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d0f71d870ec790b2b3ce5a7a2150b41eb796740d | subsection | 10 | 140 | High-Level Overview of the Algorithm | We claim that if we could find the collection \left\lbrace I_1,\ldots ,I_z \right\rbrace of the intervals of the first row, a collection {\mathcal {Q}} of sub-grids of G, a coloring \chi :{\mathcal {Q}}\rightarrow \left\lbrace c_1,\ldots ,c_z \right\rbrace , and a subset {\mathcal {M}}^{\prime }\subseteq {\mathcal {M}}... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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b8472d67c726f115cc60330f72a5df2f150c4a1e | subsection | 11 | 140 | High-Level Overview of the Algorithm | The setup for the algorithm consists of three ingredients: (i) a hierarchical decomposition \tilde{\mathcal {H}} of the grid into square sub-grids (that we refer to as squares); (ii) a hierarchical partition {\mathfrak {I}} of the first row R^* of the grid into intervals; and (iii) a hierarchical coloring f of the squa... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
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38390894e59ceb1bed41082db9960f554aea5b1c | subsection | 12 | 140 | A Hierarchical System of Squares | A hierarchical system \tilde{\mathcal {H}} of squares consists of a sequence {\mathcal {Q}}_1,{\mathcal {Q}}_2,\ldots ,{\mathcal {Q}}_{\rho } of sets of sub-grids of G. For each 1\le h\le \rho ,
{\mathcal {Q}}_h is a collection of disjoint sub-grids of G (that we refer to as level-h squares); every such square Q\in {\m... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
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5a05036b4c7bc365f7d59999a27600785e7086e8 | subsection | 13 | 140 | A Hierarchical Partition of the Top Grid Boundary | Recall that R^* denotes the first row of the grid. A hierarchical partition {\mathfrak {I}} of R^* is a sequence {\mathcal {I}}_1,{\mathcal {I}}_2,\ldots ,{\mathcal {I}}_{\rho } of sets of sub-paths of R^*, such that for each 1\le h\le \rho , the paths in {\mathcal {I}}_h (that we refer to as level-h intervals) partiti... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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f45021f88186dd2e9dffab7d12de24051e4070bc | subsection | 14 | 140 | Coloring the Squares and Selecting Demand Pairs to Route | The third ingredient of our algorithm is an assignment f of colors to the squares, and a selection of a subset of the demand pairs to be routed. For every level 1\le h\le \rho , for every level-h square Q\in {\mathcal {Q}}_h, we would like to assign a single level-h color c_h(I)\in \chi _h to Q, denoting f(Q)=c_h(I). I... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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2e1addef1abf0e1a76db73c563ca1eaa6699da96 | subsection | 15 | 140 | Coloring the Squares and Selecting Demand Pairs to Route | We say that ({\mathfrak {I}}, f,{\mathcal {M}}^{\prime }) is a good ensemble iff {\mathfrak {I}} is a hierarchical partition of R^* into intervals; f is a valid coloring of the squares in \tilde{\mathcal {H}} with respect to {\mathfrak {I}}; and {\mathcal {M}}^{\prime }\subseteq \tilde{\mathcal {M}} is a subset of the ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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5bf5ae7c9bf9104fa026d5130c1fbda4948a525c | subsection | 16 | 140 | The Routing | We show that, if we are given a good ensemble ({\mathfrak {I}}, f,{\mathcal {M}}^{\prime }), then we can route all demand pairs in {\mathcal {M}}^{\prime }. The routing itself follows the high-level idea outlined above. We gradually construct a collection {\mathcal {P}} of node-disjoint paths routing the demand pairs i... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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2eb9a5db9eb68cf433b8ee5d75570e244ec54305 | subsection | 17 | 140 | The Existence of the Ensemble | The key notion that we use in order to show that a large good ensemble ({\mathfrak {I}}, f,{\mathcal {M}}^{\prime }) exists is that of a shadow property. Suppose Q is some (d\times d) sub-grid of G, and let \hat{\mathcal {M}}\subseteq {\mathcal {M}} be some subset of the demand pairs. Among all demand pairs (s,t)\in \h... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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afae6b8e8d6a9eff7a911c843382e32307ea4965 | subsection | 18 | 140 | The Existence of the Ensemble | The intuition is that, if (s,t)\in {\mathcal {M}}^* is a demand pair whose source lies in the shadow of Q, and destination lies outside of D, then P(s,t) must cross the path P, as it needs to escape the disc D. Since path P is relatively short, only a small number of such demand pairs may exist. The main difficulty wit... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
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] | 2,018 | en | Computer Science | [
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87974914794efd29b3ce4645135ec5c93e0d7a1e | subsection | 19 | 140 | The Existence of the Ensemble | Let I be that interval. Then we color the square Q with the level-1 color c_1(I) corresponding to the interval I. This completes the first iteration. Notice that for each level-1 color c_1(I), at most d_2/16 demand pairs (s,t)\in {\mathcal {M}}^{\prime } have s\in I. In the following iteration, we similarly partition e... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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c29191aba5ccbe9c681126c373c98f72b0f78572 | subsection | 20 | 140 | Finding the Good Ensemble | In our final step, our goal is to find a good ensemble ({\mathfrak {I}}, f, {\mathcal {M}}^{\prime }) maximizing |{\mathcal {M}}^{\prime }|. We show an efficient randomized 2^{O(\sqrt{\log n}\cdot \log \log n)}-approximation algorithm for this problem. First, we show that, at the cost of losing a small factor in the ap... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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59f7e26a2332b187e2d34dd44a63e8798faa8f9e | subsection | 21 | 140 | Completing the Proof of Theorem | So far we have assumed that all source vertices lie on the top boundary of the grid, and all destination vertices are at a distance at least \Omega (\mathsf {OPT}) from the grid boundary. Let {\mathcal {A}} be the randomized efficient 2^{O(\sqrt{\log n}\cdot \log \log n)}-approximation algorithm for this special case. ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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e138ecfd4355551cb90d9592d7cdae3cf18067e2 | subsection | 22 | 140 | Completing the Proof of Theorem | First, we do not know the set {\mathcal {Z}} of sub-grids of G and the set {\mathcal {I}} of intervals of R^*. Second, it is not clear how to solve each resulting problem (G,{\mathcal {M}}_j). To address the latter problem, we define a simple mapping of all source vertices in S({\mathcal {M}}_j) to the top boundary of ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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a459c853310e3fd95ac44201be62f0188a12410a | subsection | 23 | 140 | Preliminaries | All logarithms in this paper are to the base of 2.For a pair h,\ell > 0 of integers, we let G^{h,\ell } denote the grid of height h and length \ell .
The set of its vertices is V(G^{h,\ell })=\left\lbrace v(i,j)\mid 1\le i\le h, 1\le j\le \ell \right\rbrace , and the set of its edges is the union of two subsets: the se... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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7bacb70bcd955a87c54186508d60048ed4db6b09 | subsection | 24 | 140 | Preliminaries | We say that \Upsilon =\Upsilon _G({\mathcal {R}},{\mathcal {W}}) is the sub-grid of G spanned by the set {\mathcal {R}} of rows and the set {\mathcal {W}} of columns. A sub-graph G^{\prime }\subseteq G is called a sub-grid of G iff there is a set {\mathcal {R}} of consecutive rows and a set {\mathcal {W}} of consecutiv... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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b1fa5ae3f32d04d604949abaaf24230903fb4218 | subsection | 25 | 140 | The Algorithm | Throughout, we assume that we know the value \mathsf {OPT} of the optimal solution; in order to do so, we simply guess the value \mathsf {OPT} (that is, we go over all such possible values), and run our approximation algorithm for each such guessed value. It is enough to show that the algorithm returns a factor-2^{O(\s... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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f54e8e0fe2a22edea8d936fecad2519f9a928a05 | subsection | 26 | 140 | The Algorithm | We then solve each of the resulting instances (G,{\mathcal {M}}_{q,q^{\prime }}) separately, and return the best of the resulting solutions. Since one of these instances is guaranteed to have a solution of value at least \mathsf {OPT}/16, it is enough to show a factor-2^{O(\sqrt{\log n}\log \log n)}-approximation algor... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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38acdfcf4658ab1b0d9e01f36b0d84dd96b2ecc2 | subsection | 27 | 140 | The Algorithm | Moreover, we can assume that \mathsf {OPT}^{\prime }>2^{13}d, since otherwise we can directly apply Theorem REF to instance (G,{\mathcal {M}}) with parameter \mathsf {OPT}=d to obtain an 2^{O(\sqrt{\log n}\log \log n)}-approximation. We now consider three cases, depending on the location of the boundary edges \Gamma _q... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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db4d1921ee3874f75ae150fbd5eeb1f9e2cb1c6c | subsection | 28 | 140 | Parameters. | The following parameters are used throughout the algorithm. Let c^*\ge 2 be a large constant, whose value will be set later.
Recall that \eta =2^{\left\lceil \sqrt{\log n}\right\rceil }. Let \rho be the largest integer, for which \eta ^{\rho +2}\le \mathsf {OPT}/2^{c^*\sqrt{\log n}\log \log n}.
Intuitively, we will rou... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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e3864d163f4375e67477a38add9a928a175df475 | subsection | 29 | 140 | Parameters. | Clearly, one of these instances has a solution of value at least \mathsf {OPT}/2, as for each demand pair (s,t)\in {\mathcal {M}}, both s and t must belong to either {\mathcal {M}}^{\prime } or to {\mathcal {M}}^{\prime \prime } (or both). We assume without loss of generality that the problem induced by the set {\mathc... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
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-0.009886234067380428,
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105e66ba80340ca61447f605c9a55001b130093c | subsection | 30 | 140 | Hierarchical Systems of Squares | A subset I\subseteq [\ell ^{\prime }] of consecutive integers is called an interval. We say that two intervals I,I^{\prime } are disjoint iff I\cap I^{\prime }=\emptyset , and we say that they are d-separated iff for every pair of integers i\in I, j\in I^{\prime }, |i-j|>d. A collection {\mathcal {I}} of intervals of [... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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cc34e6e8dd5ea4f04178027756920b1f82cc403d | subsection | 31 | 140 | Hierarchical Systems of Squares | A hierarchical system of squares is a sequence \tilde{\mathcal {H}}=({\mathcal {Q}}_1,{\mathcal {Q}}_2,\ldots ,{\mathcal {Q}}_{\rho }) of sets of squares, such that:for all 1\le h\le \rho , {\mathcal {Q}}_h is a d_h-canonical family of squares; and
for all 1<h\le \rho , for every square Q\in {\mathcal {Q}}_h, there is... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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e830521f2e3ae696947bc850aeae93113cee7d49 | subsection | 32 | 140 | Hierarchical Systems of Squares | Clearly, there is an index i^*, such that |{\mathcal {M}}^*_{i^*}|\ge |{\mathcal {M}}^*|/4^{\rho }. Even though we do not know the index i^*, we can run our approximation algorithm for each possible value of i^*. It is enough to show that our algorithm finds a 2^{O(\sqrt{\log n}\cdot \log \log n)}-approximate solution ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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e5fa7c404bc3383941846d9c12b491ef4d01dff5 | subsection | 33 | 140 | The Shadow Property | Definition. Let Q\subseteq \tilde{G} be a square sub-grid of \tilde{G} of size (d\times d), and let \hat{\mathcal {M}}\subseteq {\mathcal {M}} be any subset of the demand pairs, such that all vertices in S(\hat{\mathcal {M}}) are distinct. The shadow of Q with respect to \hat{\mathcal {M}}, J_{\hat{\mathcal {M}}}(Q), i... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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91d87b16d4191e54c6dd4d917b0c9eec97870e80 | subsection | 34 | 140 | The Shadow Property | Then there is an efficient algorithm to find a subset \hat{\mathcal {M}}^{\prime }\subseteq \hat{\mathcal {M}} of at least \left\lfloor \frac{\beta _1|\hat{\mathcal {M}}|}{4\beta _2}\right\rfloor demand pairs, such that every square in {\mathcal {Q}} has the \beta _1-shadow property with respect to \hat{\mathcal {M}}^{... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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27a1c8f8a53a7d420616e05104ad85d11a4ad0f2 | subsection | 35 | 140 | The Shadow Property | The input to the jth iteration is a subset {\mathcal {M}}_{j-1}\subseteq {\mathcal {M}}^{**} of demand pairs, where the input to the first iteration is {\mathcal {M}}_0= {\mathcal {M}}^{**}. In order to execute the jth iteration, we apply Theorem REF to the set {\mathcal {Q}}_j of squares and the set {\mathcal {M}}_{j-... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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495681af1afe69ea412cfca95deeea67eac234b3 | subsection | 36 | 140 | The Shadow Property | We say that a disc D in the plane is canonical, if either (i) D=D^0, or (ii) we can partition the boundary \sigma (D) of D into four contiguous segments, \sigma _1(D),\sigma _2(D),\sigma _3(D),\sigma _4(D), such that \sigma _1(D)\subseteq R^*; \sigma _3(D) is contained in the boundary of some square Q\in {\mathcal {Q}}... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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46f62b5cbfdabae424e0e107dd00b33fe34a35d4 | subsection | 37 | 140 | The Shadow Property | We say that the set {\mathcal {D}} of discs respects the active squares, iff for every active square Q\in {\mathcal {Q}}, there is a unique disc D^i\in {\mathcal {D}} with Q\subseteq R_{{\mathcal {D}}}(D^i). We will ensure that the set {\mathcal {D}} of discs we construct respects all active squares. | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.... | |
6d351e7f6b0e4756eead36175ff0db2283e7438b | subsection | 38 | 140 | Hierarchical Decomposition of Row | Recall that R^* is the first row of the grid G^{\prime }.
We start with intuition. Recall that \tilde{\mathcal {M}}^* is the set of the demand pairs from Corollary REF . In general, we would like to define a hierarchical partition ({\mathcal {J}}_1,{\mathcal {J}}_2,\ldots ,{\mathcal {J}}_{\rho }) of R^*, that has the f... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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19ed6ec628742619fc2510370181945c73d8d162 | subsection | 39 | 140 | Hierarchical Decomposition of Row | Note that for a fixed sequence L=(\ell _1,\ell _2,\ldots ,\ell _r) of integers, the corresponding hierarchical L-decomposition ({\mathcal {J}}_1,\ldots ,{\mathcal {J}}_r) of R^* is unique.Assume now that we are also given a collection {\mathcal {M}}^{\prime }\subseteq {\mathcal {M}} of demand pairs. We say that {\mathc... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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249a75dfa34e1e62e9094c91fec011b474409e22 | subsection | 40 | 140 | Hierarchical Decomposition of Row | Along the way, we will also define a sequence \gamma _{\rho ^{\prime }},\ldots ,\gamma _1 of integers that will be helpful for us later.Consider first the sequence L^{\prime }=(\ell _1^{\prime },\ell _2^{\prime },\ldots ,\ell _{\rho ^{\prime }}) of integers, where for 1\le h\le \rho ^{\prime }, \ell _h^{\prime }=\eta ^... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
bf939f448b9ce6c5925f1d74d32fa9379b52c6ac | subsection | 41 | 140 | Hierarchical Decomposition of Row | It is easy to verify that:|\tilde{\mathcal {M}}^{**}|\ge \frac{|\tilde{\mathcal {M}}^*|}{\left\lceil \log n\right\rceil ^{\rho ^{\prime }}}\ge \frac{\mathsf {OPT}}{2^{8\sqrt{\log n}\cdot \log \log n}\cdot \left\lceil \log n\right\rceil ^{\sqrt{\log n}}}\ge \frac{\mathsf {OPT}}{2^{10\sqrt{\log n}\cdot \log \log n}}.Reca... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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b4f8919aa2fc2e6fce3a3843412e2a3645ca9729 | subsection | 42 | 140 | Hierarchical Decomposition of Row | From the above discussion, such an integer exists for all 1\le j\le \rho , and we have that d_j/(16\eta )\le \gamma _h\le d_j/8. In particular for each interval I\in {\mathcal {J}}_h, if I\cap S(\tilde{\mathcal {M}}^{**})\ne \emptyset , then d_j/(16\eta )\le |I\cap S(\tilde{\mathcal {M}}^{**})|\le d_j/4.It is now easy ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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06bbaa50bfce5c35c66d5089fbe9adb5da532f74 | subsection | 43 | 140 | Hierarchical System of Colors | For every level 1\le h\le \rho , for every level-h interval I\in {\mathcal {J}}_h, we introduce a distinct level-h color c_h(I), and we color all vertices of I with this color. We let \chi _h be the set of all level-h colors, so \chi _h=\left\lbrace c_h(I)\mid I\in {\mathcal {J}}_h \right\rbrace . Every vertex of R^* i... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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9f387dbc49822118655ec17f53258c480e79a3f9 | subsection | 44 | 140 | Hierarchical System of Colors | For 1\le h\le \rho , let Q_h be the level-h square containing v, and let c_h be the color assigned to Q_h. Then we say that the level-h color of v is c_h. We also say that the level-0 color of v is c_0. Therefore, every vertex v\in U({\mathcal {H}})\cup R^* is associated with a (\rho +1)-tuple of colors (c_0(v),\ldots ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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e3ebd7f015e43b0b5cff37148a8e7c9223ab833f | subsection | 45 | 140 | Hierarchical System of Colors | The following three theorems summarize these three steps, and their proofs appear in the following sections.Theorem 4.7
There is a coloring f of \tilde{\mathcal {H}} by {\mathcal {C}}, and a set \tilde{\mathcal {M}}^{\prime }\subseteq \tilde{\mathcal {M}} of demand pairs, such that \tilde{\mathcal {M}}^{\prime } is pe... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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6475e3db30046df5a3784ed70ea6da4996b54dc7 | subsection | 46 | 140 | Hierarchical System of Colors | We assume that our algorithm is given a lower bound \mathsf {OPT} on the value of the optimal solution. It starts by constructing the family {\mathcal {F}} of hierarchical systems of squares using Claim REF . It then guesses one of the systems \tilde{\mathcal {H}}\in {\mathcal {F}}, and a sequence L=(\ell _1,\ldots ,\e... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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5222ea4c934cfa9f7ce93f49b809f9f35d19e68a | subsection | 47 | 140 | Heavy demand pairs. | Consider any demand pair (s,t)\in \hat{\mathcal {M}} and the path P(s,t) routing (s,t) in {\mathcal {P}}. We say that (s,t) covers the demand pair (s^{\prime },t^{\prime })\in \hat{\mathcal {M}} iff P(s,t) intersects the unique square Q\in {\mathcal {Q}} with t^{\prime }\in Q. Notice that (s^{\prime },t^{\prime }) may ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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4727cb6ebd43252803547ec093080d9f2f69fa10 | subsection | 48 | 140 | Partitioning algorithm. | Initially, we let \overline{{\mathcal {M}}} contain all heavy demand pairs, so |\overline{{\mathcal {M}}}|\le |\hat{\mathcal {M}}|/(128\log n). We start with {\mathcal {D}}=\left\lbrace D^0 \right\rbrace , and let the corresponding set of the demand pairs be {\mathcal {M}}_0=\hat{\mathcal {M}}\setminus \overline{{\math... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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2589147bdedf8a4863ee4926f32929dd1ee2da65 | subsection | 49 | 140 | Partitioning algorithm. | Let \sigma _2(D^{\prime })=P^{\prime }(s,t), \sigma _4(D^{\prime })=P^{\prime }(s^{\prime },t^{\prime }), and let \sigma _1(D^{\prime }) be the subpath of R^* between s and s^{\prime }. Finally, let \sigma (D^{\prime }) be the concatenation of these four curves, and let D^{\prime } be the disc whose boundary is \sigma ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.... | |
4e68de16f0849f8af92b99bc4e614761025d54ad | subsection | 50 | 140 | Partitioning algorithm. | From the above discussion, it also respects the currently active squares.We have discarded at most 2d+2^{10}d\log n\le 2^{11}d\log n demand pairs in the current iteration, but both {\mathcal {M}}^{\prime } and the new set {\mathcal {M}}_i contain at least 2^{20}d\log ^3n-2^{10}d\log n -2d\ge 2^{19}d\log ^3n demand pair... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
c4bee72b45fb936c768323b77875b111afb36179 | subsection | 51 | 140 | Partitioning algorithm. | Then the budget of every demand pair in {\mathcal {M}}^{\prime } decreases by at least \frac{1}{128\log ^2n} (as |{\mathcal {M}}^{\prime }|\le |{\mathcal {M}}_i|/2), and the total decrease in the budgets is therefore at least \frac{|{\mathcal {M}}^{\prime }|}{128\log ^2n}\ge \frac{2^{19}d\log ^3n}{128\log ^2n}\ge 2^{12... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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5d88873fbc18b3b4c9795f38e2fceee9f12a56ec | subsection | 52 | 140 | Boundary Pairs. | Consider some disc D^i\in {\mathcal {D}}, and assume first that |{\mathcal {M}}_i|\ge 2^{21}d\log ^3n.
Let a_i,b_i be the two endpoints of \sigma _1(D^i), and let {\mathcal {Z}}^i_0\subseteq {\mathcal {M}}_i contain 2^{20}d\log ^3n demand pairs whose sources lie closest to a_i and 2^{20}d\log ^3n demand pairs whose sou... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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f524287725be1e590551d0f53150bdcf84917e5c | subsection | 53 | 140 | Forest of Discs. | It would be convenient for us to organize the discs in {\mathcal {D}} into a directed forest F, in a natural way. In each arborescence of the forest, the edges will be directed towards the root.
The set of vertices of F is v_0,v_1,\ldots ,v_z, where v_i represents disc D^i. There is a directed edge (v_i,v_{i^{\prime }}... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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d25c4792a247f5714f69a8af53bc8360041d9f30 | subsection | 54 | 140 | Forest of Discs. | Consider now some vertex v_i\in Y_x. Notice that at most one child-vertex v_{i^{\prime }} of v_i in the forest F may belong to Y_x. If such a vertex v_{i^{\prime }} does not exist, then all demand pairs in {\mathcal {M}}_i are called right pairs. Otherwise, for each demand pair (s,t)\in {\mathcal {M}}_i, if s appears t... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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752ce3af40a37bc47be180af7ed18d3a30d2fc77 | subsection | 55 | 140 | Forest of Discs. | Then there are two demand pairs (s,t),(s^{\prime },t^{\prime })\in {\mathcal {M}}^{\prime \prime \prime }, with t,t^{\prime }\in Q, such that at least rd vertices of S({\mathcal {M}}^{\prime \prime \prime }) lie between s and s^{\prime } on R^*. Let S^{\prime } be the set of all these source vertices, and let {\mathcal... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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a53b1981c137df15c2686ae38e7ca5df83d21072 | subsection | 56 | 140 | The Existence of Coloring | In this section we prove Theorem REF . Recall that we assume that we are given a lower bound \mathsf {OPT} on the value of the optimal solution; a hierarchical system \tilde{\mathcal {H}}=({\mathcal {Q}}_1,\ldots ,{\mathcal {Q}}_{\rho })\in {\mathcal {F}}, and a sequence L=(\ell _1,\ldots ,\ell _{\rho }) of integers as... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.0... | |
32da62819e0c2ca2e2d72470acb2213fdbdff15c | subsection | 57 | 140 | The Existence of Coloring | Then I^{\prime \prime } is contained in the shadow L_{\hat{\mathcal {M}}}(Q). From the 1/\eta ^2-shadow property of Q, the length of this shadow must be at most d_h/\eta ^2. But from our construction, the length of the shadow is at least |S(\tilde{\mathcal {M}}^{**})\cap I^{\prime \prime }|\ge d_h/(16\eta )> d_h/\eta ^... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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42b56c2c75c7d6d0f40a6699c5c4565cf91f4e93 | subsection | 58 | 140 | The Existence of Coloring | We then set \hat{\mathcal {M}}_h=\bigcup _{Q\in {\mathcal {Q}}_h}\hat{\mathcal {M}}^{\prime \prime }(Q).Let \tilde{\mathcal {M}}^{\prime }=\hat{\mathcal {M}}_{\rho } be the set of demand pairs obtained after the last iteration. Then it is immediate to verify that:|\tilde{\mathcal {M}}^{\prime }|\ge \frac{|\tilde{\mathc... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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7ae3644d48514dfe7c08c42c24c4c7f73aa8460c | subsection | 59 | 140 | Finding the Routing | The goal of this section is to prove Theorem REF .
Recall that we are given as input a hierarchical system \tilde{\mathcal {H}}=({\mathcal {Q}}_1,\ldots ,{\mathcal {Q}}_{\rho })\in {\mathcal {F}} of squares and a hierarchical L-partition ({\mathcal {J}}_1,\ldots ,{\mathcal {J}}_{\rho }) of the row R^* of G^{\prime }.
W... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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7288765ccadace75273f651f28cebf19347e725c | subsection | 60 | 140 | Finding the Routing | Let N(c_h) be the number of demand pairs (s,t)\in \hat{\mathcal {M}}, such that s and t have level-h color c_h.
Since d_h = \eta ^{\rho -h+3}, N(c_h) \le \left\lceil d_h/(2\eta ^3)\right\rceil \le \eta ^{\rho -h}, and |\hat{\mathcal {M}}|\le d_1/(2\eta ^3).
Notice that for every level-\rho color c_\rho , N(c_\rho ) \le... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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f9248bce11695b0fc57d5fbbad4d1d53c401038c | subsection | 61 | 140 | Finding the Routing | We then let {\mathcal {I}}^{\prime }_h=\left\lbrace I^{\prime }_h\mid I_h\in {\mathcal {I}}_h \right\rbrace , so that ({\mathcal {I}}^{\prime }_1,\ldots ,{\mathcal {I}}^{\prime }_{\rho }) can be viewed as a hierarchical partition of I^{\prime }_0.
The parent-child relationship between the new intervals is defined exact... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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d1f361050620cc5e8da219d674a295b49c7907aa | subsection | 62 | 140 | Finding the Routing | A snake {\mathcal {Y}} of length z is a sequence (\Upsilon _1,\Upsilon _2,\ldots ,\Upsilon _{z}) of z corridors that are pairwise internally disjoint, such that for all 1\le z^{\prime },z^{\prime \prime } \le z, \Upsilon _{z^{\prime }} is a neighbor of \Upsilon _{z^{\prime \prime }} iff |z^{\prime }-z^{\prime \prime }|... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/3055399.3055411",
"end": 1414,
"openalex_id": "https://openalex.org/W2550041253",
"raw": "Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat. New hardness results for routing on disjoint paths. In Hamed Hatami, Pierre McKenzie, and ... | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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16570b6f63635a2a11fe1c03e58ab1e20675bf8e | subsection | 63 | 140 | Finding the Routing | Let 0\le h\le \rho be a level, let c_h\in \chi _h be a level-h color, and let {\mathcal {Y}} be a snake. We say that a snake {\mathcal {Y}} is a valid level-h snake for color c_h iff:{\mathcal {Y}}\subseteq Q_0^+;
{\mathcal {Y}} has width at least 3 \cdot N(c_h);
{\mathcal {Y}} contains the interval I^{\prime } \in {... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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c80c93f46144c00d3a14d204ccff8ecb06d3d948 | subsection | 64 | 140 | Finding the Routing | Assume now that the claim holds for some level 0\le h<\rho .
We prove that it holds for level (h+1).Fix a level-h interval I_h\in {\mathcal {I}}_h, and its corresponding level-h color c_h.
Let {\mathcal {Y}}={\mathcal {Y}}(c_h) be the valid level-h snake for c_h, given by the induction hypothesis.
Recall that for each ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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ff864649f1406cb72f3465308a987daf67a731d5 | subsection | 65 | 140 | Finding the Routing | For a non-empty level-h square Q, let \left\lbrace I^i_Q \subseteq I_Q \: | \: i \in \left\lbrace 1, \ldots , z \right\rbrace \right\rbrace be a collection of disjoint sub-paths of I_Q such that the following holds:
(i) each sub-path I^i_Q contains 3 \cdot N(c_h^i) vertices; and
(ii) for all 1 \le i < i^{\prime } \le z... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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5e8d27b420c8ec7734dd705a9d14f4d3bdfb492f | subsection | 66 | 140 | Finding the Routing | Let \Sigma _1 be some boundary edge of \Upsilon _1 that is contained in the boundary of {\mathcal {Y}}.
Similarly, let \Sigma _r \ne \Sigma _1 be some boundary edge of \Upsilon _r that is contained in the boundary of {\mathcal {Y}}.
Let (A_1, A_2, \ldots A_{j}) be disjoint sub-paths on \Sigma _1 and let (B_1, B_2, \ldo... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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fdf94ae03e1294e91bbd39ebf7568646181c67bd | subsection | 67 | 140 | Finding the Routing | From the above observation, it is now immediate to find pairwise disjoint sub-snakes {\mathcal {Y}}^1(c_h^i), \ldots , {\mathcal {Y}}^r(c_h^i) for each child-color c_h^i of c_h such that:the top boundary of the first corridor of {\mathcal {Y}}^1(c_h^i) contains the interval I^{\prime }(c_h^i) \in {\mathcal {I}}^{\prime... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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927e7baaffe237778ffaf1795e285f0ad03b4a39 | subsection | 68 | 140 | Finding the Routing | Then \hat{{\mathcal {Y}}}= {\mathcal {Y}}_1 \oplus {\mathcal {Y}}_2 is also a snake of width at least w.Notice that for each child-color c_h^i of c_h and for each 1 \le j \le r, the pair of sub-snakes {\mathcal {Y}}^j(c_h^i) and {\mathcal {Y}}_{Q_j}(c_h^i) are composable.
Similarly, for each 1 \le j < r, the pair of su... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.03... | |
bb6d25505b34210d3c6c28280811d2cf5499e7a3 | subsection | 69 | 140 | Finding the Routing | We use the term `child-snake' to distinguish them from the final combined sub-snake {\mathcal {Y}}_Q(c_h^i) that we construct.
for each child-color c_h^i of c_h, which can be combined together to obtain the desired sub-snake {\mathcal {Y}}_Q(c_h^i).
To this end, we need interfaces, which we define next.For each child-s... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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9a295a0c77343d4b5bdd7bfbdf614d3b51a07624 | subsection | 70 | 140 | Finding the Routing | Hence, portals of any pair of distinct squares are separated by at least 3 \cdot N(c_h) nodes from each other.
Exploiting this spacing, it is now immediate to construct the child-snakes of the following five types (see Figure REF for illustration).The set of child-snakes of type A contains, for each child-color c_h^i o... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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85e903f3b20197d601e513546dbba4a5a431c138 | subsection | 71 | 140 | Finding the Routing | Claim REF now follows by considering the snakes{\mathcal {Y}}_Q(c_h^i) = {\mathcal {Y}}^A(c_h^i) \oplus \hat{{\mathcal {Y}}}_1^1(c_h^i) \oplus {\mathcal {Y}}_1^1(c_h^i) \oplus \hat{{\mathcal {Y}}}_1^2(c_h^i) \oplus {\mathcal {Y}}_1^2(c_h^i) \oplus \ldots \oplus {\mathcal {Y}}_{r}^{k_r-1}(c_h^i) \oplus \hat{{\mathcal {Y... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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94c355463df6bf5fff5cee16143f971fb94075b8 | subsection | 72 | 140 | Finding the Coloring of Squares and a Perfect Set of Demand Pairs | The goal of this section is to prove Theorem REF . It would be convenient for us to reformulate the problem slightly differently. Recall that we are given a hierarchical system {\mathcal {C}} of colors, and that for all source vertices s of the demand pairs in \tilde{\mathcal {M}}, their colorings are fixed: that is, e... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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e3b78740d6b8754ad471c9fced5367eaf18c1f5f | subsection | 73 | 140 | Finding the Coloring of Squares and a Perfect Set of Demand Pairs | The main result of this section is a proof of the following theorem.Theorem 8.1
There is an efficient randomized algorithm, that, given as input an instance (\tilde{\mathcal {H}},\hat{\mathcal {M}}) of the HSC problem, with high probability returns an O(\log ^4 n)-approximate solution to it.We first show that Theorem ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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33cad75ed5e592b80ff3fbecf6afd52265e3d674 | subsection | 74 | 140 | Finding the Coloring of Squares and a Perfect Set of Demand Pairs | However, both t and t^{\prime } belong to the same level-\rho square Q_{\rho }, and their color should agree with the level-\rho color of Q_{\rho }, which is impossible.The remainder of this section is dedicated to proving Theorem REF . We formulate an LP-relaxation of the problem, and then provide a randomized LP-roun... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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087a34780ece2786704145d604edd1146598af38 | subsection | 75 | 140 | The Linear Program | For every level-\rho square Q_{\rho }\in {\mathcal {Q}}_{\rho }, and every level-\rho color c_{\rho }\in \chi _{\rho }, we let n(Q_{\rho },c_{\rho }) denote the number of vertices of U\cap V(Q_{\rho }), whose level-\rho color is c_{\rho }.Our linear program has three types of variables. Fix some level 1\le h\le \rho an... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0a92406d87cf965050513a961565a1b077e6dec2 | subsection | 76 | 140 | The Linear Program | The first set of constraints requires that for each level-\rho square Q_{\rho }, and each level-\rho color c_{\rho }, if Q_{\rho } is assigned the color c_{\rho }, then y(Q_{\rho },c_{\rho }) is bounded by n(Q_{\rho },c_{\rho }), and otherwise it is 0.y(Q_{\rho },c_{\rho })\le n(Q_{\rho },c_{\rho })\cdot x(Q_{\rho },c_... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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04e0d2b7149983e62061851a5fd8d6dea34ad0e6 | subsection | 77 | 140 | The Linear Program | If Q_h is assigned the level-h color c_h, then Q_{h+1} must be assigned some child-color of c_h. The following constraint expresses this requirement.\sum _{c_{h+1}\in \tilde{\chi }_{h+1}(c_h)}x(Q_{h+1},c_{h+1})=x(Q_h,c_h)\quad \quad \forall 1\le h<\rho ; \quad \forall Q_h\in {\mathcal {Q}}_h; \quad \forall c_h\in \chi ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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18cf44a89b6e0bf2c68fe7c39f87e47644cf3447 | subsection | 78 | 140 | The Linear Program | We now proceed to describe the LP-rounding algorithm. | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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a964e2b0208abbe0e437ad3f56df315a975b8ea6 | subsection | 79 | 140 | LP-Rounding | For convenience, we denote |U|=n.
Our LP-rounding algorithm proceeds in three stages. In the first stage we perform a simple randomized rounding, level-by-level. We will show that the expected number of vertices that we select to our solution U^{\prime } is equal to the LP-solution value. However, it is possible that w... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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928f01d9adc4f717dceee0e7fc84e82f1c04ee58 | subsection | 80 | 140 | Stage 1: Randomized Rounding | We perform \rho iterations, where in iteration h we settle the colors of all level-h squares, by suitably modifying the LP-solution.
We will maintain the following invariant:For all 1\le h\le \rho , at the beginning of iteration h, the current LP-solution (x^{\prime }, y^{\prime }) has the following properties. For eve... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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9fa0c5bbc3aa3311b995ddb7774a05bdf1f118ce | subsection | 81 | 140 | Stage 1: Randomized Rounding | Assume now that it holds at the beginning of iteration h. The iteration is executed as follows. Consider some level-h square Q_h. Suppose its parent square is Q_{h-1}, and it was assigned color c_{h-1}. Then Constraint (REF ), together with Invariant (REF ) ensures that:\sum _{c_h\in \tilde{\chi }_h(c_{h-1})}x^{\prime ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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5d2db6883bcced6fe85d9cbfb1d8d4eae337e5c9 | subsection | 82 | 140 | Stage 1: Randomized Rounding | Similarly, if h=\rho , then we set:y^{\prime \prime }(Q_{h^{\prime }},c_{h^{\prime }})=\frac{y^{\prime }(Q_{h^{\prime }},c_{h^{\prime }})}{x^{\prime }(Q_h,c_h)}=\frac{y(Q_{h^{\prime }},c_{h^{\prime }})/x(Q_{h-1},c_{h-1})}{x(Q_h,c_h)/x(Q_{h-1},c_{h-1})}=\frac{y(Q_{h^{\prime }},c_{h^{\prime }})}{x(Q_h,c_h)}.We then repla... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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8b3837cfa2f30d3b80f9ccc9fc414184e05cb350 | subsection | 83 | 140 | Stage 1: Randomized Rounding | (Recall that \frac{y^{\prime }(Q_{\rho },c_{\rho })}{x^{\prime }(Q_{\rho },c_{\rho })}=\frac{y(Q_{\rho },c_{\rho })}{x(Q_{\rho },c_{\rho })} from Invariant (REF ) — we use this fact later.) The following theorem concludes the analysis of Stage 1.Theorem 8.2
The expected number of vertices added to U^{\prime } at Stage... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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15ab3671d25199fd2091ef156ca10909464403b1 | subsection | 84 | 140 | Stage 1: Randomized Rounding | Then:\text{\bf Pr}_{}\left[{\cal {E}}_h\mid {\cal {E}}_{h-1}\right]=x^{\prime }(Q_h,c_h)=\frac{x(Q_h,c_h)}{x(Q_{h-1},c_{h-1})}.Therefore:\begin{split}
\text{\bf Pr}_{}\left[{\cal {E}}_{\rho }\right]&=\text{\bf Pr}_{}\left[{\cal {E}}_{\rho }\mid {\cal {E}}_{\rho -1}\right]\cdot \text{\bf Pr}_{}\left[{\cal {E}}_{\rho -1}... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
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e73b4e4a6f1639abfe90fb458587537d668b55e6 | subsection | 85 | 140 | Stage 1: Randomized Rounding | Then, if Q_{\rho } is assigned color c_{\rho } (which again happens with probability x(Q_{\rho },c_{\rho })), we add one vertex of Q_{\rho }, whose level-\rho color is c_{\rho }, with probability \frac{y(Q_{\rho },c_{\rho })}{x(Q_{\rho },c_{\rho })}. Clearly, in this case, \text{\bf E}_{}\left[z(Q_{\rho },c_{\rho })\ri... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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59014868d9ea90d9b7b4bf2972605e4b098b1f1d | subsection | 86 | 140 | Stage 2: Ensuring that the Solution is Almost Feasible | In this stage, for each level 1\le h\le \rho and for each level-h color c_h\in \chi _h, we inspect the number of vertices in the solution U^{\prime } that we constructed at stage 1, whose level-h color is c_h. If this number is greater than 64d_h\log ^3n, then we say that color c_h has failed. In that case, we return a... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/cbo9780511581274",
"end": 505,
"openalex_id": "https://openalex.org/W1989274820",
"raw": "Devdatt Dubhashi and Alessandro Panconesi. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, New... | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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031563d44c50ba511bb8920e5bc4adc8a9b0fe9d | subsection | 87 | 140 | Stage 2: Ensuring that the Solution is Almost Feasible | Then we set x^{\prime }(Q_{\rho },c_{\rho })=1, and y^{\prime }(Q_{\rho },c_{\rho })=y(Q_{\rho },c_{\rho })/x(Q_{\rho },c_{\rho }). For every other level-\rho color c^{\prime }_{\rho }, we set x^{\prime }(Q_{\rho },c^{\prime }_{\rho })=0, and y^{\prime }(Q_{\rho },c^{\prime }_{\rho })=0.For simplicity, we denote \chi ^... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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241a7000eb9e3f2e17e201cea45ee1dd373e1d78 | subsection | 88 | 140 | Stage 2: Ensuring that the Solution is Almost Feasible | Otherwise, with probability y^{\prime }(Q_{\rho },c_{\rho }), we add only one such vertex to our solution. Overall, from Theorem REF , it is immediate to verify that, if event {\cal {E}}_{\rho } does not happen, then with probability at least (1-1/n^4), the number of vertices in U^{\prime }, whose level-h color is c_h ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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3da25a716f63aa602795228e62fd7387cab02df1 | subsection | 89 | 140 | Stage 2: Ensuring that the Solution is Almost Feasible | We will repeatedly use Constraint (REF ) that we restate here; we slightly change the indexing to avoid confusion with the variables h and h^{\prime } that we use here.The constraint states that for all pairs 1\le \tilde{h}\le \tilde{h}^{\prime }\le \rho of levels, for every level-\tilde{h} square Q_{\tilde{h}}\in {\ma... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.0... | |
e495d30f3225a085a4c4960f85b2ed4bf02ec6cd | subsection | 90 | 140 | Stage 2: Ensuring that the Solution is Almost Feasible | Intuitively, M^{\prime }(Q_{h^{\prime }}) is the number of vertices of Q_{h^{\prime }} that are (possibly fractionally) assigned the level-h color c_h at the beginning of iteration h^{\prime }, while M^{\prime \prime }(Q_{h^{\prime }}) reflects the same quantity at the end of iteration h^{\prime }.
The values of variab... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.... | |
d8b7034e0d9107d0e44b3e4429e7dbaa79fb7deb | subsection | 91 | 140 | Case 1: | Let c_{h^{\prime }} be the unique ancestor-color of color c_h, that belongs to level h^{\prime }.
Using Constraint (REF ) with \tilde{h}= h^{\prime } and \tilde{h}^{\prime }=h, we get that of each such level-h^{\prime } square Q_{h^{\prime }}:M^{\prime }(Q_{h^{\prime }})\le d_{h}x^{\prime }(Q_{h^{\prime }},c_{h^{\prime... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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fb2dfeb44c6e3c58aaaac71ae3beba9f65008aa9 | subsection | 92 | 140 | Case 1: | Let I denote the top boundary of the grid G, and let I^{\prime } denote its bottom boundary.
Theorem REF guarantees the existence of a set \Sigma =\left\lbrace \sigma _1,\ldots ,\sigma _z \right\rbrace of disjoint sub-intervals of I, and a set \Sigma ^{\prime }=\left\lbrace \sigma _1^{\prime },\ldots ,\sigma ^{\prime }... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
b6304919c59e4a6a96e95a1a87e9c54f70850122 | subsection | 93 | 140 | Case 1: | We can then compute an approximate solution
{\mathcal {A}}(G^{\prime }_{\sigma ,\sigma ^{\prime }},{\mathcal {M}}^{\prime }_{\sigma ,\sigma ^{\prime }}) of value |{\mathcal {A}}(G^{\prime }_{\sigma ,\sigma ^{\prime }},{\mathcal {M}}^{\prime }_{\sigma ,\sigma ^{\prime }})| to this instance, using the algorithm from The... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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cb3bb393c990c6656c7812518ce710b8d493c40d | subsection | 94 | 140 | Case 1: | This can be done by using simple dynamic programming.Assume now that we have computed the collections \Sigma and \Sigma ^{\prime } of intervals as above. For each 1\le i\le z, let {\mathcal {P}}_i={\mathcal {A}}(G^{\prime }_{\sigma _i,\sigma ^{\prime }_i},{\mathcal {M}}^{\prime }_{\sigma _i,\sigma ^{\prime }_i}) be the... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0f7089310bea220c8fdf1ae1643a3e578b4af0e8 | subsection | 95 | 140 | Case 2: | Let Q_{h^{\prime }}\in {\mathcal {Q}}_{h^{\prime }} be any level-h^{\prime } square, and let c_{h^{\prime }}\in \tilde{\chi }_{h^{\prime }}(c_h) be any level-h^{\prime } descendant-color of c_h.
Using Constraint (REF ) with \tilde{h}=\tilde{h}^{\prime }=h^{\prime }, we get that:\sum _{Q_{\rho }\in {\mathcal {D}}_{\rho ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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35bb4305c1586ee3de513cd35e8bf669dfc2cb60 | subsection | 96 | 140 | Case 2: | Therefore, the expectation of M^{\prime \prime }(Q_{h^{\prime }}) is:\text{\bf E}_{}\left[M^{\prime \prime }(Q_{h^{\prime }})\right]=\sum _{c_{h^{\prime }}\in \tilde{\chi }_{h^{\prime }}(c_h)}M^{\prime }(Q_{h^{\prime }},c_{h^{\prime }})=M^{\prime }(Q_{h^{\prime }}).Overall, we conclude that variables in set \left\lbrac... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
97210f536721c894f1aaa11c0fb535c495c404c7 | subsection | 97 | 140 | Case 2: | Let {\mathcal {M}}^{\prime }\subseteq {\mathcal {M}} be the subset of all demand pairs (s,t) with \tilde{t}\in I^{\prime }. Let \mathsf {OPT}^{\prime \prime } be the value of the optimal solution to instance (G,{\mathcal {M}}^{\prime }).Observation 9.6 \mathsf {OPT}^{\prime \prime }\ge \mathsf {OPT}^{\prime }/2.Proof:
... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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499653a17c24337974892e3524a8efd7254bd9af | subsection | 98 | 140 | Stage 3: Turning the Solution into a Feasible One | If any of the colors fail, then we return an empty solution.
Assume now that no color fails. Let U^{\prime }\subseteq U be the set of all vertices chosen by the current solution.Claim 8.10 There is an efficient algorithm to compute a subset U^{\prime \prime }\subseteq U^{\prime } of vertices, with |U^{\prime \prime }|\... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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de08e778b9ce19648784829f08fab33252954dc1 | subsection | 99 | 140 | Stage 3: Turning the Solution into a Feasible One | Then:\text{\bf E}_{}\left[N_3\right] < \left( \frac{1}{1024\log ^4 n} \cdot \mathsf {OPT}_{\mathsf {LP}}\right) + \frac{\mathsf {OPT}_{\mathsf {LP}}}{1024\log ^4 n} <\left(\frac{\mathsf {OPT}_{\mathsf {LP}}}{512\log ^4 n} \right),a contradiction.We run the above algorithm c\log ^5n times independently (for some large c... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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8bca93a6cdb22e330c75da01e85d1b7bedbfcddc | subsection | 100 | 140 | Approximating | In this section we complete the proof of Theorem REF .
Suppose we are given an instance (G,{\mathcal {M}}) of Restricted NDP-Grid.
We assume that we know the value \mathsf {OPT} of the optimal solution to this instance, by going over all such possible choices, and running the algorithm on each of them.
Let \Gamma _1,\G... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
6feac6e5360e8c81ee986bdeddbb8d325f42118c | subsection | 101 | 140 | Special Instances | It will be convenient for us to define special sub-instances of instance (G,{\mathcal {M}}), that have a specific structure.
We start by defining interesting pairs of intervals.Definition. Let I,I^{\prime }\subseteq \Gamma be two disjoint intervals, and let d>0 be an integer. We say that (I,I^{\prime }) is a d-interest... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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783aae81828bfc69be79c15ca0ceaff81b9182a0 | subsection | 102 | 140 | Modified Instances | Assume that we are given an integer d>0, and a d-interesting pair (I,I^{\prime }) of intervals, together with a perfect (I,I^{\prime },d)-sub-instance (G,{\mathcal {M}}^{\prime }) of (G,{\mathcal {M}}). We define a corresponding modified instance (G^{\prime },{\mathcal {M}}^{\prime \prime }). The underlying graph G^{\p... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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85f4d1d0fa630636e7d35fbb0e7faadf7ad609b3 | subsection | 103 | 140 | Modified Instances | Similarly, a traversal of \Gamma (G^{\prime }) in the clock-wise direction, starting from v, defines an ordering \pi ^{\prime } of the vertices in X. Consider some vertex s\in S({\mathcal {M}}^{\prime }), and assume that it is the ith vertex of S({\mathcal {M}}^{\prime }) according to the ordering \pi . We map it to th... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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dede37990933777aad7f0926ad1fd2954065dd93 | subsection | 104 | 140 | Modified Instances | We let {\mathcal {A}}(G^{\prime },{\mathcal {M}}^{\prime \prime }) denote the resulting solution, and we let |{\mathcal {A}}(G^{\prime },{\mathcal {M}}^{\prime \prime })| denote its value. | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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9154705052c2d07a3385b02def3a0a0a57cddf2b | subsection | 105 | 140 | Main Partitioning Theorem | The following theorem is central to our proof.Theorem 9.2
Suppose we are given two disjoint intervals \pi ,\pi ^{\prime } of \Gamma , each of which is contained in a single boundary edge of G, an integer d>0, and a valid (\pi ,\pi ^{\prime },d)-instance (G,{\mathcal {M}}^{\prime }). Assume further that |{\mathcal {M}}... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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459d2d9fd14811f8653d6267ca90598691de10f9 | subsection | 106 | 140 | Main Partitioning Theorem | Then for all 1\le i\le z, (G,\tilde{\mathcal {M}}_{\sigma _i,\sigma ^{\prime }_i}) is a perfect (\sigma _i,\sigma ^{\prime }_i,d)-instance. Let (G_i,\tilde{\mathcal {M}}^{\prime }_{\sigma _i,\sigma ^{\prime }_i}) denote the corresponding modified instance for (G,\tilde{\mathcal {M}}_{\sigma _i,\sigma ^{\prime }_i}).
Th... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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ccb55eda39d530b0618f301abc7e8150cc151ed1 | subsection | 107 | 140 | Main Partitioning Theorem | If no such interval exists, then I(v) is undefined.Observation 9.3
For each vertex v\in \pi ^{\prime }, if I(v) is defined, then |{\mathcal {M}}^{\prime }(I(v))|\le 20d.Proof:
Let v^{\prime } be the rightmost vertex of I(v), and let I^{\prime }=I(v)\setminus \left\lbrace v^{\prime } \right\rbrace . Since we have chose... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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68580a756df601b6c14e81111c77ccae9034ceb2 | subsection | 108 | 140 | Main Partitioning Theorem | Since z=\left\lceil \frac{|{\mathcal {M}}^{\prime }|}{160d}\right\rceil -1, all intervals \mu _1,\ldots ,\mu _{2z+2} are well-defined.
For convenience, we denote, for each 1\le i\le 2z, {\mathcal {M}}^{\prime }(\mu _i) by {\mathcal {M}}_i.
For every vertex t\in T({\mathcal {M}}^{\prime }), let Q_t be the shortest path ... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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43a50c77e06c70a230f4e78785a65c7a082ceff1 | subsection | 109 | 140 | Main Partitioning Theorem | We let \sigma _i be the smallest sub-interval of \pi , containing all vertices of S({\mathcal {M}}^{\prime }_{i}).It is immediate to verify that, for each 1\le i\le z, the value of the optimal solution of the NDP instance (G,{\mathcal {M}}^{\prime }_{\sigma _i,\sigma ^{\prime }_i}) is at least d, since we can route all... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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