.0, "start": 915.0, "text": "of"}, {"end": 915.28, "start": 915.0, "text": "perfect"}, {"end": 916.32, "start": 915.28, "text": "matchings."}, {"end": 916.72, "start": 916.32, "text": "Namely,"}, {"end": 917.04, "start": 916.72, "text": "this"}, {"end": 917.32, "start": 917.04, "text": "set"}, {"end": 917.96, "start": 917.32, "text": "Pn"}, {"end": 918.24, "start": 917.96, "text": "is"}, {"end": 918.4, "start": 918.24, "text": "the"}, {"end": 918.56, "start": 918.4, "text": "set"}, {"end": 918.56, "start": 918.56, "text": "of"}, {"end": 918.84, "start": 918.56, "text": "perfect"}, {"end": 919.28, "start": 918.84, "text": "matchings"}, {"end": 919.4, "start": 919.28, "text": "of"}, {"end": 919.68, "start": 919.4, "text": "this"}, {"end": 920.2, "start": 919.68, "text": "graph"}, {"end": 920.36, "start": 920.2, "text": "here"}, {"end": 920.64, "start": 920.36, "text": "that"}, {"end": 920.72, "start": 920.64, "text": "you"}, {"end": 922.24, "start": 920.72, "text": "can"}, {"end": 922.72, "start": 922.24, "text": "see."}, {"end": 923.56, "start": 922.72, "text": "And"}, {"end": 923.92, "start": 923.56, "text": "it's"}, {"end": 924.16, "start": 923.92, "text": "not"}, {"end": 924.8, "start": 924.16, "text": "hard"}, {"end": 924.88, "start": 924.8, "text": "to"}, {"end": 925.08, "start": 924.88, "text": "see"}, {"end": 925.44, "start": 925.08, "text": "that"}, {"end": 925.88, "start": 925.44, "text": "the"}, {"end": 926.44, "start": 925.88, "text": "cardinality"}, {"end": 926.6, "start": 926.44, "text": "of"}, {"end": 927.36, "start": 926.6, "text": "this"}, {"end": 927.72, "start": 927.36, "text": "family"}, {"end": 927.96, "start": 927.72, "text": "is"}, {"end": 928.16, "start": 927.96, "text": "only"}, {"end": 928.2, "start": 928.16, "text": "2"}, {"end": 928.36, "start": 928.2, "text": "to"}, {"end": 928.56, "start": 928.36, "text": "the"}, {"end": 928.8, "start": 928.56, "text": "n"}, {"end": 929.0, "start": 928.8, "text": "over"}, {"end": 929.24, "start": 929.0, "text": "2"}, {"end": 929.52, "start": 929.24, "text": "minus"}, {"end": 929.96, "start": 929.52, "text": "1."}], "text": " So what we showed in 2013 is that the matching conductivity matrix can be factorized in this way. So these are just the sub-matrices of the matching conductivity matrix restricted to a special set of perfect matchings. Namely, this set Pn is the set of perfect matchings of this graph here that you can see. And it's not hard to see that the cardinality of this family is only 2 to the n over 2 minus 1."}, {"chunks": [{"end": 930.24, "start": 930.0, "text": "And"}, {"end": 930.48, "start": 930.24, "text": "that"}, {"end": 931.2, "start": 930.48, "text": "tells"}, {"end": 931.48, "start": 931.2, "text": "us"}, {"end": 931.48, "start": 931.48, "text": "that"}, {"end": 931.68, "start": 931.48, "text": "the"}, {"end": 932.44, "start": 931.68, "text": "factorization"}, {"end": 933.4, "start": 932.44, "text": "that"}, {"end": 933.68, "start": 933.4, "text": "we"}, {"end": 934.0, "start": 933.68, "text": "have,"}, {"end": 934.4, "start": 934.0, "text": "it"}, {"end": 934.6, "start": 934.4, "text": "is"}, {"end": 935.36, "start": 934.6, "text": "factorizing"}, {"end": 936.12, "start": 935.36, "text": "matrices."}, {"end": 936.36, "start": 936.12, "text": "They"}, {"end": 936.44, "start": 936.36, "text": "are"}, {"end": 936.72, "start": 936.44, "text": "very"}, {"end": 936.96, "start": 936.72, "text": "narrow"}, {"end": 937.76, "start": 936.96, "text": "matrices."}, {"end": 938.32, "start": 937.76, "text": "The"}, {"end": 939.52, "start": 938.32, "text": "inner"}, {"end": 940.0, "start": 939.52, "text": "dimension"}, {"end": 940.24, "start": 940.0, "text": "is"}, {"end": 940.68, "start": 940.24, "text": "only"}, {"end": 941.0, "start": 940.68, "text": "2"}, {"end": 941.16, "start": 941.0, "text": "to"}, {"end": 941.36, "start": 941.16, "text": "the"}, {"end": 941.52, "start": 941.36, "text": "n"}, {"end": 941.64, "start": 941.52, "text": "over"}, {"end": 941.84, "start": 941.64, "text": "2"}, {"end": 942.08, "start": 941.84, "text": "minus"}, {"end": 942.4, "start": 942.08, "text": "1."}, {"end": 942.84, "start": 942.4, "text": "And"}, {"end": 943.28, "start": 942.84, "text": "therefore,"}, {"end": 943.52, "start": 943.28, "text": "the"}, {"end": 943.96, "start": 943.52, "text": "rank"}, {"end": 944.28, "start": 943.96, "text": "of"}, {"end": 944.48, "start": 944.28, "text": "the"}, {"end": 944.76, "start": 944.48, "text": "MdCM"}, {"end": 945.0, "start": 944.76, "text": "over"}, {"end": 945.08, "start": 945.0, "text": "G"}, {"end": 945.28, "start": 945.08, "text": "is"}, {"end": 945.48, "start": 945.28, "text": "only"}, {"end": 946.04, "start": 945.48, "text": "2"}, {"end": 946.64, "start": 946.04, "text": "to"}, {"end": 947.16, "start": 946.64, "text": "the"}, {"end": 947.28, "start": 947.16, "text": "n"}, {"end": 947.84, "start": 947.28, "text": "over"}, {"end": 948.48, "start": 947.84, "text": "2"}, {"end": 948.72, "start": 948.48, "text": "minus"}, {"end": 950.6, "start": 948.72, "text": "1."}, {"end": 950.8, "start": 950.6, "text": "And"}, {"end": 951.68, "start": 950.8, "text": "getting"}, {"end": 952.0, "start": 951.68, "text": "back"}, {"end": 952.08, "start": 952.0, "text": "to"}, {"end": 952.6, "start": 952.08, "text": "step"}, {"end": 952.88, "start": 952.6, "text": "1,"}, {"end": 953.2, "start": 952.88, "text": "so"}, {"end": 953.4, "start": 953.2, "text": "what"}, {"end": 953.6, "start": 953.4, "text": "we"}, {"end": 954.0, "start": 953.6, "text": "can"}, {"end": 954.12, "start": 954.0, "text": "now"}, {"end": 954.24, "start": 954.12, "text": "do"}, {"end": 954.24, "start": 954.24, "text": "to"}, {"end": 954.52, "start": 954.24, "text": "compute"}, {"end": 954.76, "start": 954.52, "text": "the"}, {"end": 955.68, "start": 954.76, "text": "representative"}, {"end": 955.92, "start": 955.68, "text": "set"}, {"end": 956.16, "start": 955.92, "text": "is"}, {"end": 956.2, "start": 956.16, "text": "that"}, {"end": 956.48, "start": 956.2, "text": "we"}, {"end": 956.88, "start": 956.48, "text": "can"}, {"end": 957.2, "start": 956.88, "text": "look"}, {"end": 957.6, "start": 957.2, "text": "at"}, {"end": 958.0, "start": 957.6, "text": "the"}, {"end": 958.84, "start": 958.0, "text": "submatrix"}, {"end": 959.0, "start": 958.84, "text": "of"}, {"end": 959.4, "start": 959.0, "text": "the"}, {"end": 959.96, "start": 959.4, "text": "MdCM"}], "text": " And that tells us that the factorization that we have, it is factorizing matrices. They are very narrow matrices. The inner dimension is only 2 to the n over 2 minus 1. And therefore, the rank of the MdCM over G is only 2 to the n over 2 minus 1. And getting back to step 1, so what we can now do to compute the representative set is that we can look at the submatrix of the MdCM"}, {"chunks": [{"end": 960.4, "start": 960.0, "text": "And"}, {"end": 960.72, "start": 960.4, "text": "what"}, {"end": 960.76, "start": 960.72, "text": "we"}, {"end": 960.96, "start": 960.76, "text": "do"}, {"end": 961.48, "start": 960.96, "text": "here"}, {"end": 961.6, "start": 961.48, "text": "on"}, {"end": 961.84, "start": 961.6, "text": "top"}, {"end": 962.0, "start": 961.84, "text": "is"}, {"end": 962.2, "start": 962.0, "text": "we"}, {"end": 962.32, "start": 962.2, "text": "have"}, {"end": 962.96, "start": 962.32, "text": "this"}, {"end": 963.84, "start": 962.96, "text": "family"}, {"end": 964.4, "start": 963.84, "text": "P4"}, {"end": 964.48, "start": 964.4, "text": "in"}, {"end": 964.72, "start": 964.48, "text": "this"}, {"end": 965.48, "start": 964.72, "text": "case."}, {"end": 965.56, "start": 965.48, "text": "So"}, {"end": 965.76, "start": 965.56, "text": "this"}, {"end": 966.0, "start": 965.76, "text": "gives"}, {"end": 966.16, "start": 966.0, "text": "us"}, {"end": 966.48, "start": 966.16, "text": "this"}, {"end": 966.84, "start": 966.48, "text": "factorizing"}, {"end": 967.4, "start": 966.84, "text": "matrix"}, {"end": 968.4, "start": 967.4, "text": "H6"}, {"end": 968.6, "start": 968.4, "text": "with"}, {"end": 968.8, "start": 968.6, "text": "all"}, {"end": 968.92, "start": 968.8, "text": "the"}, {"end": 969.52, "start": 968.92, "text": "rows"}, {"end": 969.64, "start": 969.52, "text": "and"}, {"end": 970.28, "start": 969.64, "text": "P6,"}, {"end": 970.44, "start": 970.28, "text": "which"}, {"end": 971.04, "start": 970.44, "text": "is"}, {"end": 971.32, "start": 971.04, "text": "this"}, {"end": 971.56, "start": 971.32, "text": "guy"}, {"end": 971.8, "start": 971.56, "text": "here."}, {"end": 971.8, "start": 971.8, "text": "And"}, {"end": 971.84, "start": 971.8, "text": "what"}, {"end": 972.16, "start": 971.84, "text": "we"}, {"end": 973.04, "start": 972.16, "text": "can"}, {"end": 973.4, "start": 973.04, "text": "do"}, {"end": 973.6, "start": 973.4, "text": "is"}, {"end": 973.64, "start": 973.6, "text": "we"}, {"end": 974.08, "start": 973.64, "text": "can"}, {"end": 974.48, "start": 974.08, "text": "compute"}, {"end": 974.84, "start": 974.48, "text": "the"}, {"end": 975.6, "start": 974.84, "text": "matrix"}, {"end": 975.76, "start": 975.6, "text": "with"}, {"end": 975.88, "start": 975.76, "text": "all"}, {"end": 976.08, "start": 975.88, "text": "the"}, {"end": 976.72, "start": 976.08, "text": "rows"}, {"end": 977.4, "start": 976.72, "text": "that"}, {"end": 978.24, "start": 977.4, "text": "are"}, {"end": 978.52, "start": 978.24, "text": "in"}, {"end": 978.88, "start": 978.52, "text": "the"}, {"end": 979.12, "start": 978.88, "text": "set"}, {"end": 979.48, "start": 979.12, "text": "family"}, {"end": 979.6, "start": 979.48, "text": "of"}, {"end": 980.08, "start": 979.6, "text": "which"}, {"end": 980.32, "start": 980.08, "text": "we"}, {"end": 980.44, "start": 980.32, "text": "want"}, {"end": 980.68, "start": 980.44, "text": "to"}, {"end": 980.84, "start": 980.68, "text": "compute"}, {"end": 981.0, "start": 980.84, "text": "the"}, {"end": 981.44, "start": 981.0, "text": "representative"}, {"end": 981.56, "start": 981.44, "text": "set."}, {"end": 982.44, "start": 981.56, "text": "And"}, {"end": 982.88, "start": 982.44, "text": "now"}, {"end": 982.88, "start": 982.88, "text": "what"}, {"end": 982.88, "start": 982.88, "text": "we"}, {"end": 983.36, "start": 982.88, "text": "can"}, {"end": 983.56, "start": 983.36, "text": "do"}, {"end": 983.72, "start": 983.56, "text": "is"}, {"end": 983.76, "start": 983.72, "text": "we"}, {"end": 983.96, "start": 983.76, "text": "can"}, {"end": 984.2, "start": 983.96, "text": "compute"}, {"end": 984.36, "start": 984.2, "text": "the"}, {"end": 984.56, "start": 984.36, "text": "row"}, {"end": 985.24, "start": 984.56, "text": "basis"}, {"end": 985.4, "start": 985.24, "text": "of"}, {"end": 985.44, "start": 985.4, "text": "the"}, {"end": 986.2, "start": 985.44, "text": "set"}, {"end": 987.08, "start": 986.2, "text": "of"}, {"end": 987.92, "start": 987.08, "text": "rows"}, {"end": 988.04, "start": 987.92, "text": "in"}, {"end": 988.76, "start": 988.04, "text": "calligraphic"}, {"end": 989.96, "start": 988.76, "text": "A."}], "text": " And what we do here on top is we have this family P4 in this case. So this gives us this factorizing matrix H6 with all the rows and P6, which is this guy here. And what we can do is we can compute the matrix with all the rows that are in the set family of which we want to compute the representative set. And now what we can do is we can compute the row basis of the set of rows in calligraphic A."}, {"chunks": [{"end": 991.84, "start": 990.0, "text": "And"}, {"end": 993.04, "start": 991.84, "text": "the"}, {"end": 993.68, "start": 993.04, "text": "minimum"}, {"end": 994.0, "start": 993.68, "text": "weight"}, {"end": 994.2, "start": 994.0, "text": "row"}, {"end": 994.72, "start": 994.2, "text": "basis"}, {"end": 995.0, "start": 994.72, "text": "turns"}, {"end": 995.2, "start": 995.0, "text": "out"}, {"end": 995.24, "start": 995.2, "text": "to"}, {"end": 995.44, "start": 995.24, "text": "be"}, {"end": 995.56, "start": 995.44, "text": "a"}, {"end": 996.12, "start": 995.56, "text": "representative"}, {"end": 996.6, "start": 996.12, "text": "set"}, {"end": 996.96, "start": 996.6, "text": "of"}, {"end": 997.2, "start": 996.96, "text": "the"}, {"end": 998.32, "start": 997.2, "text": "input."}, {"end": 998.88, "start": 998.32, "text": "So"}, {"end": 999.04, "start": 998.88, "text": "here"}, {"end": 999.32, "start": 999.04, "text": "we"}, {"end": 999.56, "start": 999.32, "text": "can"}, {"end": 999.56, "start": 999.56, "text": "see"}, {"end": 999.56, "start": 999.56, "text": "again"}, {"end": 999.84, "start": 999.56, "text": "the"}, {"end": 1000.16, "start": 999.84, "text": "minimum"}, {"end": 1000.36, "start": 1000.16, "text": "weight"}, {"end": 1000.56, "start": 1000.36, "text": "row"}, {"end": 1001.32, "start": 1000.56, "text": "basis"}, {"end": 1001.64, "start": 1001.32, "text": "is"}, {"end": 1001.88, "start": 1001.64, "text": "where"}, {"end": 1002.08, "start": 1001.88, "text": "we"}, {"end": 1002.28, "start": 1002.08, "text": "remove"}, {"end": 1002.92, "start": 1002.28, "text": "this"}, {"end": 1003.2, "start": 1002.92, "text": "row"}, {"end": 1004.04, "start": 1003.2, "text": "and"}, {"end": 1004.28, "start": 1004.04, "text": "then"}, {"end": 1004.56, "start": 1004.28, "text": "indeed"}, {"end": 1004.72, "start": 1004.56, "text": "we"}, {"end": 1005.16, "start": 1004.72, "text": "get"}, {"end": 1005.48, "start": 1005.16, "text": "this"}, {"end": 1005.96, "start": 1005.48, "text": "set"}, {"end": 1006.36, "start": 1005.96, "text": "of"}, {"end": 1007.2, "start": 1006.36, "text": "matchings."}, {"end": 1007.8, "start": 1007.2, "text": "And"}, {"end": 1008.08, "start": 1007.8, "text": "the"}, {"end": 1008.68, "start": 1008.08, "text": "output"}, {"end": 1009.36, "start": 1008.68, "text": "that"}, {"end": 1009.76, "start": 1009.36, "text": "we"}, {"end": 1010.24, "start": 1009.76, "text": "get"}, {"end": 1010.72, "start": 1010.24, "text": "in"}, {"end": 1010.84, "start": 1010.72, "text": "this"}, {"end": 1011.44, "start": 1010.84, "text": "procedure,"}, {"end": 1011.64, "start": 1011.44, "text": "it"}, {"end": 1011.68, "start": 1011.64, "text": "will"}, {"end": 1011.8, "start": 1011.68, "text": "always"}, {"end": 1011.84, "start": 1011.8, "text": "be"}, {"end": 1012.36, "start": 1011.84, "text": "at"}, {"end": 1012.64, "start": 1012.36, "text": "most"}, {"end": 1012.84, "start": 1012.64, "text": "two"}, {"end": 1012.84, "start": 1012.84, "text": "to"}, {"end": 1013.04, "start": 1012.84, "text": "the"}, {"end": 1013.2, "start": 1013.04, "text": "n"}, {"end": 1013.44, "start": 1013.2, "text": "over"}, {"end": 1013.92, "start": 1013.44, "text": "two"}, {"end": 1014.28, "start": 1013.92, "text": "because"}, {"end": 1014.32, "start": 1014.28, "text": "that"}, {"end": 1014.56, "start": 1014.32, "text": "is"}, {"end": 1014.88, "start": 1014.56, "text": "the"}, {"end": 1015.0, "start": 1014.88, "text": "amount"}, {"end": 1015.0, "start": 1015.0, "text": "on"}, {"end": 1015.12, "start": 1015.0, "text": "the"}, {"end": 1015.24, "start": 1015.12, "text": "rank."}, {"end": 1015.4, "start": 1015.24, "text": "Yeah,"}, {"end": 1015.56, "start": 1015.4, "text": "so"}, {"end": 1017.8, "start": 1015.56, "text": "this"}, {"end": 1018.36, "start": 1017.8, "text": "was"}, {"end": 1019.32, "start": 1018.36, "text": "all"}, {"end": 1019.44, "start": 1019.32, "text": "in"}, {"end": 1019.56, "start": 1019.44, "text": "the"}, {"end": 1019.96, "start": 1019.56, "text": "previous"}], "text": " And the minimum weight row basis turns out to be a representative set of the input. So here we can see again the minimum weight row basis is where we remove this row and then indeed we get this set of matchings. And the output that we get in this procedure, it will always be at most two to the n over two because that is the amount on the rank. Yeah, so this was all in the previous"}, {"chunks": [{"end": 1020.36, "start": 1020.0, "text": "By"}, {"end": 1020.68, "start": 1020.36, "text": "Bodlander,"}, {"end": 1021.16, "start": 1020.68, "text": "Sigan,"}, {"end": 1021.96, "start": 1021.16, "text": "Kratz"}, {"end": 1024.2, "start": 1021.96, "text": "and"}, {"end": 1024.88, "start": 1024.2, "text": "myself."}, {"end": 1025.12, "start": 1024.88, "text": "So"}, {"end": 1025.2, "start": 1025.12, "text": "in"}, {"end": 1025.48, "start": 1025.2, "text": "this,"}, {"end": 1025.88, "start": 1025.48, "text": "for"}, {"end": 1026.16, "start": 1025.88, "text": "our"}, {"end": 1026.6, "start": 1026.16, "text": "particular"}, {"end": 1027.44, "start": 1026.6, "text": "purposes,"}, {"end": 1027.64, "start": 1027.44, "text": "this"}, {"end": 1028.12, "start": 1027.64, "text": "is"}, {"end": 1028.6, "start": 1028.12, "text": "not"}, {"end": 1028.72, "start": 1028.6, "text": "at"}, {"end": 1029.04, "start": 1028.72, "text": "all"}, {"end": 1029.36, "start": 1029.04, "text": "fast"}, {"end": 1030.0, "start": 1029.36, "text": "enough."}, {"end": 1031.0, "start": 1030.0, "text": "And"}, {"end": 1031.96, "start": 1031.0, "text": "actually"}, {"end": 1032.2, "start": 1031.96, "text": "in"}, {"end": 1032.36, "start": 1032.2, "text": "the"}, {"end": 1032.68, "start": 1032.36, "text": "paper"}, {"end": 1033.08, "start": 1032.68, "text": "itself"}, {"end": 1033.24, "start": 1033.08, "text": "to"}, {"end": 1033.32, "start": 1033.24, "text": "do"}, {"end": 1033.68, "start": 1033.32, "text": "this,"}, {"end": 1033.92, "start": 1033.68, "text": "to"}, {"end": 1034.32, "start": 1033.92, "text": "find"}, {"end": 1034.52, "start": 1034.32, "text": "the"}, {"end": 1035.28, "start": 1034.52, "text": "representative,"}, {"end": 1035.48, "start": 1035.28, "text": "said"}, {"end": 1036.0, "start": 1035.48, "text": "quickly"}, {"end": 1036.48, "start": 1036.0, "text": "there"}, {"end": 1036.72, "start": 1036.48, "text": "are"}, {"end": 1037.04, "start": 1036.72, "text": "a"}, {"end": 1037.36, "start": 1037.04, "text": "lot"}, {"end": 1037.96, "start": 1037.36, "text": "of"}, {"end": 1038.28, "start": 1037.96, "text": "other"}, {"end": 1038.6, "start": 1038.28, "text": "things"}, {"end": 1038.8, "start": 1038.6, "text": "going"}, {"end": 1038.84, "start": 1038.8, "text": "on"}, {"end": 1039.0, "start": 1038.84, "text": "where"}, {"end": 1039.08, "start": 1039.0, "text": "we"}, {"end": 1039.36, "start": 1039.08, "text": "also"}, {"end": 1040.12, "start": 1039.36, "text": "exploit"}, {"end": 1040.48, "start": 1040.12, "text": "the"}, {"end": 1040.6, "start": 1040.48, "text": "biopetiteness."}, {"end": 1040.84, "start": 1040.6, "text": "But"}, {"end": 1041.08, "start": 1040.84, "text": "I'm"}, {"end": 1041.4, "start": 1041.08, "text": "not"}, {"end": 1041.64, "start": 1041.4, "text": "going"}, {"end": 1041.92, "start": 1041.64, "text": "to"}, {"end": 1042.12, "start": 1041.92, "text": "go"}, {"end": 1044.04, "start": 1042.12, "text": "into"}, {"end": 1044.36, "start": 1044.04, "text": "detail"}, {"end": 1044.36, "start": 1044.36, "text": "on"}, {"end": 1044.44, "start": 1044.36, "text": "that."}, {"end": 1044.64, "start": 1044.44, "text": "Okay,"}, {"end": 1045.12, "start": 1044.64, "text": "now"}, {"end": 1045.32, "start": 1045.12, "text": "let's"}, {"end": 1045.48, "start": 1045.32, "text": "go"}, {"end": 1045.8, "start": 1045.48, "text": "to"}, {"end": 1046.12, "start": 1045.8, "text": "step"}, {"end": 1046.56, "start": 1046.12, "text": "two,"}, {"end": 1046.84, "start": 1046.56, "text": "which"}, {"end": 1046.96, "start": 1046.84, "text": "is"}, {"end": 1047.28, "start": 1046.96, "text": "actually"}, {"end": 1047.64, "start": 1047.28, "text": "the"}, {"end": 1048.76, "start": 1047.64, "text": "more"}, {"end": 1049.56, "start": 1048.76, "text": "innovative"}, {"end": 1049.96, "start": 1049.56, "text": "part."}], "text": " By Bodlander, Sigan, Kratz and myself. So in this, for our particular purposes, this is not at all fast enough. And actually in the paper itself to do this, to find the representative, said quickly there are a lot of other things going on where we also exploit the biopetiteness. But I'm not going to go into detail on that. Okay, now let's go to step two, which is actually the more innovative part."}, {"chunks": [{"end": 1050.16, "start": 1050.0, "text": "This"}, {"end": 1050.32, "start": 1050.16, "text": "is"}, {"end": 1050.52, "start": 1050.32, "text": "step"}, {"end": 1050.96, "start": 1050.52, "text": "2a"}, {"end": 1051.16, "start": 1050.96, "text": "where"}, {"end": 1051.28, "start": 1051.16, "text": "we"}, {"end": 1051.48, "start": 1051.28, "text": "just"}, {"end": 1051.64, "start": 1051.48, "text": "get"}, {"end": 1051.84, "start": 1051.64, "text": "rid"}, {"end": 1051.84, "start": 1051.84, "text": "of"}, {"end": 1051.88, "start": 1051.84, "text": "the"}, {"end": 1052.56, "start": 1051.88, "text": "weights,"}, {"end": 1052.72, "start": 1052.56, "text": "so"}, {"end": 1052.72, "start": 1052.72, "text": "we"}, {"end": 1052.76, "start": 1052.72, "text": "get"}, {"end": 1052.92, "start": 1052.76, "text": "to"}, {"end": 1053.0, "start": 1052.92, "text": "an"}, {"end": 1053.36, "start": 1053.0, "text": "unweighted"}, {"end": 1054.04, "start": 1053.36, "text": "variant."}, {"end": 1054.36, "start": 1054.04, "text": "I"}, {"end": 1054.6, "start": 1054.36, "text": "will"}, {"end": 1054.84, "start": 1054.6, "text": "not"}, {"end": 1054.96, "start": 1054.84, "text": "get"}, {"end": 1055.24, "start": 1054.96, "text": "into"}, {"end": 1055.64, "start": 1055.24, "text": "that."}, {"end": 1056.04, "start": 1055.64, "text": "And"}, {"end": 1056.48, "start": 1056.04, "text": "then"}, {"end": 1056.8, "start": 1056.48, "text": "in"}, {"end": 1057.32, "start": 1056.8, "text": "2b,"}, {"end": 1057.48, "start": 1057.32, "text": "we"}, {"end": 1057.68, "start": 1057.48, "text": "are"}, {"end": 1057.68, "start": 1057.68, "text": "going"}, {"end": 1057.72, "start": 1057.68, "text": "to"}, {"end": 1057.96, "start": 1057.72, "text": "solve"}, {"end": 1058.12, "start": 1057.96, "text": "the"}, {"end": 1058.6, "start": 1058.12, "text": "unweighted"}, {"end": 1060.32, "start": 1058.6, "text": "variant."}, {"end": 1061.68, "start": 1060.32, "text": "Specifically,"}, {"end": 1062.16, "start": 1061.68, "text": "one"}, {"end": 1064.56, "start": 1062.16, "text": "could"}, {"end": 1065.4, "start": 1064.56, "text": "obtain"}, {"end": 1065.48, "start": 1065.4, "text": "the"}, {"end": 1065.84, "start": 1065.48, "text": "following"}, {"end": 1066.92, "start": 1065.84, "text": "theorem."}, {"end": 1067.32, "start": 1066.92, "text": "Given"}, {"end": 1067.6, "start": 1067.32, "text": "a"}, {"end": 1068.2, "start": 1067.6, "text": "family"}, {"end": 1068.32, "start": 1068.2, "text": "of"}, {"end": 1068.88, "start": 1068.32, "text": "perfect"}, {"end": 1069.24, "start": 1068.88, "text": "matchings"}, {"end": 1070.12, "start": 1069.24, "text": "calligraphic"}, {"end": 1070.6, "start": 1070.12, "text": "a,"}, {"end": 1070.92, "start": 1070.6, "text": "we"}, {"end": 1071.28, "start": 1070.92, "text": "can"}, {"end": 1071.44, "start": 1071.28, "text": "detect"}, {"end": 1071.56, "start": 1071.44, "text": "if"}, {"end": 1071.72, "start": 1071.56, "text": "there"}, {"end": 1071.84, "start": 1071.72, "text": "are"}, {"end": 1072.36, "start": 1071.84, "text": "two"}, {"end": 1072.56, "start": 1072.36, "text": "perfect"}, {"end": 1072.8, "start": 1072.56, "text": "matchings"}, {"end": 1073.0, "start": 1072.8, "text": "that"}, {"end": 1073.2, "start": 1073.0, "text": "form"}, {"end": 1073.36, "start": 1073.2, "text": "a"}, {"end": 1073.68, "start": 1073.36, "text": "Hamiltonian"}, {"end": 1074.72, "start": 1073.68, "text": "cycle"}, {"end": 1075.12, "start": 1074.72, "text": "in"}, {"end": 1075.64, "start": 1075.12, "text": "time"}, {"end": 1075.72, "start": 1075.64, "text": "the"}, {"end": 1076.12, "start": 1075.72, "text": "size"}, {"end": 1076.12, "start": 1076.12, "text": "of"}, {"end": 1076.52, "start": 1076.12, "text": "calligraphic"}, {"end": 1076.96, "start": 1076.52, "text": "a"}, {"end": 1077.52, "start": 1076.96, "text": "times"}, {"end": 1077.68, "start": 1077.52, "text": "2"}, {"end": 1077.8, "start": 1077.68, "text": "to"}, {"end": 1078.04, "start": 1077.8, "text": "the"}, {"end": 1078.28, "start": 1078.04, "text": "n"}, {"end": 1078.48, "start": 1078.28, "text": "over"}, {"end": 1079.96, "start": 1078.48, "text": "2."}], "text": " This is step 2a where we just get rid of the weights, so we get to an unweighted variant. I will not get into that. And then in 2b, we are going to solve the unweighted variant. Specifically, one could obtain the following theorem. Given a family of perfect matchings calligraphic a, we can detect if there are two perfect matchings that form a Hamiltonian cycle in time the size of calligraphic a times 2 to the n over 2."}, {"chunks": [{"end": 1080.0, "start": 1080.0, "text": "This"}, {"end": 1080.32, "start": 1080.0, "text": "is"}, {"end": 1080.56, "start": 1080.32, "text": "not"}, {"end": 1081.0, "start": 1080.56, "text": "what"}, {"end": 1081.04, "start": 1081.0, "text": "I"}, {"end": 1081.04, "start": 1081.04, "text": "promised,"}, {"end": 1081.04, "start": 1081.04, "text": "I"}, {"end": 1081.4, "start": 1081.04, "text": "promised"}, {"end": 1081.52, "start": 1081.4, "text": "a"}, {"end": 1082.12, "start": 1081.52, "text": "faster"}, {"end": 1082.28, "start": 1082.12, "text": "running"}, {"end": 1082.92, "start": 1082.28, "text": "time."}, {"end": 1083.68, "start": 1082.92, "text": "But"}, {"end": 1084.0, "start": 1083.68, "text": "let's"}, {"end": 1084.4, "start": 1084.0, "text": "first"}, {"end": 1084.4, "start": 1084.4, "text": "do"}, {"end": 1084.76, "start": 1084.4, "text": "this"}, {"end": 1085.16, "start": 1084.76, "text": "easier"}, {"end": 1085.32, "start": 1085.16, "text": "running"}, {"end": 1085.64, "start": 1085.32, "text": "time"}, {"end": 1085.76, "start": 1085.64, "text": "and"}, {"end": 1086.04, "start": 1085.76, "text": "then"}, {"end": 1086.24, "start": 1086.04, "text": "let's"}, {"end": 1086.48, "start": 1086.24, "text": "later"}, {"end": 1086.8, "start": 1086.48, "text": "improve"}, {"end": 1088.96, "start": 1086.8, "text": "it."}, {"end": 1089.16, "start": 1088.96, "text": "So"}, {"end": 1089.52, "start": 1089.16, "text": "an"}, {"end": 1090.32, "start": 1089.52, "text": "important"}, {"end": 1090.4, "start": 1090.32, "text": "part"}, {"end": 1090.6, "start": 1090.4, "text": "here"}, {"end": 1091.0, "start": 1090.6, "text": "is"}, {"end": 1091.64, "start": 1091.0, "text": "an"}, {"end": 1092.0, "start": 1091.64, "text": "easy"}, {"end": 1092.56, "start": 1092.0, "text": "extension"}, {"end": 1092.84, "start": 1092.56, "text": "of"}, {"end": 1094.36, "start": 1092.84, "text": "Freifalt's"}, {"end": 1095.88, "start": 1094.36, "text": "verification"}, {"end": 1096.2, "start": 1095.88, "text": "of"}, {"end": 1096.64, "start": 1096.2, "text": "matrix"}, {"end": 1097.64, "start": 1096.64, "text": "multiplication."}, {"end": 1097.72, "start": 1097.64, "text": "So"}, {"end": 1097.8, "start": 1097.72, "text": "if"}, {"end": 1098.08, "start": 1097.8, "text": "you"}, {"end": 1098.68, "start": 1098.08, "text": "have"}, {"end": 1099.2, "start": 1098.68, "text": "a"}, {"end": 1099.64, "start": 1099.2, "text": "matrix"}, {"end": 1099.96, "start": 1099.64, "text": "M"}, {"end": 1100.0, "start": 1099.96, "text": "that"}, {"end": 1100.28, "start": 1100.0, "text": "is"}, {"end": 1100.52, "start": 1100.28, "text": "maybe"}, {"end": 1100.8, "start": 1100.52, "text": "claimed"}, {"end": 1101.0, "start": 1100.8, "text": "to"}, {"end": 1101.2, "start": 1101.0, "text": "be"}, {"end": 1101.28, "start": 1101.2, "text": "the"}, {"end": 1101.6, "start": 1101.28, "text": "product"}, {"end": 1101.8, "start": 1101.6, "text": "of"}, {"end": 1101.96, "start": 1101.8, "text": "L"}, {"end": 1102.28, "start": 1101.96, "text": "times"}, {"end": 1102.76, "start": 1102.28, "text": "R,"}, {"end": 1103.04, "start": 1102.76, "text": "then"}, {"end": 1103.2, "start": 1103.04, "text": "we"}, {"end": 1103.36, "start": 1103.2, "text": "can"}, {"end": 1103.6, "start": 1103.36, "text": "check"}, {"end": 1104.08, "start": 1103.6, "text": "this"}, {"end": 1104.76, "start": 1104.08, "text": "because"}, {"end": 1105.04, "start": 1104.76, "text": "if"}, {"end": 1105.12, "start": 1105.04, "text": "you"}, {"end": 1105.24, "start": 1105.12, "text": "pick"}, {"end": 1105.48, "start": 1105.24, "text": "two"}, {"end": 1105.84, "start": 1105.48, "text": "vectors"}, {"end": 1106.16, "start": 1105.84, "text": "X"}, {"end": 1106.24, "start": 1106.16, "text": "and"}, {"end": 1106.56, "start": 1106.24, "text": "Y"}, {"end": 1106.56, "start": 1106.56, "text": "of"}, {"end": 1106.8, "start": 1106.56, "text": "the"}, {"end": 1107.84, "start": 1106.8, "text": "appropriate"}, {"end": 1108.88, "start": 1107.84, "text": "dimension,"}, {"end": 1109.96, "start": 1108.88, "text": "then"}], "text": " This is not what I promised, I promised a faster running time. But let's first do this easier running time and then let's later improve it. So an important part here is an easy extension of Freifalt's verification of matrix multiplication. So if you have a matrix M that is maybe claimed to be the product of L times R, then we can check this because if you pick two vectors X and Y of the appropriate dimension, then"}, {"chunks": [{"end": 1110.48, "start": 1110.0, "text": "If"}, {"end": 1110.6, "start": 1110.48, "text": "m"}, {"end": 1110.8, "start": 1110.6, "text": "is"}, {"end": 1111.0, "start": 1110.8, "text": "not"}, {"end": 1111.04, "start": 1111.0, "text": "equal"}, {"end": 1111.24, "start": 1111.04, "text": "to"}, {"end": 1111.52, "start": 1111.24, "text": "L"}, {"end": 1111.84, "start": 1111.52, "text": "times"}, {"end": 1112.44, "start": 1111.84, "text": "r,"}, {"end": 1112.88, "start": 1112.44, "text": "then"}, {"end": 1113.56, "start": 1112.88, "text": "we"}, {"end": 1113.84, "start": 1113.56, "text": "get"}, {"end": 1113.84, "start": 1113.84, "text": "some"}, {"end": 1114.32, "start": 1113.84, "text": "probabilistic"}, {"end": 1114.88, "start": 1114.32, "text": "guarantee"}, {"end": 1115.24, "start": 1114.88, "text": "that"}, {"end": 1115.44, "start": 1115.24, "text": "a"}, {"end": 1115.6, "start": 1115.44, "text": "certain"}, {"end": 1115.96, "start": 1115.6, "text": "check"}, {"end": 1116.44, "start": 1115.96, "text": "is"}, {"end": 1116.96, "start": 1116.44, "text": "going"}, {"end": 1117.36, "start": 1116.96, "text": "to"}, {"end": 1117.52, "start": 1117.36, "text": "fail."}, {"end": 1117.56, "start": 1117.52, "text": "In"}, {"end": 1117.96, "start": 1117.56, "text": "particular,"}, {"end": 1118.16, "start": 1117.96, "text": "y"}, {"end": 1118.68, "start": 1118.16, "text": "transpose"}, {"end": 1119.12, "start": 1118.68, "text": "times"}, {"end": 1119.2, "start": 1119.12, "text": "m"}, {"end": 1119.52, "start": 1119.2, "text": "times"}, {"end": 1120.08, "start": 1119.52, "text": "x"}, {"end": 1120.44, "start": 1120.08, "text": "is"}, {"end": 1121.6, "start": 1120.44, "text": "equal"}, {"end": 1122.12, "start": 1121.6, "text": "to"}, {"end": 1122.6, "start": 1122.12, "text": "y"}, {"end": 1123.12, "start": 1122.6, "text": "transpose"}, {"end": 1123.28, "start": 1123.12, "text": "times"}, {"end": 1123.4, "start": 1123.28, "text": "L"}, {"end": 1123.72, "start": 1123.4, "text": "times"}, {"end": 1123.88, "start": 1123.72, "text": "r"}, {"end": 1124.16, "start": 1123.88, "text": "times"}, {"end": 1124.44, "start": 1124.16, "text": "x."}, {"end": 1124.48, "start": 1124.44, "text": "The"}, {"end": 1124.88, "start": 1124.48, "text": "probability"}, {"end": 1125.32, "start": 1124.88, "text": "of"}, {"end": 1125.64, "start": 1125.32, "text": "this"}, {"end": 1125.8, "start": 1125.64, "text": "is"}, {"end": 1125.84, "start": 1125.8, "text": "at"}, {"end": 1126.08, "start": 1125.84, "text": "most"}, {"end": 1126.24, "start": 1126.08, "text": "1"}, {"end": 1126.44, "start": 1126.24, "text": "over"}, {"end": 1128.32, "start": 1126.44, "text": "4."}, {"end": 1128.76, "start": 1128.32, "text": "So"}, {"end": 1129.32, "start": 1128.76, "text": "that's"}, {"end": 1129.72, "start": 1129.32, "text": "something"}, {"end": 1130.04, "start": 1129.72, "text": "very"}, {"end": 1130.52, "start": 1130.04, "text": "useful"}, {"end": 1130.68, "start": 1130.52, "text": "to"}, {"end": 1131.2, "start": 1130.68, "text": "know"}, {"end": 1132.44, "start": 1131.2, "text": "that"}, {"end": 1132.88, "start": 1132.44, "text": "we"}, {"end": 1133.2, "start": 1132.88, "text": "will"}, {"end": 1133.36, "start": 1133.2, "text": "use."}, {"end": 1133.4, "start": 1133.36, "text": "In"}, {"end": 1134.44, "start": 1133.4, "text": "particular,"}, {"end": 1134.84, "start": 1134.44, "text": "so"}, {"end": 1135.32, "start": 1134.84, "text": "what"}, {"end": 1135.36, "start": 1135.32, "text": "we"}, {"end": 1135.36, "start": 1135.36, "text": "are"}, {"end": 1135.36, "start": 1135.36, "text": "set"}, {"end": 1135.36, "start": 1135.36, "text": "to"}, {"end": 1135.56, "start": 1135.36, "text": "do"}, {"end": 1135.72, "start": 1135.56, "text": "in"}, {"end": 1136.08, "start": 1135.72, "text": "this"}, {"end": 1136.44, "start": 1136.08, "text": "problem"}, {"end": 1136.92, "start": 1136.44, "text": "again"}, {"end": 1137.12, "start": 1136.92, "text": "is"}, {"end": 1137.36, "start": 1137.12, "text": "we"}, {"end": 1137.36, "start": 1137.36, "text": "are"}, {"end": 1137.76, "start": 1137.36, "text": "given"}, {"end": 1138.08, "start": 1137.76, "text": "this"}, {"end": 1138.52, "start": 1138.08, "text": "family"}, {"end": 1138.64, "start": 1138.52, "text": "of"}, {"end": 1138.96, "start": 1138.64, "text": "perfect"}, {"end": 1139.96, "start": 1138.96, "text": "matchings."}], "text": " If m is not equal to L times r, then we get some probabilistic guarantee that a certain check is going to fail. In particular, y transpose times m times x is equal to y transpose times L times r times x. The probability of this is at most 1 over 4. So that's something very useful to know that we will use. In particular, so what we are set to do in this problem again is we are given this family of perfect matchings."}, {"chunks": [{"end": 1140.12, "start": 1140.0, "text": "we"}, {"end": 1140.28, "start": 1140.12, "text": "would"}, {"end": 1140.48, "start": 1140.28, "text": "like"}, {"end": 1140.56, "start": 1140.48, "text": "to"}, {"end": 1140.84, "start": 1140.56, "text": "know"}, {"end": 1141.28, "start": 1140.84, "text": "whether"}, {"end": 1141.64, "start": 1141.28, "text": "there"}, {"end": 1141.76, "start": 1141.64, "text": "are"}, {"end": 1142.12, "start": 1141.76, "text": "two"}, {"end": 1142.28, "start": 1142.12, "text": "perfect"}, {"end": 1142.8, "start": 1142.28, "text": "matchings"}, {"end": 1142.8, "start": 1142.8, "text": "in"}, {"end": 1142.84, "start": 1142.8, "text": "the"}, {"end": 1143.2, "start": 1142.84, "text": "family"}, {"end": 1143.36, "start": 1143.2, "text": "that"}, {"end": 1143.6, "start": 1143.36, "text": "form"}, {"end": 1144.16, "start": 1143.6, "text": 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like to know whether there are two perfect matchings in the family that form a Hamiltonian cycle. So we can again think of this task in terms of the matching's conductivity matrix. So here's again the family we had before, and now they're all, the ones in the family have weight zero and the other ones have infinity. And we would just like to know whether there's actually at least one one in these intersections of the orange rows and orange columns. So that actually you could also write it as,"}, {"chunks": [{"end": 1170.24, "start": 1170.0, "text": "The"}, {"end": 1170.72, "start": 1170.24, "text": "matrix"}, {"end": 1171.04, "start": 1170.72, "text": "of"}, {"end": 1171.28, "start": 1171.04, "text": "the"}, {"end": 1171.48, "start": 1171.28, "text": "matching"}, {"end": 1171.88, "start": 1171.48, "text": "connectivity"}, {"end": 1172.48, "start": 1171.88, "text": "matrix"}, {"end": 1172.96, "start": 1172.48, "text": "induced"}, {"end": 1172.96, "start": 1172.96, "text": "by"}, {"end": 1173.36, "start": 1172.96, "text": "the"}, {"end": 1173.8, "start": 1173.36, "text": "rows"}, {"end": 1173.88, "start": 1173.8, "text": "and"}, {"end": 1174.28, "start": 1173.88, "text": "columns"}, {"end": 1175.44, "start": 1174.28, "text": "calligraphic"}, {"end": 1175.6, "start": 1175.44, "text": "A"}, {"end": 1176.04, "start": 1175.6, "text": "Is"}, {"end": 1176.04, "start": 1176.04, "text": "that"}, {"end": 1176.12, "start": 1176.04, "text": "the"}, {"end": 1176.32, "start": 1176.12, "text": "all"}, {"end": 1176.6, "start": 1176.32, "text": "zero"}, {"end": 1177.52, "start": 1176.6, "text": "matrix"}, {"end": 1177.68, "start": 1177.52, "text": "or"}, {"end": 1177.8, "start": 1177.68, "text": "is"}, {"end": 1178.04, "start": 1177.8, "text": "there"}, {"end": 1178.08, "start": 1178.04, "text": "a"}, {"end": 1178.28, "start": 1178.08, "text": "non"}, {"end": 1178.56, "start": 1178.28, "text": "zero"}, {"end": 1178.72, "start": 1178.56, "text": "somewhere?"}, {"end": 1178.76, "start": 1178.72, "text": "So"}, {"end": 1179.56, "start": 1178.76, "text": "now"}, {"end": 1179.84, "start": 1179.56, "text": "we"}, {"end": 1180.4, "start": 1179.84, "text": "can"}, {"end": 1180.72, "start": 1180.4, "text": "actually"}, {"end": 1180.92, "start": 1180.72, "text": "use"}, {"end": 1181.4, "start": 1180.92, "text": "Freyval's"}, {"end": 1182.24, "start": 1181.4, "text": "test"}, {"end": 1182.56, "start": 1182.24, "text": "with"}, {"end": 1183.04, "start": 1182.56, "text": "the"}, {"end": 1183.52, "start": 1183.04, "text": "matrix"}, {"end": 1183.72, "start": 1183.52, "text": "M"}, {"end": 1183.92, "start": 1183.72, "text": "being"}, {"end": 1184.04, "start": 1183.92, "text": "the"}, {"end": 1184.16, "start": 1184.04, "text": "all"}, {"end": 1184.4, "start": 1184.16, "text": "zero"}, {"end": 1186.32, "start": 1184.4, "text": "matrix"}, {"end": 1186.8, "start": 1186.32, "text": "So"}, {"end": 1187.24, "start": 1186.8, "text": "then"}, {"end": 1187.68, "start": 1187.24, "text": "what"}, {"end": 1188.52, "start": 1187.68, "text": "we"}, {"end": 1188.8, "start": 1188.52, "text": "need"}, {"end": 1189.12, "start": 1188.8, "text": "to"}, {"end": 1189.4, "start": 1189.12, "text": "do"}, {"end": 1189.64, "start": 1189.4, "text": "here"}, {"end": 1190.12, "start": 1189.64, "text": "is"}, {"end": 1190.8, "start": 1190.12, "text": "we"}, {"end": 1191.16, "start": 1190.8, "text": "need"}, {"end": 1191.68, "start": 1191.16, "text": "to"}, {"end": 1192.64, "start": 1191.68, "text": "compute"}, {"end": 1192.8, "start": 1192.64, "text": "Y"}, {"end": 1193.84, "start": 1192.8, "text": "transpose"}, {"end": 1194.36, "start": 1193.84, "text": "times"}, {"end": 1194.72, "start": 1194.36, "text": "the"}, {"end": 1196.08, "start": 1194.72, "text": "matrix"}, {"end": 1196.6, "start": 1196.08, "text": "we"}, {"end": 1197.24, "start": 1196.6, "text": "would"}, {"end": 1197.6, "start": 1197.24, "text": "like"}, {"end": 1198.0, "start": 1197.6, "text": "to"}, {"end": 1198.72, "start": 1198.0, "text": "test"}, {"end": 1199.44, "start": 1198.72, "text": "times"}, {"end": 1199.96, "start": 1199.44, "text": "X"}], "text": " The matrix of the matching connectivity matrix induced by the rows and columns calligraphic A Is that the all zero matrix or is there a non zero somewhere? So now we can actually use Freyval's test with the matrix M being the all zero matrix So then what we need to do here is we need to compute Y transpose times the matrix we would like to test times X"}, {"chunks": [{"end": 1200.4, "start": 1200.0, "text": "times"}, {"end": 1200.4, "start": 1200.4, "text": "R."}, {"end": 1201.68, "start": 1200.4, "text": "So"}, {"end": 1201.8, "start": 1201.68, "text": "the"}, {"end": 1202.84, "start": 1201.8, "text": "factorization"}, {"end": 1202.96, "start": 1202.84, "text": "I"}, {"end": 1203.16, "start": 1202.96, "text": "told"}, {"end": 1203.36, "start": 1203.16, "text": "you"}, {"end": 1203.8, "start": 1203.36, "text": "earlier"}, {"end": 1204.32, "start": 1203.8, "text": "tells"}, {"end": 1204.72, "start": 1204.32, "text": "us"}, {"end": 1205.0, "start": 1204.72, "text": "that"}, {"end": 1205.24, "start": 1205.0, "text": "this"}, {"end": 1205.92, "start": 1205.24, "text": "quantity"}, {"end": 1205.96, "start": 1205.92, "text": "that"}, {"end": 1206.24, "start": 1205.96, "text": "we"}, {"end": 1207.12, "start": 1206.24, "text": "have"}, {"end": 1207.52, "start": 1207.12, "text": "here"}, {"end": 1207.72, "start": 1207.52, "text": "is"}, {"end": 1208.0, "start": 1207.72, "text": "actually"}, {"end": 1208.32, "start": 1208.0, "text": "equal"}, {"end": 1208.6, "start": 1208.32, "text": "to"}, {"end": 1208.8, "start": 1208.6, "text": "what"}, {"end": 1208.92, "start": 1208.8, "text": "I"}, {"end": 1209.16, "start": 1208.92, "text": "wrote"}, {"end": 1209.36, "start": 1209.16, "text": "here."}, {"end": 1209.56, "start": 1209.36, "text": "So"}, {"end": 1209.8, "start": 1209.56, "text": "here"}, {"end": 1209.84, "start": 1209.8, "text": "I"}, {"end": 1210.04, "start": 1209.84, "text": "just"}, {"end": 1210.28, "start": 1210.04, "text": "plugged"}, {"end": 1210.44, "start": 1210.28, "text": "in"}, {"end": 1211.2, "start": 1210.44, "text": "this"}, {"end": 1212.16, "start": 1211.2, "text": "factorization"}, {"end": 1212.16, "start": 1212.16, "text": "that"}, {"end": 1212.16, "start": 1212.16, "text": "I"}, {"end": 1212.4, "start": 1212.16, "text": "told"}, {"end": 1212.56, "start": 1212.4, "text": "you"}, {"end": 1213.44, "start": 1212.56, "text": "before."}, {"end": 1213.6, "start": 1213.44, "text": "So"}, {"end": 1213.88, "start": 1213.6, "text": "here"}, {"end": 1214.08, "start": 1213.88, "text": "again,"}, {"end": 1214.12, "start": 1214.08, "text": "we"}, {"end": 1214.32, "start": 1214.12, "text": "have"}, {"end": 1214.56, "start": 1214.32, "text": "this"}, {"end": 1214.92, "start": 1214.56, "text": "special"}, {"end": 1215.4, "start": 1214.92, "text": "family"}, {"end": 1215.4, "start": 1215.4, "text": "of"}, {"end": 1215.72, "start": 1215.4, "text": "perfect"}, {"end": 1216.2, "start": 1215.72, "text": "matchings"}, {"end": 1216.36, "start": 1216.2, "text": "of"}, {"end": 1216.6, "start": 1216.36, "text": "this"}, {"end": 1216.88, "start": 1216.6, "text": "narrow"}, {"end": 1217.12, "start": 1216.88, "text": "graph"}, {"end": 1217.2, "start": 1217.12, "text": "that"}, {"end": 1217.4, "start": 1217.2, "text": "you"}, {"end": 1218.96, "start": 1217.4, "text": "saw."}, {"end": 1220.12, "start": 1218.96, "text": "And"}, {"end": 1220.92, "start": 1220.12, "text": "then"}, {"end": 1221.08, "start": 1220.92, "text": "what"}, {"end": 1221.44, "start": 1221.08, "text": "the"}, {"end": 1221.92, "start": 1221.44, "text": "suggested"}, {"end": 1222.68, "start": 1221.92, "text": "algorithm"}, {"end": 1222.76, "start": 1222.68, "text": "is,"}, {"end": 1222.88, "start": 1222.76, "text": "is"}, {"end": 1222.96, "start": 1222.88, "text": "just"}, {"end": 1223.12, "start": 1222.96, "text": "compute"}, {"end": 1223.48, "start": 1223.12, "text": "this"}, {"end": 1223.92, "start": 1223.48, "text": "quantity"}, {"end": 1224.08, "start": 1223.92, "text": "as"}, {"end": 1224.4, "start": 1224.08, "text": "quick"}, {"end": 1224.72, "start": 1224.4, "text": "as"}, {"end": 1224.8, "start": 1224.72, "text": "you"}, {"end": 1225.24, "start": 1224.8, "text": "can,"}, {"end": 1225.44, "start": 1225.24, "text": "and"}, {"end": 1225.76, "start": 1225.44, "text": "then"}, {"end": 1226.08, "start": 1225.76, "text": "output"}, {"end": 1226.96, "start": 1226.08, "text": "yes"}, {"end": 1227.2, "start": 1226.96, "text": "if"}, {"end": 1227.52, "start": 1227.2, "text": "what"}, {"end": 1227.64, "start": 1227.52, "text": "you"}, {"end": 1227.88, "start": 1227.64, "text": "get"}, {"end": 1228.32, "start": 1227.88, "text": "is"}, {"end": 1228.92, "start": 1228.32, "text": "non-zero."}, {"end": 1229.16, "start": 1228.92, "text": "So"}, {"end": 1229.48, "start": 1229.16, "text": "you"}, {"end": 1229.56, "start": 1229.48, "text": "will"}, {"end": 1229.68, "start": 1229.56, "text": "get"}, {"end": 1229.68, "start": 1229.68, "text": "the"}, {"end": 1229.96, "start": 1229.68, "text": "number"}], "text": " times R. So the factorization I told you earlier tells us that this quantity that we have here is actually equal to what I wrote here. So here I just plugged in this factorization that I told you before. So here again, we have this special family of perfect matchings of this narrow graph that you saw. And then what the suggested algorithm is, is just compute this quantity as quick as you can, and then output yes if what you get is non-zero. So you will get the number"}, {"chunks": [{"end": 1230.56, "start": 1230.0, "text": "If"}, {"end": 1230.84, "start": 1230.56, "text": "it's"}, {"end": 1231.4, "start": 1230.84, "text": "non-zero"}, {"end": 1231.56, "start": 1231.4, "text": "you"}, {"end": 1231.72, "start": 1231.56, "text": "output"}, {"end": 1232.2, "start": 1231.72, "text": "yes,"}, {"end": 1232.6, "start": 1232.2, "text": "otherwise"}, {"end": 1232.72, "start": 1232.6, "text": "you"}, {"end": 1234.04, "start": 1232.72, "text": "output"}, {"end": 1234.76, "start": 1234.04, "text": "no"}, {"end": 1235.0, "start": 1234.76, "text": "And"}, {"end": 1235.4, "start": 1235.0, "text": "the"}, {"end": 1236.04, "start": 1235.4, "text": "point"}, {"end": 1236.32, "start": 1236.04, "text": "is"}, {"end": 1237.04, "start": 1236.32, "text": "that"}, {"end": 1237.48, "start": 1237.04, "text": "since"}, {"end": 1237.8, "start": 1237.48, "text": "this"}, {"end": 1238.04, "start": 1237.8, "text": "was"}, {"end": 1238.08, "start": 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"start": 1244.92, "text": "doing"}, {"end": 1245.6, "start": 1245.28, "text": "matrix"}, {"end": 1246.12, "start": 1245.6, "text": "vector"}, {"end": 1247.08, "start": 1246.12, "text": "multiplication"}, {"end": 1247.28, "start": 1247.08, "text": "here"}, {"end": 1247.28, "start": 1247.28, "text": "and"}, {"end": 1247.64, "start": 1247.28, "text": "the"}, {"end": 1248.36, "start": 1247.64, "text": "matrices"}, {"end": 1249.16, "start": 1248.36, "text": "are"}, {"end": 1250.2, "start": 1249.16, "text": "of"}, {"end": 1251.4, "start": 1250.2, "text": "dimensions"}, {"end": 1251.76, "start": 1251.4, "text": "The"}, {"end": 1252.4, "start": 1251.76, "text": "cardinality"}, {"end": 1252.4, "start": 1252.4, "text": "of"}, {"end": 1253.0, "start": 1252.4, "text": "calligraphic"}, {"end": 1253.76, "start": 1253.0, "text": "A"}, {"end": 1254.36, "start": 1253.76, "text": "and"}, {"end": 1256.24, "start": 1254.36, "text": "2"}, {"end": 1256.44, "start": 1256.24, "text": "to"}, {"end": 1256.88, "start": 1256.44, "text": "the"}, {"end": 1257.16, "start": 1256.88, "text": "n"}, {"end": 1257.64, "start": 1257.16, "text": "over"}, {"end": 1257.96, "start": 1257.64, "text": "2"}, {"end": 1258.2, "start": 1257.96, "text": "And"}, {"end": 1258.4, "start": 1258.2, "text": "this"}, {"end": 1259.12, "start": 1258.4, "text": "can"}, {"end": 1259.28, "start": 1259.12, "text": "be"}, {"end": 1259.48, "start": 1259.28, "text": "easily"}, {"end": 1259.64, "start": 1259.48, "text": "done"}, {"end": 1259.64, "start": 1259.64, "text": "in"}, {"end": 1259.96, "start": 1259.64, "text": "time"}], "text": " If it's non-zero you output yes, otherwise you output no And the point is that since this was a narrow factorization, so the size of this Pn was only 2 to the n over 2 We are just doing matrix vector multiplication here and the matrices are of dimensions The cardinality of calligraphic A and 2 to the n over 2 And this can be easily done in time"}, {"chunks": [{"end": 1260.16, "start": 1260.0, 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"text": "also"}, {"end": 1277.72, "start": 1277.4, "text": "modulo"}, {"end": 1277.96, "start": 1277.72, "text": "2"}, {"end": 1278.24, "start": 1277.96, "text": "equal"}, {"end": 1278.52, "start": 1278.24, "text": "to"}, {"end": 1279.0, "start": 1278.52, "text": "Cn"}, {"end": 1279.36, "start": 1279.0, "text": "times"}, {"end": 1279.52, "start": 1279.36, "text": "Cn"}, {"end": 1280.68, "start": 1279.52, "text": "transpose,"}, {"end": 1281.0, "start": 1280.68, "text": "where"}, {"end": 1281.52, "start": 1281.0, "text": "Cn"}, {"end": 1281.72, "start": 1281.52, "text": "is"}, {"end": 1282.24, "start": 1281.72, "text": "defined"}, {"end": 1282.52, "start": 1282.24, "text": "for"}, {"end": 1282.96, "start": 1282.52, "text": "every"}, {"end": 1283.48, "start": 1282.96, "text": "subset"}, {"end": 1284.04, "start": 1283.48, "text": "X"}, {"end": 1284.24, "start": 1284.04, "text": "that"}, {"end": 1285.56, "start": 1284.24, "text": "contains"}, {"end": 1286.04, "start": 1285.56, "text": "the"}, {"end": 1286.96, "start": 1286.04, "text": "element"}, {"end": 1287.28, "start": 1286.96, "text": "n"}, {"end": 1287.48, "start": 1287.28, "text": "as"}, {"end": 1288.24, "start": 1287.48, "text": "this."}, {"end": 1288.4, "start": 1288.24, "text": "So"}, {"end": 1288.6, "start": 1288.4, "text": "you"}, {"end": 1289.0, "start": 1288.6, "text": "should"}, {"end": 1289.64, "start": 1289.0, "text": "think"}, {"end": 1289.72, "start": 1289.64, "text": "of"}, {"end": 1289.96, "start": 1289.72, "text": "the"}], "text": " So we get back to this in a second, because there are more factorizations of the matching connectivity matrix. So here's another one, it's called the cut factorization, followed from a bunch of previous works. So Hn is also modulo 2 equal to Cn times Cn transpose, where Cn is defined for every subset X that contains the element n as this. So you should think of the"}, {"chunks": [{"end": 1290.24, "start": 1290.0, "text": "of"}, {"end": 1290.72, "start": 1290.24, "text": "x"}, {"end": 1291.0, "start": 1290.72, "text": "as"}, {"end": 1291.2, "start": 1291.0, "text": "a"}, {"end": 1292.04, "start": 1291.2, "text": "cut"}, {"end": 1292.2, "start": 1292.04, "text": "and"}, {"end": 1292.68, "start": 1292.2, "text": "then"}, {"end": 1293.08, "start": 1292.68, "text": "cn"}, {"end": 1293.48, "start": 1293.08, "text": "of"}, {"end": 1293.92, "start": 1293.48, "text": "ax"}, {"end": 1294.12, "start": 1293.92, "text": "is"}, {"end": 1294.52, "start": 1294.12, "text": "one"}, {"end": 1294.72, "start": 1294.52, "text": "if"}, {"end": 1295.48, "start": 1294.72, "text": "x,"}, {"end": 1295.6, "start": 1295.48, "text": "if"}, {"end": 1295.68, "start": 1295.6, "text": "the"}, {"end": 1296.0, "start": 1295.68, "text": "cut"}, {"end": 1296.28, "start": 1296.0, "text": "splits"}, {"end": 1296.32, "start": 1296.28, "text": "the"}, {"end": 1296.8, "start": 1296.32, "text": "matching"}, {"end": 1296.84, "start": 1296.8, "text": "and"}, {"end": 1297.2, "start": 1296.84, "text": "zero"}, {"end": 1298.32, "start": 1297.2, "text": "otherwise."}, {"end": 1298.6, "start": 1298.32, "text": "And"}, {"end": 1298.88, "start": 1298.6, "text": "x"}, {"end": 1299.48, "start": 1298.88, "text": "splits"}, {"end": 1299.84, "start": 1299.48, "text": "a"}, {"end": 1300.32, "start": 1299.84, "text": "means"}, {"end": 1300.8, "start": 1300.32, "text": "that"}, {"end": 1301.2, "start": 1300.8, "text": "there"}, {"end": 1301.64, "start": 1301.2, "text": "is"}, {"end": 1301.92, "start": 1301.64, "text": "no"}, {"end": 1302.32, "start": 1301.92, "text": "edge"}, {"end": 1302.4, "start": 1302.32, "text": "in"}, {"end": 1302.68, "start": 1302.4, "text": "a"}, {"end": 1303.08, "start": 1302.68, "text": "that"}, {"end": 1303.32, "start": 1303.08, "text": "has"}, {"end": 1303.84, "start": 1303.32, "text": "exactly"}, {"end": 1304.08, "start": 1303.84, "text": "one"}, {"end": 1304.28, "start": 1304.08, "text": "end"}, {"end": 1304.36, "start": 1304.28, "text": "in"}, {"end": 1305.24, "start": 1304.36, "text": "x."}, {"end": 1305.48, "start": 1305.24, "text": "So"}, {"end": 1305.68, "start": 1305.48, "text": "for"}, {"end": 1306.12, "start": 1305.68, "text": "example,"}, {"end": 1306.2, "start": 1306.12, "text": "if"}, {"end": 1306.6, "start": 1306.2, "text": "this"}, {"end": 1306.96, "start": 1306.6, "text": "is"}, {"end": 1307.08, "start": 1306.96, "text": "your"}, {"end": 1307.44, "start": 1307.08, "text": "matching,"}, {"end": 1307.6, "start": 1307.44, "text": "then"}, {"end": 1308.08, "start": 1307.6, "text": "this"}, {"end": 1308.44, "start": 1308.08, "text": "is"}, {"end": 1308.56, "start": 1308.44, "text": "a"}, {"end": 1308.68, "start": 1308.56, "text": "cut"}, {"end": 1309.0, "start": 1308.68, "text": "that"}, {"end": 1309.4, "start": 1309.0, "text": "splits"}, {"end": 1309.48, "start": 1309.4, "text": "the"}, {"end": 1309.92, "start": 1309.48, "text": "matching,"}, {"end": 1310.0, "start": 1309.92, "text": "but"}, {"end": 1310.52, "start": 1310.0, "text": "this"}, {"end": 1310.76, "start": 1310.52, "text": "is"}, {"end": 1310.96, "start": 1310.76, "text": "not"}, {"end": 1311.08, "start": 1310.96, "text": "the"}, {"end": 1311.08, "start": 1311.08, "text": "cut"}, {"end": 1311.28, "start": 1311.08, "text": "that"}, {"end": 1311.48, "start": 1311.28, "text": "splits"}, {"end": 1311.6, "start": 1311.48, "text": "the"}, {"end": 1311.92, "start": 1311.6, "text": "matching"}, {"end": 1312.24, "start": 1311.92, "text": "because"}, {"end": 1312.28, "start": 1312.24, "text": "there"}, {"end": 1312.52, "start": 1312.28, "text": "is"}, {"end": 1312.76, "start": 1312.52, "text": "this"}, {"end": 1312.96, "start": 1312.76, "text": "one"}, {"end": 1313.36, "start": 1312.96, "text": "edge"}, {"end": 1313.76, "start": 1313.36, "text": "that"}, {"end": 1314.64, "start": 1313.76, "text": "has"}, {"end": 1315.12, "start": 1314.64, "text": "endpoint"}, {"end": 1315.68, "start": 1315.12, "text": "in"}, {"end": 1316.2, "start": 1315.68, "text": "a"}, {"end": 1316.76, "start": 1316.2, "text": "and"}, {"end": 1317.16, "start": 1316.76, "text": "one"}, {"end": 1317.36, "start": 1317.16, "text": "not"}, {"end": 1317.4, "start": 1317.36, "text": "in"}, {"end": 1317.56, "start": 1317.4, "text": "a."}, {"end": 1319.56, "start": 1317.56, "text": "So"}, {"end": 1319.96, "start": 1319.56, "text": "that's"}], "text": " of x as a cut and then cn of ax is one if x, if the cut splits the matching and zero otherwise. And x splits a means that there is no edge in a that has exactly one end in x. So for example, if this is your matching, then this is a cut that splits the matching, but this is not the cut that splits the matching because there is this one edge that has endpoint in a and one not in a. So that's"}, {"chunks": [{"end": 1320.52, "start": 1320.0, "text": "Another"}, {"end": 1320.96, "start": 1320.52, "text": "factorization,"}, {"end": 1321.68, "start": 1320.96, "text": "so"}, {"end": 1322.08, "start": 1321.68, "text": "let's"}, {"end": 1322.28, "start": 1322.08, "text": "try"}, {"end": 1322.28, "start": 1322.28, "text": "to"}, {"end": 1322.52, "start": 1322.28, "text": "compare"}, {"end": 1323.36, "start": 1322.52, "text": "them."}, {"end": 1323.56, "start": 1323.36, "text": "So"}, {"end": 1323.8, "start": 1323.56, "text": "one,"}, {"end": 1324.16, "start": 1323.8, "text": "there"}, {"end": 1324.64, "start": 1324.16, "text": "are"}, {"end": 1324.8, "start": 1324.64, "text": "two"}, {"end": 1324.8, "start": 1324.8, "text": "to"}, {"end": 1325.0, "start": 1324.8, "text": "the"}, {"end": 1325.24, "start": 1325.0, "text": "n"}, {"end": 1325.6, "start": 1325.24, "text": "minus"}, {"end": 1325.8, "start": 1325.6, "text": "one"}, {"end": 1326.36, "start": 1325.8, "text": "cuts,"}, {"end": 1326.52, "start": 1326.36, "text": "so"}, {"end": 1327.04, "start": 1326.52, "text": "this"}, {"end": 1328.0, "start": 1327.04, "text": "factorization"}, {"end": 1328.52, "start": 1328.0, "text": "has"}, {"end": 1329.4, "start": 1328.52, "text": "quite"}, {"end": 1329.6, "start": 1329.4, "text": "a"}, {"end": 1330.56, "start": 1329.6, "text": "bit"}, {"end": 1330.72, "start": 1330.56, "text": "larger"}, {"end": 1330.84, "start": 1330.72, "text": "rank."}, {"end": 1331.2, "start": 1330.84, "text": "But"}, {"end": 1331.76, "start": 1331.2, "text": "if"}, {"end": 1331.84, "start": 1331.76, "text": "you"}, {"end": 1332.32, "start": 1331.84, "text": "look"}, {"end": 1332.4, "start": 1332.32, "text": "a"}, {"end": 1332.84, "start": 1332.4, "text": "bit"}, {"end": 1333.04, "start": 1332.84, "text": "more"}, {"end": 1333.48, "start": 1333.04, "text": "about"}, {"end": 1333.6, "start": 1333.48, "text": "it,"}, {"end": 1333.76, "start": 1333.6, "text": "the"}, {"end": 1334.4, "start": 1333.76, "text": "factorizing"}, {"end": 1335.08, "start": 1334.4, "text": "matrices"}, {"end": 1335.28, "start": 1335.08, "text": "in"}, {"end": 1335.32, "start": 1335.28, "text": "the"}, {"end": 1335.84, "start": 1335.32, "text": "previous"}, {"end": 1336.2, "start": 1335.84, "text": "matching"}, {"end": 1337.04, "start": 1336.2, "text": "factorization,"}, {"end": 1337.52, "start": 1337.04, "text": "actually"}, {"end": 1337.64, "start": 1337.52, "text": "it"}, {"end": 1337.92, "start": 1337.64, "text": "will"}, {"end": 1337.96, "start": 1337.92, "text": "be"}, {"end": 1338.32, "start": 1337.96, "text": "very"}, {"end": 1338.92, "start": 1338.32, "text": "dense."}, {"end": 1339.16, "start": 1338.92, "text": "Two"}, {"end": 1339.92, "start": 1339.16, "text": "perfect"}, {"end": 1340.32, "start": 1339.92, "text": "matching"}, {"end": 1340.88, "start": 1340.32, "text": "have"}, {"end": 1341.28, "start": 1340.88, "text": "a"}, {"end": 1341.52, "start": 1341.28, "text": "fairly"}, {"end": 1341.92, "start": 1341.52, "text": "large"}, {"end": 1342.68, "start": 1341.92, "text": "probability"}, {"end": 1342.84, "start": 1342.68, "text": "that"}, {"end": 1343.0, "start": 1342.84, "text": "they"}, {"end": 1343.16, "start": 1343.0, "text": "form"}, {"end": 1343.44, "start": 1343.16, "text": "a"}, {"end": 1344.4, "start": 1343.44, "text": "Hamiltonian"}, {"end": 1345.08, "start": 1344.4, "text": "cycle,"}, {"end": 1345.48, "start": 1345.08, "text": "namely"}, {"end": 1345.64, "start": 1345.48, "text": "1"}, {"end": 1345.92, "start": 1345.64, "text": "over"}, {"end": 1346.28, "start": 1345.92, "text": "square"}, {"end": 1346.68, "start": 1346.28, "text": "root"}, {"end": 1346.68, "start": 1346.68, "text": "n,"}, {"end": 1347.08, "start": 1346.68, "text": "therefore"}, {"end": 1348.36, "start": 1347.08, "text": "this"}, {"end": 1348.6, "start": 1348.36, "text": "is"}, {"end": 1348.72, "start": 1348.6, "text": "a"}, {"end": 1349.12, "start": 1348.72, "text": "dense"}, {"end": 1349.96, "start": 1349.12, "text": "matrix."}], "text": " Another factorization, so let's try to compare them. So one, there are two to the n minus one cuts, so this factorization has quite a bit larger rank. But if you look a bit more about it, the factorizing matrices in the previous matching factorization, actually it will be very dense. Two perfect matching have a fairly large probability that they form a Hamiltonian cycle, namely 1 over square root n, therefore this is a dense matrix."}, {"chunks": [{"end": 1350.16, "start": 1350.0, "text": "But"}, {"end": 1350.4, "start": 1350.16, "text": "this"}, {"end": 1350.84, "start": 1350.4, "text": "matrix"}, {"end": 1350.84, "start": 1350.84, "text": "that"}, {"end": 1350.84, "start": 1350.84, "text": "we"}, {"end": 1350.88, "start": 1350.84, "text": "have"}, {"end": 1351.28, "start": 1350.88, "text": "here,"}, {"end": 1351.48, "start": 1351.28, "text": "it's"}, {"end": 1351.8, "start": 1351.48, "text": "very"}, {"end": 1352.28, "start": 1351.8, "text": "sparse."}, {"end": 1352.64, "start": 1352.28, "text": "If"}, {"end": 1352.8, "start": 1352.64, "text": "you"}, {"end": 1353.2, "start": 1352.8, "text": "look"}, {"end": 1353.48, "start": 1353.2, "text": "at"}, {"end": 1353.72, "start": 1353.48, "text": "one"}, {"end": 1354.36, "start": 1353.72, "text": "matching,"}, {"end": 1354.68, "start": 1354.36, "text": "you"}, {"end": 1355.16, "start": 1354.68, "text": "only"}, {"end": 1355.32, "start": 1355.16, "text": "have"}, {"end": 1355.36, "start": 1355.32, "text": "one"}, {"end": 1355.92, "start": 1355.36, "text": "choice"}, {"end": 1356.44, "start": 1355.92, "text": "for"}, {"end": 1356.72, "start": 1356.44, "text": "every"}, {"end": 1357.2, "start": 1356.72, "text": "edge"}, {"end": 1357.36, "start": 1357.2, "text": "where"}, {"end": 1357.48, "start": 1357.36, "text": "to"}, {"end": 1358.04, "start": 1357.48, "text": "put"}, {"end": 1358.36, "start": 1358.04, "text": "it"}, {"end": 1358.92, "start": 1358.36, "text": "in"}, {"end": 1359.24, "start": 1358.92, "text": "the"}, {"end": 1360.04, "start": 1359.24, "text": "cut"}, {"end": 1360.16, "start": 1360.04, "text": "if"}, {"end": 1360.28, "start": 1360.16, "text": "you"}, {"end": 1360.64, "start": 1360.28, "text": "want"}, {"end": 1360.92, "start": 1360.64, "text": "to"}, {"end": 1361.12, "start": 1360.92, "text": "split"}, {"end": 1361.36, "start": 1361.12, "text": "it."}, {"end": 1362.32, "start": 1361.36, "text": "So"}, {"end": 1362.64, "start": 1362.32, "text": "there"}, {"end": 1362.76, "start": 1362.64, "text": "are"}, {"end": 1362.92, "start": 1362.76, "text": "only"}, {"end": 1363.04, "start": 1362.92, "text": "two"}, {"end": 1363.04, "start": 1363.04, "text": "to"}, {"end": 1363.16, "start": 1363.04, "text": "the"}, {"end": 1363.44, "start": 1363.16, "text": "n"}, {"end": 1363.76, "start": 1363.44, "text": "minus"}, {"end": 1364.04, "start": 1363.76, "text": "one,"}, {"end": 1364.08, "start": 1364.04, "text": "two"}, {"end": 1364.72, "start": 1364.08, "text": "to"}, {"end": 1364.92, "start": 1364.72, "text": "the"}, {"end": 1365.04, "start": 1364.92, "text": "n"}, {"end": 1365.44, "start": 1365.04, "text": "over"}, {"end": 1365.6, "start": 1365.44, "text": "two"}, {"end": 1365.84, "start": 1365.6, "text": "minus"}, {"end": 1366.28, "start": 1365.84, "text": "one,"}, {"end": 1366.96, "start": 1366.28, "text": "once"}, {"end": 1367.12, "start": 1366.96, "text": "per"}, {"end": 1367.44, "start": 1367.12, "text": "row"}, {"end": 1367.48, "start": 1367.44, "text": "in"}, {"end": 1367.68, "start": 1367.48, "text": "this"}, {"end": 1369.04, "start": 1367.68, "text": "matching."}, {"end": 1369.2, "start": 1369.04, "text": "And"}, {"end": 1369.56, "start": 1369.2, "text": "then"}, {"end": 1369.92, "start": 1369.56, "text": "if"}, {"end": 1370.12, "start": 1369.92, "text": "you"}, {"end": 1370.56, "start": 1370.12, "text": "think"}, {"end": 1370.6, "start": 1370.56, "text": "back"}, {"end": 1370.92, "start": 1370.6, "text": "in"}, {"end": 1371.32, "start": 1370.92, "text": "the"}, {"end": 1371.68, "start": 1371.32, "text": "previous"}, {"end": 1372.92, "start": 1371.68, "text": "slide,"}, {"end": 1373.44, "start": 1372.92, "text": "so"}, {"end": 1373.68, "start": 1373.44, "text": "if"}, {"end": 1373.76, "start": 1373.68, "text": "you"}, {"end": 1373.92, "start": 1373.76, "text": "would"}, {"end": 1374.12, "start": 1373.92, "text": "like"}, {"end": 1374.2, "start": 1374.12, "text": "to"}, {"end": 1374.44, "start": 1374.2, "text": "use"}, {"end": 1374.6, "start": 1374.44, "text": "this"}, {"end": 1374.72, "start": 1374.6, "text": "cut"}, {"end": 1375.52, "start": 1374.72, "text": "factorization"}, {"end": 1375.72, "start": 1375.52, "text": "in"}, {"end": 1375.76, "start": 1375.72, "text": "the"}, {"end": 1376.24, "start": 1375.76, "text": "previous"}, {"end": 1376.52, "start": 1376.24, "text": "5"}, {"end": 1377.2, "start": 1376.52, "text": "volts"}, {"end": 1377.44, "start": 1377.2, "text": "base"}, {"end": 1378.04, "start": 1377.44, "text": "template,"}, {"end": 1378.4, "start": 1378.04, "text": "actually"}, {"end": 1378.72, "start": 1378.4, "text": "we"}, {"end": 1379.0, "start": 1378.72, "text": "will"}, {"end": 1379.56, "start": 1379.0, "text": "get"}, {"end": 1379.96, "start": 1379.56, "text": "a,"}], "text": " But this matrix that we have here, it's very sparse. If you look at one matching, you only have one choice for every edge where to put it in the cut if you want to split it. So there are only two to the n minus one, two to the n over two minus one, once per row in this matching. And then if you think back in the previous slide, so if you would like to use this cut factorization in the previous 5 volts base template, actually we will get a,"}, {"chunks": [{"end": 1380.24, "start": 1380.0, "text": "Even"}, {"end": 1381.08, "start": 1380.24, "text": "the"}, {"end": 1381.32, "start": 1381.08, "text": "same"}, {"end": 1381.52, "start": 1381.32, "text": "running"}, {"end": 1381.8, "start": 1381.52, "text": "time,"}, {"end": 1381.96, "start": 1381.8, "text": "but"}, {"end": 1382.2, "start": 1381.96, "text": "for"}, {"end": 1382.44, "start": 1382.2, "text": "a"}, {"end": 1382.88, "start": 1382.44, "text": "different"}, {"end": 1383.24, "start": 1382.88, "text": "reason."}, {"end": 1383.56, "start": 1383.24, "text": "Because"}, {"end": 1384.28, "start": 1383.56, "text": "matrix"}, {"end": 1384.6, "start": 1384.28, "text": "vector"}, {"end": 1384.96, "start": 1384.6, "text": "multiplication"}, {"end": 1385.16, "start": 1384.96, "text": "can"}, {"end": 1385.44, "start": 1385.16, "text": "also"}, {"end": 1385.48, "start": 1385.44, "text": "be"}, {"end": 1385.68, "start": 1385.48, "text": "done"}, {"end": 1386.2, "start": 1385.68, "text": "quickly"}, {"end": 1386.52, "start": 1386.2, "text": "if"}, {"end": 1387.04, "start": 1386.52, "text": "the"}, {"end": 1387.6, "start": 1387.04, "text": "matrix"}, {"end": 1388.0, "start": 1387.6, "text": "is"}, {"end": 1388.12, "start": 1388.0, "text": "very"}, {"end": 1388.68, "start": 1388.12, "text": "sparse."}, {"end": 1389.48, "start": 1388.68, "text": "So"}, {"end": 1389.8, "start": 1389.48, "text": "we"}, {"end": 1390.32, "start": 1389.8, "text": "have"}, {"end": 1391.2, "start": 1390.32, "text": "these"}, {"end": 1391.8, "start": 1391.2, "text": "two"}, {"end": 1392.68, "start": 1391.8, "text": "matrices"}, {"end": 1393.04, "start": 1392.68, "text": "that"}, {"end": 1393.4, "start": 1393.04, "text": "because"}, {"end": 1393.44, "start": 1393.4, "text": "of"}, {"end": 1393.68, "start": 1393.44, "text": "different"}, {"end": 1394.08, "start": 1393.68, "text": "reasons,"}, {"end": 1394.6, "start": 1394.08, "text": "they"}, {"end": 1394.68, "start": 1394.6, "text": "are"}, {"end": 1395.76, "start": 1394.68, "text": "pretty"}, {"end": 1395.96, "start": 1395.76, "text": "good."}, {"end": 1396.2, "start": 1395.96, "text": "And"}, {"end": 1396.44, "start": 1396.2, "text": "now"}, {"end": 1396.68, "start": 1396.44, "text": "what"}, {"end": 1396.68, "start": 1396.68, "text": "the"}, {"end": 1396.72, "start": 1396.68, "text": "main"}, {"end": 1397.24, "start": 1396.72, "text": "contribution"}, {"end": 1397.4, "start": 1397.24, "text": "is"}, {"end": 1397.4, "start": 1397.4, "text": "of"}, {"end": 1397.72, "start": 1397.4, "text": "this"}, {"end": 1398.28, "start": 1397.72, "text": "paper"}, {"end": 1398.4, "start": 1398.28, "text": "is"}, {"end": 1398.68, "start": 1398.4, "text": "that"}, {"end": 1399.08, "start": 1398.68, "text": "we"}, {"end": 1399.36, "start": 1399.08, "text": "show"}, {"end": 1399.56, "start": 1399.36, "text": "that"}, {"end": 1399.76, "start": 1399.56, "text": "you"}, {"end": 1399.92, "start": 1399.76, "text": "can"}, {"end": 1400.2, "start": 1399.92, "text": "combine"}, {"end": 1400.4, "start": 1400.2, "text": "these"}, {"end": 1400.84, "start": 1400.4, "text": "both"}, {"end": 1401.88, "start": 1400.84, "text": "vectorizations"}, {"end": 1402.24, "start": 1401.88, "text": "to"}, {"end": 1402.52, "start": 1402.24, "text": "get"}, {"end": 1402.76, "start": 1402.52, "text": "yet"}, {"end": 1403.0, "start": 1402.76, "text": "another"}, {"end": 1403.72, "start": 1403.0, "text": "vectorization"}, {"end": 1403.8, "start": 1403.72, "text": "that"}, {"end": 1404.04, "start": 1403.8, "text": "we"}, {"end": 1404.6, "start": 1404.04, "text": "call"}, {"end": 1404.64, "start": 1404.6, "text": "the"}, {"end": 1404.88, "start": 1404.64, "text": "narrow"}, {"end": 1405.24, "start": 1404.88, "text": "cut"}, {"end": 1405.88, "start": 1405.24, "text": "vectorization."}, {"end": 1406.16, "start": 1405.88, "text": "And"}, {"end": 1406.36, "start": 1406.16, "text": "again,"}, {"end": 1406.72, "start": 1406.36, "text": "if"}, {"end": 1406.88, "start": 1406.72, "text": "I'm"}, {"end": 1407.52, "start": 1406.88, "text": "simplifying"}, {"end": 1407.76, "start": 1407.52, "text": "here"}, {"end": 1408.08, "start": 1407.76, "text": "a"}, {"end": 1408.28, "start": 1408.08, "text": "bit,"}, {"end": 1408.88, "start": 1408.28, "text": "so"}, {"end": 1409.0, "start": 1408.88, "text": "we"}, {"end": 1409.52, "start": 1409.0, "text": "let"}, {"end": 1409.96, "start": 1409.52, "text": "C"}], "text": " Even the same running time, but for a different reason. Because matrix vector multiplication can also be done quickly if the matrix is very sparse. So we have these two matrices that because of different reasons, they are pretty good. And now what the main contribution is of this paper is that we show that you can combine these both vectorizations to get yet another vectorization that we call the narrow cut vectorization. And again, if I'm simplifying here a bit, so we let C"}, {"chunks": [{"end": 1410.36, "start": 1410.0, "text": "and"}, {"end": 1410.64, "start": 1410.36, "text": "be"}, {"end": 1410.84, "start": 1410.64, "text": "all"}, {"end": 1410.84, "start": 1410.84, "text": "the"}, {"end": 1410.88, "start": 1410.84, "text": "cuts"}, {"end": 1411.16, "start": 1410.88, "text": "that"}, {"end": 1411.68, "start": 1411.16, "text": "split"}, {"end": 1411.8, "start": 1411.68, "text": "some"}, {"end": 1412.36, "start": 1411.8, "text": "matching"}, {"end": 1412.56, "start": 1412.36, "text": "in"}, {"end": 1413.56, "start": 1412.56, "text": "the"}, {"end": 1414.84, "start": 1413.56, "text": "special"}, {"end": 1415.28, "start": 1414.84, "text": "family"}, {"end": 1415.48, "start": 1415.28, "text": "of"}, {"end": 1415.68, "start": 1415.48, "text": "matching"}, {"end": 1416.2, "start": 1415.68, "text": "Pn,"}, {"end": 1416.64, "start": 1416.2, "text": "then"}, {"end": 1417.08, "start": 1416.64, "text": "there"}, {"end": 1417.64, "start": 1417.08, "text": "is"}, {"end": 1418.28, "start": 1417.64, "text": "a"}, {"end": 1419.28, "start": 1418.28, "text": "factorization"}, {"end": 1419.76, "start": 1419.28, "text": "also"}, {"end": 1420.12, "start": 1419.76, "text": "that"}, {"end": 1420.48, "start": 1420.12, "text": "is"}, {"end": 1420.8, "start": 1420.48, "text": "only"}, {"end": 1421.24, "start": 1420.8, "text": "indexed"}, {"end": 1421.4, "start": 1421.24, "text": "by"}, {"end": 1421.68, "start": 1421.4, "text": "this"}, {"end": 1422.04, "start": 1421.68, "text": "special"}, {"end": 1422.52, "start": 1422.04, "text": "family"}, {"end": 1422.52, "start": 1422.52, "text": "of"}, {"end": 1424.52, "start": 1422.52, "text": "cuts."}, {"end": 1424.8, "start": 1424.52, "text": "So"}, {"end": 1425.2, "start": 1424.8, "text": "the"}, {"end": 1426.04, "start": 1425.2, "text": "rank"}, {"end": 1426.44, "start": 1426.04, "text": "is"}, {"end": 1427.28, "start": 1426.44, "text": "kind"}, {"end": 1427.44, "start": 1427.28, "text": "of"}, {"end": 1427.64, "start": 1427.44, "text": "in"}, {"end": 1427.88, "start": 1427.64, "text": "between"}, {"end": 1428.4, "start": 1427.88, "text": "these"}, {"end": 1428.96, "start": 1428.4, "text": "two"}, {"end": 1430.56, "start": 1428.96, "text": "factorizations,"}, {"end": 1431.2, "start": 1430.56, "text": "but"}, {"end": 1431.68, "start": 1431.2, "text": "this"}, {"end": 1432.12, "start": 1431.68, "text": "matrix"}, {"end": 1432.92, "start": 1432.12, "text": "actually"}, {"end": 1433.36, "start": 1432.92, "text": "has"}, {"end": 1433.76, "start": 1433.36, "text": "only"}, {"end": 1433.76, "start": 1433.76, "text": "2"}, {"end": 1433.92, "start": 1433.76, "text": "to"}, {"end": 1434.16, "start": 1433.92, "text": "the"}, {"end": 1434.64, "start": 1434.16, "text": "power"}, {"end": 1435.44, "start": 1434.64, "text": "0.3n"}, {"end": 1435.8, "start": 1435.44, "text": "once"}, {"end": 1435.88, "start": 1435.8, "text": "per"}, {"end": 1436.24, "start": 1435.88, "text": "row"}, {"end": 1436.4, "start": 1436.24, "text": "on"}, {"end": 1437.08, "start": 1436.4, "text": "average."}, {"end": 1437.2, "start": 1437.08, "text": "So"}, {"end": 1437.84, "start": 1437.2, "text": "therefore"}, {"end": 1438.2, "start": 1437.84, "text": "it's"}, {"end": 1438.36, "start": 1438.2, "text": "actually"}, {"end": 1439.08, "start": 1438.36, "text": "pretty"}, {"end": 1439.96, "start": 1439.08, "text": "good."}], "text": " and be all the cuts that split some matching in the special family of matching Pn, then there is a factorization also that is only indexed by this special family of cuts. So the rank is kind of in between these two factorizations, but this matrix actually has only 2 to the power 0.3n once per row on average. So therefore it's actually pretty good."}, {"chunks": [{"end": 1440.08, "start": 1440.0, "text": "We"}, {"end": 1440.24, "start": 1440.08, "text": "go"}, {"end": 1440.48, "start": 1440.24, "text": "back"}, {"end": 1440.6, "start": 1440.48, "text": "to"}, {"end": 1440.8, "start": 1440.6, "text": "this"}, {"end": 1441.6, "start": 1440.8, "text": "slide,"}, {"end": 1441.96, "start": 1441.6, "text": "then"}, {"end": 1442.12, "start": 1441.96, "text": "we"}, {"end": 1442.24, "start": 1442.12, "text": "get"}, {"end": 1442.48, "start": 1442.24, "text": "this"}, {"end": 1442.84, "start": 1442.48, "text": "promised"}, {"end": 1443.04, "start": 1442.84, "text": "running"}, {"end": 1443.8, "start": 1443.04, "text": "time"}, {"end": 1444.0, "start": 1443.8, "text": "by"}, {"end": 1444.48, "start": 1444.0, "text": "just"}, {"end": 1444.84, "start": 1444.48, "text": "using"}, {"end": 1445.28, "start": 1444.84, "text": "the"}, {"end": 1445.48, "start": 1445.28, "text": "new"}, {"end": 1446.4, "start": 1445.48, "text": "factorization."}, {"end": 1447.0, "start": 1446.4, "text": "There"}, {"end": 1447.32, "start": 1447.0, "text": "are"}, {"end": 1447.48, "start": 1447.32, "text": "some"}, {"end": 1448.08, "start": 1447.48, "text": "issues"}, {"end": 1448.36, "start": 1448.08, "text": "that"}, {"end": 1448.44, "start": 1448.36, "text": "one"}, {"end": 1448.6, "start": 1448.44, "text": "has"}, {"end": 1448.76, "start": 1448.6, "text": "to"}, {"end": 1449.08, "start": 1448.76, "text": "deal"}, {"end": 1449.12, "start": 1449.08, "text": "with."}, {"end": 1449.24, "start": 1449.12, "text": "So"}, {"end": 1449.68, "start": 1449.24, "text": "to"}, {"end": 1450.16, "start": 1449.68, "text": "actually"}, {"end": 1450.52, "start": 1450.16, "text": "get"}, {"end": 1450.92, "start": 1450.52, "text": "this"}, {"end": 1451.2, "start": 1450.92, "text": "running"}, {"end": 1451.48, "start": 1451.2, "text": "time,"}, {"end": 1451.92, "start": 1451.48, "text": "one"}, {"end": 1452.24, "start": 1451.92, "text": "has"}, {"end": 1452.4, "start": 1452.24, "text": "to"}, {"end": 1452.84, "start": 1452.4, "text": "randomly"}, {"end": 1453.24, "start": 1452.84, "text": "order"}, {"end": 1453.4, "start": 1453.24, "text": "the"}, {"end": 1454.44, "start": 1453.4, "text": "vertices"}, {"end": 1455.04, "start": 1454.44, "text": "and"}, {"end": 1455.4, "start": 1455.04, "text": "then"}, {"end": 1455.76, "start": 1455.4, "text": "throw"}, {"end": 1456.04, "start": 1455.76, "text": "away"}, {"end": 1456.12, "start": 1456.04, "text": "all"}, {"end": 1456.24, "start": 1456.12, "text": "the"}, {"end": 1456.72, "start": 1456.24, "text": "matchings"}, {"end": 1456.84, "start": 1456.72, "text": "with"}, {"end": 1457.36, "start": 1456.84, "text": "too"}, {"end": 1457.76, "start": 1457.36, "text": "many"}, {"end": 1458.28, "start": 1457.76, "text": "ones"}, {"end": 1458.28, "start": 1458.28, "text": "in"}, {"end": 1458.44, "start": 1458.28, "text": "the"}, {"end": 1458.96, "start": 1458.44, "text": "matrix"}, {"end": 1459.48, "start": 1458.96, "text": "and"}, {"end": 1460.04, "start": 1459.48, "text": "also"}, {"end": 1460.64, "start": 1460.04, "text": "enumerate"}, {"end": 1460.92, "start": 1460.64, "text": "all"}, {"end": 1460.92, "start": 1460.92, "text": "the"}, {"end": 1461.24, "start": 1460.92, "text": "ones"}, {"end": 1461.32, "start": 1461.24, "text": "in"}, {"end": 1461.32, "start": 1461.32, "text": "the"}, {"end": 1461.56, "start": 1461.32, "text": "matrix"}, {"end": 1462.12, "start": 1461.56, "text": "carefully."}, {"end": 1462.6, "start": 1462.12, "text": "But"}, {"end": 1463.0, "start": 1462.6, "text": "I"}, {"end": 1463.08, "start": 1463.0, "text": "will"}, {"end": 1463.24, "start": 1463.08, "text": "not"}, {"end": 1463.4, "start": 1463.24, "text": "get"}, {"end": 1464.24, "start": 1463.4, "text": "into"}, {"end": 1464.52, "start": 1464.24, "text": "that."}, {"end": 1465.12, "start": 1464.52, "text": "So"}, {"end": 1465.32, "start": 1465.12, "text": "in"}, {"end": 1465.88, "start": 1465.32, "text": "conclusion,"}, {"end": 1466.36, "start": 1465.88, "text": "the"}, {"end": 1466.6, "start": 1466.36, "text": "rank"}, {"end": 1466.88, "start": 1466.6, "text": "upper"}, {"end": 1467.36, "start": 1466.88, "text": "bounds"}, {"end": 1467.6, "start": 1467.36, "text": "imply"}, {"end": 1468.2, "start": 1467.6, "text": "powerful"}, {"end": 1469.96, "start": 1468.2, "text": "algorithms."}], "text": " We go back to this slide, then we get this promised running time by just using the new factorization. There are some issues that one has to deal with. So to actually get this running time, one has to randomly order the vertices and then throw away all the matchings with too many ones in the matrix and also enumerate all the ones in the matrix carefully. But I will not get into that. So in conclusion, the rank upper bounds imply powerful algorithms."}, {"chunks": [{"end": 1470.16, "start": 1470.0, "text": "And"}, {"end": 1470.44, "start": 1470.16, "text": "also"}, {"end": 1470.52, "start": 1470.44, "text": "for"}, {"end": 1470.8, "start": 1470.52, "text": "weighted"}, {"end": 1471.2, "start": 1470.8, "text": "problems"}, {"end": 1471.44, "start": 1471.2, "text": "via"}, {"end": 1472.08, "start": 1471.44, "text": "Gaussian"}, {"end": 1472.32, "start": 1472.08, "text": "elimination,"}, {"end": 1472.52, "start": 1472.32, "text": "as"}, {"end": 1472.56, "start": 1472.52, "text": "I"}, {"end": 1473.12, "start": 1472.56, "text": "told"}, {"end": 1473.4, "start": 1473.12, "text": "you."}, {"end": 1473.56, "start": 1473.4, "text": "Not"}, {"end": 1473.8, "start": 1473.56, "text": "only"}, {"end": 1474.0, "start": 1473.8, "text": "the"}, {"end": 1474.52, "start": 1474.0, "text": "rank"}, {"end": 1474.76, "start": 1474.52, "text": "is"}, {"end": 1475.16, "start": 1474.76, "text": "actually"}, {"end": 1475.68, "start": 1475.16, "text": "important,"}, {"end": 1475.88, "start": 1475.68, "text": "but"}, {"end": 1476.12, "start": 1475.88, "text": "also"}, {"end": 1476.2, "start": 1476.12, "text": "the"}, {"end": 1476.88, "start": 1476.2, "text": "sparsity"}, {"end": 1476.92, "start": 1476.88, "text": "can"}, {"end": 1477.32, "start": 1476.92, "text": "be"}, {"end": 1477.92, "start": 1477.32, "text": "an"}, {"end": 1478.24, "start": 1477.92, "text": "important"}, {"end": 1478.52, "start": 1478.24, "text": "measure"}, {"end": 1478.72, "start": 1478.52, "text": "of"}, {"end": 1479.12, "start": 1478.72, "text": "a"}, {"end": 1480.88, "start": 1479.12, "text": "factorization."}, {"end": 1481.6, "start": 1480.88, "text": "And"}, {"end": 1482.04, "start": 1481.6, "text": "I"}, {"end": 1482.64, "start": 1482.04, "text": "want"}, {"end": 1482.72, "start": 1482.64, "text": "to"}, {"end": 1483.56, "start": 1482.72, "text": "advertise"}, {"end": 1483.96, "start": 1483.56, "text": "here"}, {"end": 1484.28, "start": 1483.96, "text": "some"}, {"end": 1484.6, "start": 1484.28, "text": "survey"}, {"end": 1484.8, "start": 1484.6, "text": "of"}, {"end": 1485.44, "start": 1484.8, "text": "myself"}, {"end": 1486.68, "start": 1485.44, "text": "in"}, {"end": 1487.12, "start": 1486.68, "text": "a"}, {"end": 1487.44, "start": 1487.12, "text": "recent"}, {"end": 1487.8, "start": 1487.44, "text": "fetch"}, {"end": 1488.48, "start": 1487.8, "text": "script."}, {"end": 1488.8, "start": 1488.48, "text": "Yeah,"}, {"end": 1489.12, "start": 1488.8, "text": "so"}, {"end": 1489.44, "start": 1489.12, "text": "there"}, {"end": 1489.64, "start": 1489.44, "text": "are"}, {"end": 1490.0, "start": 1489.64, "text": "plenty"}, {"end": 1490.28, "start": 1490.0, "text": "open"}, {"end": 1490.76, "start": 1490.28, "text": "questions,"}, {"end": 1490.8, "start": 1490.76, "text": "but"}, {"end": 1490.92, "start": 1490.8, "text": "I"}, {"end": 1491.48, "start": 1490.92, "text": "will"}, {"end": 1492.44, "start": 1491.48, "text": "not"}, {"end": 1492.72, "start": 1492.44, "text": "mention"}, {"end": 1492.88, "start": 1492.72, "text": "them"}, {"end": 1493.32, "start": 1492.88, "text": "explicitly"}, {"end": 1494.2, "start": 1493.32, "text": "now."}, {"end": 1494.72, "start": 1494.2, "text": "Thank"}, {"end": 1494.84, "start": 1494.72, "text": "you"}, {"end": 1495.44, "start": 1494.84, "text": "for"}, {"end": 1495.52, "start": 1495.44, "text": "your"}, {"end": 1499.72, "start": 1495.52, "text": "attention."}], "text": " And also for weighted problems via Gaussian elimination, as I told you. Not only the rank is actually important, but also the sparsity can be an important measure of a factorization. And I want to advertise here some survey of myself in a recent fetch script. Yeah, so there are plenty open questions, but I will not mention them explicitly now. Thank you for your attention."}]}} |