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23,600 | sorting ( / ( + / this equation is valid for any that is power of so we may also write ( / ( / + / / and ( / ( / + / / ( ( + now add up all the equations this means that we add all of the terms on the left-hand side and set the result equal to the sum of all of the terms on the right-hand side observe that the term ( /... |
23,601 | continuing in this mannerwe obtain ( ( / using log nwe obtain (nnt( log log the choice of which method to use is matter of taste the first method tends to produce scrap work that fits better on standard / sheet of paper leading to fewer mathematical errorsbut it requires certain amount of experience to apply the second... |
23,602 | sorting with heapsortwe can achieve quicksort' fast running time on almost all inputswith heapsort' ( log nworst-case running time exercise describes this approach the quicksort algorithm is simple to understand and prove correctalthough for many years it had the reputation of being an algorithm that could in theory be... |
23,603 | respectable on most inputs in factif the list contains large numbers of duplicates with relatively few distinct itemsas is sometimes the casethen the performance is extremely good the algorithm we have described forms the basis of the quicksort howeverby making the extra listsand doing so recursivelyit is hard to see h... |
23,604 | sorting select pivot partition quicksort small quicksort large figure the steps of quicksort illustrated by example wrong way the popularuninformed choice is to use the first element as the pivot this is acceptable if the input is randombut if the input is presorted or in reverse orderthen the pivot provides poor parti... |
23,605 | the same bad properties as merely choosing the first element do not use that pivoting strategyeither safe maneuver safe course is merely to choose the pivot randomly this strategy is generally perfectly safeunless the random number generator has flaw (which is not as uncommon as you might think)since it is very unlikel... |
23,606 | sorting is to the left of jthose elements are swapped the effect is to push large element to the right and small element to the left in the example abovei would not move and would slide over one place the situation is as follows we then swap the elements pointed to by and and repeat the process until and crossafter fir... |
23,607 | to start withor the large element originally in position was replaced during swap similar argument shows that elements in positions must be large one important detail we must consider is how to handle elements that are equal to the pivot the questions are whether or not should stop when it sees an element equal to the ... |
23,608 | sorting /*quicksort algorithm (driver*template void quicksortvector quicksorta size )figure driver for quicksort the general form of the routines will be to pass the array and the range of the array (left and rightto be sorted the first routine to deal with is pivot selection the easiest way to do this is to sort [left... |
23,609 | figure does the median-of-three partitioning with all the side effects described it may seem that it is only slightly inefficient to compute the pivot by method that does not actually sort [left] [center]and [right]butsurprisinglythis produces bad results (see exercise the real heart of the quicksort routine is in figu... |
23,610 | sorting int left right forwhileai pivot ++whilepivot aj --ifi std::swapai ]aj )else breakfigure small change to quicksortwhich breaks the algorithm some side effectsthis program will not work if you try to use it without change with simple pivoting strategybecause and start in the wrong place and there is no longer sen... |
23,611 | we telescopeusing equation ( repeatedly thust( ( ( ( ( ( ( ( ( ( ( ( adding up all these equations yields (nt( ( ( = as claimed earlier to see that this is the worst possible casenote that the total cost of all the partitions in recursive calls at depth must be at most since the recursion depth is at most nthis gives ... |
23,612 | sorting average-case analysis this is the most difficult part for the average casewe assume that each of the sizes for is equally likelyand hence has probability / this assumption is actually valid for our pivoting and partitioning strategybut it is not valid for some others partitioning strategies that do not preserve... |
23,613 | adding equations ( through ( yields (nt( + + ( = the sum is about loge ( where is known as euler' constantso (no(log nn+ ( (no( log ( and so although this analysis seems complicatedit really is not--the steps are natural once you have seen some recurrence relations the analysis can actually be taken further the highly ... |
23,614 | sorting really saving recursive call the average running timehoweveris (nthe analysis is similar to quicksort' and is left as an exercise the implementation of quickselect is even simpler than the abstract description might imply the code to do this is shown in figure when the algorithm terminatesthe /*internal selecti... |
23,615 | kth smallest element is in position (because arrays start at index this destroys the original orderingif this is not desirablethen copy must be made using median-of-three pivoting strategy makes the chance of the worst case occurring almost negligible by carefully choosing the pivothoweverwe can eliminate the quadratic... |
23,616 | sorting < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < figure decision tree for three-element sort leaf in our casethis algorithm uses three comparisons in the worst case the average number of comparisons used is equal to the average depth of the leaves since... |
23,617 | theorem any sorting algorithm that uses only comparisons between elements requires at least log( !)comparisons in the worst case proof decision tree to sort elements must have nleaves the theorem follows from the preceding lemma theorem any sorting algorithm that uses only comparisons between elements requires ( log n... |
23,618 | sorting the lower bounds for all these problemswith the exception of finding the medianare tightalgorithms exist that use exactly the specified number of comparisons in all our proofswe assume all items are unique lemma if all the leaves in decision tree are at depth or higherthe decision tree must have at least leaves... |
23,619 | yes < no tn ty tree ty tree figure smallest three elements are abc }largest four elements are defg }the comparison between and for this choice of and can be eliminated when forming tree tkth smallest form new decision treetby purging any comparisons in between an element in and an element in since any element in is sma... |
23,620 | sorting theorem any comparison-based algorithm to find the median must use at least / (log ncomparisons proof apply theorem with / the lower bound for selection is not tightnor is it the best knownsee the references for details adversary lower bounds although decision-tree arguments allowed us to show lower bounds ... |
23,621 | new proof begin by marking each item as (for unknownwhen an item is declared larger than another itemwe will change its marking to (for eliminatedthis change represents one unit of information initially each unknown item has value of but there have been no comparisonsso this ordering is consistent with prior answers co... |
23,622 | sorting theorem any comparison-based algorithm to find the minimum and maximum must use at least / comparisons proof the basic idea is that if two items are unmarkedthe adversary must give out two pieces of information otherwiseone of the items has either or an (perhaps bothin that casewith reasonable carethe adversa... |
23,623 | otherwiseboth items involved in the comparison are marked if both are wlthen we can answer consistently with the current assignmentyielding no information otherwiseat least one of the items has only an or only we will allow that item to compare redundantly (if it is an then it loses againif it is then it wins again)and... |
23,624 | sorting make decisions based on ordering information only naturallyif there is extra information availablewe should expect to find more efficient algorithmsince otherwise the extra information would be wasted although bucket sort seems like much too trivial an algorithm to be usefulit turns out that there are many case... |
23,625 | /radix sort an array of strings assume all are all ascii assume all have same length *void radixsortavector arrint stringlen const int buckets vectorbucketsbuckets )forint pos stringlen pos > --pos forstring arr bucketsspos push_backstd::moves )int idx forauto thisbucket buckets forstring thisbucket arridx+std::moves )... |
23,626 | sorting /counting radix sort an array of strings assume all are all ascii assume all have same length *void countingradixsortvectro arrint stringlen const int buckets int arr size)vector buffern )vector *in &arrvector *out &bufferforint pos stringlen pos > --pos vector countbuckets )forint ++ ++count(*in) pos ]forint <... |
23,627 | /radix sort an array of strings assume all are all ascii assume all have length bounded by maxlen *void radixsortvector arrint maxlen const int buckets vectorwordsbylengthmaxlen )vectorbucketsbuckets )forstring arr wordsbylengths lengthpush_backstd::moves )int idx forauto wordlist wordsbylength forstring wordlist arrid... |
23,628 | sorting posby making use of the variable startingindexwhich is maintained at lines and except for thatlines to in figure are the same as lines to in figure the running time of this version of radix sort is linear in the total number of characters in all the strings (each character appears exactly once at line and the s... |
23,629 | the simple algorithm the basic external sorting algorithm uses the merging algorithm from mergesort suppose we have four tapesta ta tb tb which are two input and two output tapes depending on the point in the algorithmthe and tapes are either input tapes or output tapes suppose the data are initially on ta suppose furt... |
23,630 | ta ta tb tb ta ta tb tb sorting multiway merge if we have extra tapesthen we can expect to reduce the number of passes required to sort our input we do this by extending the basic (two-waymerge to -way merge merging two runs is done by winding each input tape to the beginning of each run then the smaller element is fou... |
23,631 | ta ta ta tb tb tb after the initial run construction phasethe number of passes required using -way merging is logk ( / )because the runs get times as large in each pass for the example abovethe formula is verifiedsince log ( / ) if we have tapesthen and our large example from the previous section would require log ... |
23,632 | sorting runs and the other tapes are empty we must copy one run to another tapeand then we can finish the merge it turns out that the first distribution we gave is optimal if the number of runs is fibonacci number fn then the best way to distribute them is to split them into two fibonacci numbers fn- and fn- otherwisei... |
23,633 | elements in heap array output next element read end of run rebuild heap end of run end of tape [ [ [ run run run rebuild heap figure example of run construction summary sorting is one of the oldest and most well-studied problems in computing for most general internal sorting applicationsan insertion sortshellsortmerges... |
23,634 | sorting suppose we exchange elements [iand [ + ]which were originally out of order prove that at least and at most inversions are removed show the result of running shellsort on the input using the increments { what is the running time of shellsort using the two-increment sequence { } show that for any nthere exists ... |
23,635 | sort using quicksort with median-of-three partitioning and cutoff of using the quicksort implementation in this determine the running time of quicksort for sorted input reverse-ordered input random input repeat exercise when the pivot is chosen as the first element the larger of the first two distinct elements random e... |
23,636 | sorting prove that the worst-case running time of this algorithm is ( log nc conduct experiments to determine how often heapsort gets called implement this technique in conjunction with tail-recursion removal in exercise explain why the technique in exercise would no longer be needed when implementing quicksortif the a... |
23,637 | show that this algorithm is suboptimalregardless of the choices for algorithms aband write program that reads points in plane and outputs any group of four or more colinear points ( points on the same linethe obvious brute-force algorithm requires ( time howeverthere is better algorithm that makes use of sorting and ru... |
23,638 | sorting prove that comparisons are required to sort elements using any comparisonbased algorithm give an algorithm to sort elements with comparisons write an efficient version of shellsort and compare performance when the following increment sequences are useda shell' original sequence hibbard' increments knuth' increm... |
23,639 | possiblewithout testing whether the new cell can be inserted this would place the new cell in leaf and probably violate the heap orderto fix the heap orderpercolate the new cell up in the normal manner write routine to include this ideaand compare the running time with standard implementation of heapsort propose an alg... |
23,640 | sorting heapsort was invented by williams [ ]floyd [ provided the linear-time algorithm for heap construction theorem is from [ an exact average-case analysis of mergesort has been described in [ an algorithm to perform merging in linear time without extra space is described in [ quicksort is from hoare [ this paper an... |
23,641 | janson and knuth"shellsort with three increments,random structures and algorithms ( ) - jiangm liand vitanyi" lower bound on the average-case complexity of shellsort,journal of the acm ( ) - knuththe art of computer programming volume sorting and searching ed addison-wesleyreadingmass manacher"the ford-johnson sorting ... |
23,642 | |
23,643 | the disjoint sets class in this we describe an efficient data structure to solve the equivalence problem the data structure is simple to implement each routine requires only few lines of codeand simple array can be used the implementation is also extremely fastrequiring constant average time per operation this data str... |
23,644 | the disjoint sets class the dynamic equivalence problem given an equivalence relation ~the natural problem is to decidefor any and bif if the relation is stored as two-dimensional array of boolean variablesthenof coursethis can be done in constant time the problem is that the relation is usually not explicitlybut rathe... |
23,645 | be determined easily by some hashing scheme thusinitially we have si {ifor through our second observation is that the name of the set returned by find is actually fairly arbitrary all that really matters is that find( )==find(bis true if and only if and are in the same set these operations are important in many graph t... |
23,646 | the disjoint sets class figure eight elementsinitially in different sets easybecause the only information we will need is parent link the name of set is given by the node at the root since only the name of the parent is requiredwe can assume that this tree is stored implicitly in an arrayeach entry [iin the array repre... |
23,647 | figure after union( , - - - - - figure implicit representation of previous tree time of find is (ntypicallythe running time is computed for sequence of intermixed instructions in this casem consecutive operations could take (mntime in the worst case the code in figures through represents an implementation of the basi... |
23,648 | the disjoint sets class /*construct the disjoint sets object numelements is the initial number of disjoint sets *disjsets::disjsetsint numelements snumelements figure disjoint sets initialization routine /*union two disjoint sets for simplicitywe assume root and root are distinct and represent set names root is the roo... |
23,649 | of coursethe implication of this model is that there is only chance that the next union will involve the large tree another model might say that all unions between any two elements in different trees are equally likelyso larger tree is more likely to be involved in the chance that the large next union than smaller tree... |
23,650 | the disjoint sets class figure result of an arbitrary union figure worst-case tree for the size of its tree thusinitially the array representation of the tree is all - when union is performedcheck the sizesthe new size is the sum of the old thusunion-by-size is not at all difficult to implement and requires no extra sp... |
23,651 | - - - - - - - - figure forest with implicit representation for union-by-size and union-by-height /*union two disjoint sets for simplicitywe assume root and root are distinct and represent set names root is the root of set root is the root of set *void disjsets::unionsetsint root int root ifsroot sroot /root is deeper s... |
23,652 | the disjoint sets class path compression the union/find algorithmas described so faris quite acceptable for most cases it is very simple and linear on average for sequence of instructions (under all modelshoweverthe worst case of ( log ncan occur fairly easily and naturally for instanceif we put all the sets on queue a... |
23,653 | /*perform find with path compression error checks omitted again for simplicity return the set containing *int disjsets::findint ifsx return xelse return sx findsx )figure code for disjoint sets find with path compression operations requires at most ( log ntime it is still an open problem to determine what the average-c... |
23,654 | the disjoint sets class slowly growing functions consider the recurrence ( tf( <= > ( in this equationt(nrepresents the number of timesstarting at nthat we must iteratively apply (nuntil we reach (or lesswe assume that (nis nicely defined function that reduces call the solution to the equation (nwe have already encount... |
23,655 | figure large disjoint set tree (numbers below nodes are rankswe begin by establishing two lemmas concerning the properties of the ranks figure gives visual picture of both lemmas lemma when executing sequence of union instructionsa node of rank must have at least one child of rank proof by induction the basis is clearl... |
23,656 | the disjoint sets class partial path compression algorithm is our standard sequence of union-by-rank and find with path compression operations we design an algorithm that will perform the exact same sequence of path compression operations as algorithm in algorithm bwe perform all the unions prior to any find then each ... |
23,657 | bottom figure recursive decompositioncase partial find is entirely in bottom top bottom figure recursive decompositioncase partial find is entirely in top top bottom figure recursive decompositioncase partial find goes from bottom to top |
23,658 | the disjoint sets class in figure the partial find resides entirely in the top half thus one partial find in the top half corresponds to one original partial findand the charges can be recursively assigned to the top half howeverwe run into lots of trouble when we reach figure here is in the bottom halfand is in the to... |
23,659 | top bottom figure recursive decompositioncase the bottom node new parents are the nodes themselves mwould no longer be the same fortunatelythere is simpler argumentsince each node on the bottom can have its parent set to itself only oncethe number of charges are limited by the number of nodes on the bottom whose parent... |
23,660 | the disjoint sets class for the number of parent changes by splitting the charges into the top and bottom groups one of the key ideas is that recursive formula is written not only in terms of and nwhich would be obviousbut also in terms of the maximum rank in the group lemma let (mnrbe the number of parent changes for ... |
23,661 | theorem (mnrm log proof we start with lemma (mnrc(mt nt rc(mb nb smt ( )nt ( observe that in the top halfthere are only nodes of rank + + rand thus no node can have its parent change more than ( - - times this yields trivial bound of nt ( - - for (mt nt rthusc(mnrnt ( (mb nb smt ( )nt ( combining termsc(mnrnt ( (mb nb ... |
23,662 | the disjoint sets class proof from lemma we havec(mnrc(mt nt rc(mb nb smt ( )nt ( and by theorem (mt nt rmt nt log thusc(mnrmt nt log (mb nb smt ( )nt ( rearranging and combining terms yields (mnrc(mb nb mt ( log )nt ( so choose log clearlythis choice implies that ( log and thus we obtain (mnrc(mb nb log mt ( rearrangi... |
23,663 | rearranging as in theorems and we obtain (mnr (mb nb logr mb ( this timelet (mnrc(mnr mthen (mnrd(mb nb logrn ( which implies (mnrn log* this yields (mnr log* needless to saywe could continue this ad infinitim thus with bit of mathwe get progression of boundsc(mnr logr (mnr log* (mnr log** (mnr log*** (mnr log**** each... |
23,664 | the disjoint sets class an application an example of the use of the union/find data structure is the generation of mazessuch as the one shown in figure in figure the starting point is the top-left cornerand the ending point is the bottom-right corner we can view the maze as -by- rectangle of cells in which the top-left... |
23,665 | { { { { { { { { { { { { { { { { { { { { { { { { { figure initial stateall walls upall cells in their own set containing and are combined via union operation this is because everything that was connected to is now connected to everything that was connected to at the end of the algorithmdepicted in figure everything is c... |
23,666 | the disjoint sets class { { { { { { { { { { { { figure wall between squares and is randomly selected in figure this wall is knocked downbecause and are not already connectedtheir sets are merged { figure eventually walls are knocked downall elements are in the same set summary we have seen very simple data structure to... |
23,667 | exercises show the result of the following sequence of instructionsunion( , )union( , )union( , )union( , )union( , )union( , )union( , )union( , )union ( , )union( , )union( , )union( , )union( , )union( , )union ( , )union( when the unions are performed arbitrarily performed by height performed by size for each of th... |
23,668 | the disjoint sets class show how to implement all three operations so that the sequence of operations takes ( log /log log ntime suppose we want to add an extra operationremove( )which removes from its current set and places it in its own show how to modify the union/find algorithm so that the running time of seque... |
23,669 | in by factor of (mnotherssuch as [ and the graph connectivity problem in this are unaffected the paper lists examples tarjan has used path compression to obtain efficient algorithms for several graph problems [ average-case results for the union/find problem appear in [ ][ ][ ]and [ results bounding the running time of... |
23,670 | the disjoint sets class tarjan"applications of path compression on balanced trees,journal of the acm ( ) - tarjan and van leeuwen"worst-case analysis of set union algorithms,journal of the acm ( ) - van kreveld and overmars"union-copy structures and dynamic segment trees,journal of the acm ( ) - westbrook and tarjan"am... |
23,671 | graph algorithms in this we discuss several common problems in graph theory not only are these algorithms useful in practicethey are also interesting because in many real-life applications they are too slow unless careful attention is paid to the choice of data structures we will show several real-life problemswhich ca... |
23,672 | graph algorithms an undirected graph is connected if there is path from every vertex to every other vertex directed graph with this property is called strongly connected if directed graph is not strongly connectedbut the underlying graph (without direction to the arcsis connectedthen the graph is said to be weakly conn... |
23,673 | one simple way to represent graph is to use two-dimensional array this is known as an adjacency matrix representation for each edge (uv)we set [ ][vto trueotherwise the entry in the array is false if the edge has weight associated with itthen we can set [ ][vequal to the weight and use either very large or very small w... |
23,674 | graph algorithms (empty figure an adjacency list representation of graph in most of the we present the graph algorithms using pseudocode we will do this to save space andof courseto make the presentation of the algorithms much clearer at the end of section we provide working +implementation of routine that makes underl... |
23,675 | cap mad cop mac mad mad cop cop cop cop cis cop cda cop cop cda cop figure an acyclic graph representing course prerequisite structure figure an acyclic graph vertex is storedand that the graph is read into an adjacency listwe can then apply the algorithm in figure to generate topological ordering the function findnewv... |
23,676 | graph algorithms void graph::topsortforint counter counter num_verticescounter+vertex findnewvertexofindegreezero)ifv =not_a_vertex throw cyclefoundexception} topnum counterfor each vertex adjacent to indegree--figure simple topological sort pseudocode graph is sparsewe would expect that only few vertices have their in... |
23,677 | void graph::topsortqueue qint counter makeempty)for each vertex ifv indegree = enqueuev )while! isemptyvertex dequeue) topnum ++counter/assign next number for each vertex adjacent to if-- indegree = enqueuew )ifcounter !num_vertices throw cyclefoundexception}figure pseudocode to perform topological sort pseudocode impl... |
23,678 | graph algorithms shortest-path algorithms in this section we examine various shortest-path problems the input is weighted graphassociated with each edge (vi vj is cost ci, to traverse the edge the cost of path vn is - = ci, + this is referred to as the weighted path length the unweighted path length is merely the numbe... |
23,679 | cost but shorter path exists by following the loop which has cost - this path is still not the shortestbecause we could stay in the loop arbitrarily long thusthe shortest path between these two points is undefined similarlythe shortest path from to is undefinedbecause we can get into the same loop this loop is known as... |
23,680 | figure an unweighted directed graph figure graph after marking the start node as reachable in zero edges figure graph after finding all vertices whose path length from is |
23,681 | figure graph after finding all vertices whose shortest path is already known this search tells us that the shortest path to and is figure shows the progress that has been made so far finally we can findby examining vertices adjacent to the recently evaluated and that and have shortest path of three edges all vertices h... |
23,682 | graph algorithms known dv pv figure initial configuration of table used in unweighted shortest-path computation guarantee that no cheaper path will ever be foundand so processing for that vertex is essentially complete the basic algorithm can be described in figure the algorithm in figure mimics the diagrams by declari... |
23,683 | figure bad case for unweighted shortest-path algorithm using figure by tracing back through the pv variablethe actual path can be printed we will see how when we discuss the weighted case the running time of the algorithm is (| | )because of the doubly nested for loops an obvious inefficiency is that the outside loop c... |
23,684 | graph algorithms void graph::unweightedvertex queue qfor each vertex dist infinitys dist enqueues )while! isemptyvertex dequeue)for each vertex adjacent to ifw dist =infinity dist dist path vq enqueuew )figure psuedocode for unweighted shortest-path algorithm we keep all of the same information as before thuseach verte... |
23,685 | initial state dequeued dequeued dequeued known dv pv known dv pv known dv pv known dv pv qv dequeued dequeued dequeued dequeued known dv pv known dv pv known dv pv known dv pv qv empty figure how the data change during the unweighted shortest-path algorithm casethen we should set dw dv cv, if this new value for dw woul... |
23,686 | graph algorithms figure the directed graph (againv known dv pv figure initial configuration of table used in dijkstra' algorithm known dv pv figure after is declared known the next vertex selected is at cost is the only adjacent vertexbut it is not adjustedbecause then is selectedand the distance for is adjusted down t... |
23,687 | known dv pv figure after is declared known known dv pv figure after is declared known known dv pv figure after and then are declared known finallyv is selected the final table is shown in figure figure graphically shows how edges are marked known and vertices updated during dijkstra' algorithm to print out the actual p... |
23,688 | graph algorithms known dv pv figure after is declared known known dv pv figure after is declared known and algorithm terminates the path can be printed out using the recursive routine in figure the routine recursively prints the path all the way up to the vertex before on the pathand then just prints this works because... |
23,689 | figure stages of dijkstra' algorithm selection of the vertex is deletemin operationsince once the unknown minimum vertex is foundit is no longer unknown and must be removed from future consideration the update of ' distance can be implemented two ways one way treats the update as decreasekey operation the time to find ... |
23,690 | graph algorithms /*pseudocode sketch of the vertex structure in real ++path would be of type vertex *and many of the code fragments that we describe require either dereferencing or use the -operator instead of the operator needless to saythis obscures the basic algorithmic ideas *struct vertex list adj/adjacency list b... |
23,691 | void graph::dijkstravertex for each vertex dist infinityv known falses dist whilethere is an unknown distance vertex vertex smallest unknown distance vertexv known truefor each vertex adjacent to if! known disttype cvw cost of edge from to wifv dist cvw dist /update decreasew dist to dist cvw ) path vfigure pseudocode ... |
23,692 | graph algorithms fibonacci heap when this is usedthe running time is (| |+|vlog | |fibonacci heaps have good theoretical time bounds but fair amount of overheadso it is not clear whether using fibonacci heaps is actually better in practice than dijkstra' algorithm with binary heaps to datethere are no meaningful averag... |
23,693 | void graph::weightednegativevertex queue qfor each vertex dist infinitys dist enqueues )while! isemptyvertex dequeue)for each vertex adjacent to ifv dist cvw dist /update dist dist cvww path vifw is not already in enqueuew )figure pseudocode for weighted shortest-path algorithm with negative edge costs application migh... |
23,694 | graph algorithms ( ( ( start ( ( ( finish ( ( ( figure activity-node graph time past units on the other handactivity is less critical and can be delayed up to two time units without affecting the final completion time to perform these calculationswe convert the activity-node graph to an event-node graph each event corr... |
23,695 | completion time for all nodes in the graph if eci is the earliest completion time for node ithen the applicable rules are ec ecw max (ecv cv, ( , ) figure shows the earliest completion time for each event in our example event-node graph we can also compute the latest timelci that each event can finish without affecting... |
23,696 | graph algorithms / / / / / / / / / / / / / / / / / / figure earliest completion timelatest completion timeand slack figure shows the slack (as the third entryfor each activity in the event-node graph for each nodethe top number is the earliest completion time and the bottom entry is the latest completion time some acti... |
23,697 | /runs the shortest path calculation from the adjacency mapreturning vector /that contains the sequence of word changes to get from first to second unordered_map findchainconst unordered_mapadjacentwordsconst string firstconst string second unordered_map previouswordqueue qq pushfirst )while! emptystring current front) ... |
23,698 | graph algorithms single-source shortest-path algorithm has completed these two routines are both shown in figure the first routine is findchainwhich takes the map representing the adjacency lists and the two words to be connected and returns map in which the keys are wordsand the corresponding value is the word prior t... |
23,699 | figure graph (leftand its maximum flow combine and distribute flow in any manner that it likesas long as edge capacities are not violated and as long as flow conservation is maintained (what goes in must come outlooking at the graphwe see that has edges of capacities and leaving itand has edges of capacities and enteri... |
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