F. Fomin, Sudeshna Kolay, D. Lokshtanov, Fahad Panolan, Saket Saurabh
A rectilinear Steiner tree for a set K of points in the plane is a tree that connects k using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, the input is a set K={z1,z2,…, zn} of n points in the Euclidean plane (R2), and the goal is to find a rectilinear Steiner tree for k of smallest possible total length. A rectilinear Steiner arborescence for a set k of points and a root r ∈ K is a rectilinear Steiner tree T for K such that the path in T from r to any point z ∈ K is a shortest path. In the Rectilinear Steiner Arborescence problem, the input is a set K of n points in R2, and a root r ∈ K, and the task is to find a rectilinear Steiner arborescence for K, rooted at r of smallest possible total length. In this article, we design deterministic algorithms for these problems that run in 2O(√ nlog n) time.
{"title":"Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems","authors":"F. Fomin, Sudeshna Kolay, D. Lokshtanov, Fahad Panolan, Saket Saurabh","doi":"10.1145/3381420","DOIUrl":"https://doi.org/10.1145/3381420","url":null,"abstract":"A rectilinear Steiner tree for a set K of points in the plane is a tree that connects k using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, the input is a set K={z1,z2,…, zn} of n points in the Euclidean plane (R2), and the goal is to find a rectilinear Steiner tree for k of smallest possible total length. A rectilinear Steiner arborescence for a set k of points and a root r ∈ K is a rectilinear Steiner tree T for K such that the path in T from r to any point z ∈ K is a shortest path. In the Rectilinear Steiner Arborescence problem, the input is a set K of n points in R2, and a root r ∈ K, and the task is to find a rectilinear Steiner arborescence for K, rooted at r of smallest possible total length. In this article, we design deterministic algorithms for these problems that run in 2O(√ nlog n) time.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127297182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group Γ over domain V, and asking whether there is a permutation γ ε Γ that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time (n + m)O((log d)c) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for this problem due to Schweitzer and Wiebking (STOC 2019) runs in time nO(d)mO(1). As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K3,h (h ≥ 3) as a minor in time nO((log h)c). In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time nO((log g)c).
{"title":"Hypergraph Isomorphism for Groups with Restricted Composition Factors","authors":"Daniel Neuen","doi":"10.1145/3527667","DOIUrl":"https://doi.org/10.1145/3527667","url":null,"abstract":"We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group Γ over domain V, and asking whether there is a permutation γ ε Γ that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time (n + m)O((log d)c) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for this problem due to Schweitzer and Wiebking (STOC 2019) runs in time nO(d)mO(1). As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K3,h (h ≥ 3) as a minor in time nO((log h)c). In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time nO((log g)c).","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134131354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Agarwal, Hsien-Chih Chang, S. Suri, Allen Xiao, J. Xue
We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our article are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in ℝ1 and ℝ2. The first framework uses bootstrapping and gives a (1 + ε)-approximate data structure for dynamic interval set cover in ℝ1 with O(nα / ε) amortized update time for any constant α > 0; in ℝ2, this method gives O(1)-approximate data structures for unit-square set cover and hitting set with O(n1/2+α) amortized update time. The second framework uses local modification and leads to a (1 + ε)-approximate data structure for dynamic interval hitting set in ℝ1 with Õ(1/ε) amortized update time; in ℝ2, it gives O(1)-approximate data structures for unit-square set cover and hitting set in the partially dynamic settings with Õ(1) amortized update time.
{"title":"Dynamic Geometric Set Cover and Hitting Set","authors":"P. Agarwal, Hsien-Chih Chang, S. Suri, Allen Xiao, J. Xue","doi":"10.1145/3551639","DOIUrl":"https://doi.org/10.1145/3551639","url":null,"abstract":"We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our article are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in ℝ1 and ℝ2. The first framework uses bootstrapping and gives a (1 + ε)-approximate data structure for dynamic interval set cover in ℝ1 with O(nα / ε) amortized update time for any constant α > 0; in ℝ2, this method gives O(1)-approximate data structures for unit-square set cover and hitting set with O(n1/2+α) amortized update time. The second framework uses local modification and leads to a (1 + ε)-approximate data structure for dynamic interval hitting set in ℝ1 with Õ(1/ε) amortized update time; in ℝ2, it gives O(1)-approximate data structures for unit-square set cover and hitting set in the partially dynamic settings with Õ(1) amortized update time.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114713010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Lokshtanov, P. Misra, Joydeep Mukherjee, Fahad Panolan, Geevarghese Philip, Saket Saurabh
A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T − S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here, the input is a tournament T and a weight function w : V(T) → N, and the task is to find a feedback vertex set S in T minimizing w(S) = ∑v∈S w(v). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this article, we give the first polynomial time factor 2-approximation algorithm for this problem. Assuming the Unique Games Conjecture, this is the best possible approximation ratio achievable in polynomial time.
{"title":"2-Approximating Feedback Vertex Set in Tournaments","authors":"D. Lokshtanov, P. Misra, Joydeep Mukherjee, Fahad Panolan, Geevarghese Philip, Saket Saurabh","doi":"10.1145/3446969","DOIUrl":"https://doi.org/10.1145/3446969","url":null,"abstract":"A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T − S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here, the input is a tournament T and a weight function w : V(T) → N, and the task is to find a feedback vertex set S in T minimizing w(S) = ∑v∈S w(v). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this article, we give the first polynomial time factor 2-approximation algorithm for this problem. Assuming the Unique Games Conjecture, this is the best possible approximation ratio achievable in polynomial time.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129983585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the Feedback Vertex Set (FVS) problem, one is given an undirected graph G and an integer k, and one needs to determine whether there exists a set of k vertices that intersects all cycles of G (a so-called feedback vertex set). Feedback Vertex Set is one of the most central problems in parameterized complexity: It served as an excellent testbed for many important algorithmic techniques in the field such as Iterative Compression [Guo et al. (JCSS’06)], Randomized Branching [Becker et al. (J. Artif. Intell. Res’00)] and Cut&Count [Cygan et al. (FOCS’11)]. In particular, there has been a long race for the smallest dependence f(k) in run times of the type O⋆ (f(k)), where the O⋆ notation omits factors polynomial in n. This race seemed to have reached a conclusion in 2011, when a randomized O⋆ (3k) time algorithm based on Cut&Count was introduced. In this work, we show the contrary and give a O⋆ (2.7k) time randomized algorithm. Our algorithm combines all mentioned techniques with substantial new ideas: First, we show that, given a feedback vertex set of size k of bounded average degree, a tree decomposition of width (1-Ω (1))k can be found in polynomial time. Second, we give a randomized branching strategy inspired by the one from [Becker et al. (J. Artif. Intell. Res’00)] to reduce to the aforementioned bounded average degree setting. Third, we obtain significant run time improvements by employing fast matrix multiplication.
{"title":"Detecting Feedback Vertex Sets of Size k in O⋆ (2.7k) Time","authors":"Jason Li, Jesper Nederlof","doi":"10.1145/3504027","DOIUrl":"https://doi.org/10.1145/3504027","url":null,"abstract":"In the Feedback Vertex Set (FVS) problem, one is given an undirected graph G and an integer k, and one needs to determine whether there exists a set of k vertices that intersects all cycles of G (a so-called feedback vertex set). Feedback Vertex Set is one of the most central problems in parameterized complexity: It served as an excellent testbed for many important algorithmic techniques in the field such as Iterative Compression [Guo et al. (JCSS’06)], Randomized Branching [Becker et al. (J. Artif. Intell. Res’00)] and Cut&Count [Cygan et al. (FOCS’11)]. In particular, there has been a long race for the smallest dependence f(k) in run times of the type O⋆ (f(k)), where the O⋆ notation omits factors polynomial in n. This race seemed to have reached a conclusion in 2011, when a randomized O⋆ (3k) time algorithm based on Cut&Count was introduced. In this work, we show the contrary and give a O⋆ (2.7k) time randomized algorithm. Our algorithm combines all mentioned techniques with substantial new ideas: First, we show that, given a feedback vertex set of size k of bounded average degree, a tree decomposition of width (1-Ω (1))k can be found in polynomial time. Second, we give a randomized branching strategy inspired by the one from [Becker et al. (J. Artif. Intell. Res’00)] to reduce to the aforementioned bounded average degree setting. Third, we obtain significant run time improvements by employing fast matrix multiplication.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128030364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The weighted k-server problem is a generalization of the k-server problem wherein the cost of moving a server of weight βi through a distance d is βi⋅ d. On uniform metric spaces, this models caching with caches having different page replacement costs. A memoryless algorithm is an online algorithm whose behavior is independent of the history given the positions of its k servers. In this article, we develop a framework to analyze the competitiveness of randomized memoryless algorithms. The key technical contribution is a method for working with potential functions defined implicitly as the solution of a linear system. Using this, we establish tight bounds on the competitive ratio achievable by randomized memoryless algorithms for the weighted k-server problem on uniform metrics. We first prove that there is an αk-competitive memoryless algorithm for this problem, where αk=αk− 12+ 3αk− 1+1; α1 = 1. We complement this result by proving that no randomized memoryless algorithm can have a competitive ratio less than αk. Finally, we prove that the above bounds also hold for the generalized k-server problem on weighted uniform metrics.
{"title":"Randomized Memoryless Algorithms for the Weighted and the Generalized k-server Problems","authors":"Ashish Chiplunkar, S. Vishwanathan","doi":"10.1145/3365002","DOIUrl":"https://doi.org/10.1145/3365002","url":null,"abstract":"The weighted k-server problem is a generalization of the k-server problem wherein the cost of moving a server of weight βi through a distance d is βi⋅ d. On uniform metric spaces, this models caching with caches having different page replacement costs. A memoryless algorithm is an online algorithm whose behavior is independent of the history given the positions of its k servers. In this article, we develop a framework to analyze the competitiveness of randomized memoryless algorithms. The key technical contribution is a method for working with potential functions defined implicitly as the solution of a linear system. Using this, we establish tight bounds on the competitive ratio achievable by randomized memoryless algorithms for the weighted k-server problem on uniform metrics. We first prove that there is an αk-competitive memoryless algorithm for this problem, where αk=αk− 12+ 3αk− 1+1; α1 = 1. We complement this result by proving that no randomized memoryless algorithm can have a competitive ratio less than αk. Finally, we prove that the above bounds also hold for the generalized k-server problem on weighted uniform metrics.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116367472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We are delighted to present a Special Issue of ACM Transactions on Algorithms, containing full versions of six papers that were presented at the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, in New Orleans on January 7–10, 2018. These papers, selected on the basis of their high rating by the conference program committee, have been thoroughly reviewed according to the journal’s highest standards. In “A Faster Algorithm for the Minimum-Cost Bipartite Perfect Matching in Planar Graphs,” Mudabir Kabir Asathulla, Sanjeev Khanna, Nathaniel Lahn, and Sharath Raghvendra provide a new algorithm with running time ̃ O (n log(nC )) for maximum-weight matching on n-vertex planar bipartite graphs with positive integer edge-weights not exceeding C . The algorithm is a blend of the bit-scaling approach of Gabow and Tarjan with a speed-up achieved by an involved use of the r -divisions of planar graphs. In the classic distinct elements problem, given a stream of elements from {1, 2, . . . ,n}, one asks for a (1 + ε )-approximation to the number of distinct elements of the stream. Since 2010, we know that an optimal amount of space needed for a constant success probability is Θ(ε−2 + logn). Jarosław Błasiok, in “Optimal Streaming and Tracking Distinct Elements with High Probability,” shows that if one wants to boost the success probability to (1 − δ ), only O (ε−2 log(δ−1) + logn) space is needed, instead of O (log(δ−1) · (ε−2 + logn)) needed for log(δ−1) parallel and independent runs. The space complexity is asymptotically optimal with respect to all three parameters. In “A Fast Generalized DFT for Finite Groups of Lie Type,” Chloe Ching-Yun Hsu and Chris Umans give a O ( |G | (1) )-time algorithm for the generalized Discrete Fourier Transform over group G for finite groups of Lie type. If the matrix multiplication exponent ω is 2, then running time of the algorithm is essentially optimal. An algorithm of Papadimitriou from 1981 solves an integer linear program in standard form max{cx |Ax = b,x ≥ 0,x ∈ Z } where A ∈ Zm×n , b ∈ Z , and a ∈ Z in time (m · (‖A‖∞ + ‖B‖∞)) (m 2 ) . Friedrich Eisenbrand and Robert Weismantel, in “Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma,” improve this bound to (m · ‖A‖∞) (m) · ‖B‖ ∞ using the classic Steinitz Lemma. In “Tight Analysis of Parallel Randomized Greedy MIS,” Manuela Fischer and Andreas Noever study the parallel randomized greedy algorithm for Maximum Independent Set: In each round order, the vertices, at random, select to the independent set every vertex appearing in the order before its neighbors and delete the neighborhoods of the chosen vertices from the graph. They prove that with high probability the algorithm finishes afterO (logn) rounds, finishing the analysis of an algorithm that was initiated in 1987. Finally, in “More Logarithmic-factor Speedups for 3SUM, (median,+)-Convolution, and Some Geometric 3SUM-Hard Problems,” Timothy M. Chan improves an
{"title":"Introduction to the Special Issue on SODA’18","authors":"Y. Lee, Marcin Pilipczuk, David P. Woodruff","doi":"10.1145/3368307","DOIUrl":"https://doi.org/10.1145/3368307","url":null,"abstract":"We are delighted to present a Special Issue of ACM Transactions on Algorithms, containing full versions of six papers that were presented at the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, in New Orleans on January 7–10, 2018. These papers, selected on the basis of their high rating by the conference program committee, have been thoroughly reviewed according to the journal’s highest standards. In “A Faster Algorithm for the Minimum-Cost Bipartite Perfect Matching in Planar Graphs,” Mudabir Kabir Asathulla, Sanjeev Khanna, Nathaniel Lahn, and Sharath Raghvendra provide a new algorithm with running time ̃ O (n log(nC )) for maximum-weight matching on n-vertex planar bipartite graphs with positive integer edge-weights not exceeding C . The algorithm is a blend of the bit-scaling approach of Gabow and Tarjan with a speed-up achieved by an involved use of the r -divisions of planar graphs. In the classic distinct elements problem, given a stream of elements from {1, 2, . . . ,n}, one asks for a (1 + ε )-approximation to the number of distinct elements of the stream. Since 2010, we know that an optimal amount of space needed for a constant success probability is Θ(ε−2 + logn). Jarosław Błasiok, in “Optimal Streaming and Tracking Distinct Elements with High Probability,” shows that if one wants to boost the success probability to (1 − δ ), only O (ε−2 log(δ−1) + logn) space is needed, instead of O (log(δ−1) · (ε−2 + logn)) needed for log(δ−1) parallel and independent runs. The space complexity is asymptotically optimal with respect to all three parameters. In “A Fast Generalized DFT for Finite Groups of Lie Type,” Chloe Ching-Yun Hsu and Chris Umans give a O ( |G | (1) )-time algorithm for the generalized Discrete Fourier Transform over group G for finite groups of Lie type. If the matrix multiplication exponent ω is 2, then running time of the algorithm is essentially optimal. An algorithm of Papadimitriou from 1981 solves an integer linear program in standard form max{cx |Ax = b,x ≥ 0,x ∈ Z } where A ∈ Zm×n , b ∈ Z , and a ∈ Z in time (m · (‖A‖∞ + ‖B‖∞)) (m 2 ) . Friedrich Eisenbrand and Robert Weismantel, in “Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma,” improve this bound to (m · ‖A‖∞) (m) · ‖B‖ ∞ using the classic Steinitz Lemma. In “Tight Analysis of Parallel Randomized Greedy MIS,” Manuela Fischer and Andreas Noever study the parallel randomized greedy algorithm for Maximum Independent Set: In each round order, the vertices, at random, select to the independent set every vertex appearing in the order before its neighbors and delete the neighborhoods of the chosen vertices from the graph. They prove that with high probability the algorithm finishes afterO (logn) rounds, finishing the analysis of an algorithm that was initiated in 1987. Finally, in “More Logarithmic-factor Speedups for 3SUM, (median,+)-Convolution, and Some Geometric 3SUM-Hard Problems,” Timothy M. Chan improves an ","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128690285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shortest paths computation is one of the most fundamental problems in computer science. An important variant of the problem is when edges can fail, and one needs to compute shortest paths that avoid a (failing) edge. More formally, given a source node s, a target node t, and an edge e, a replacement path for the triple (s,t,e) is a shortest s-t path avoiding edge e. Replacement paths computation can be seen either as a static problem or as a data structure problem. In the static setting, a typical goal is to compute for fixed s and t, for every possible failed edge e, the length of the best replacement path around e (replacement paths problem). In the data structure setting, a typical goal is to design a data structure (distance sensitivity oracle) that, after some preprocessing, quickly answers queries of the form: What is the length of the replacement path for the triple (s,t,e)? In this article, we focus on n-node directed graphs with integer edge weights in [−M,M], and present improved replacement paths algorithms and distance sensitivity oracles based on fast matrix multiplication. In more detail, we obtain the following main results: • We describe a replacement paths algorithm with runtime Õ(Mnω), where ω < 2.373 is the fast matrix multiplication exponent. For a comparison, the previous fastest algorithms have runtime õ(Mn1+2ω /3) [Weimann,Yuster—FOCS’10] and, in the unweighted case, õ(n2.5) [Roditty, Zwick—ICALP’05]. Our result shows that, at least for small integer weights, the replacement paths problem in directed graphs may be easier than the related all-pairs shortest paths problem, as the current best runtime for the latter is õ(M14−ω n2+1 4−ω): this is Ω (n2.5) even if ω = 2. Our algorithm also implies that the k shortest simple s-t paths can be computed in õ(kMnω) time. • We consider the single-source generalization of the replacement paths problem, where only the source s is fixed. We show how to solve this problem in all-pairs shortest paths time, currently õ(M14−ω n2+14−ω). Our runtime reduces to õ(Mnω) for positive weights, hence matching our mentioned result for the simpler replacement paths case (that, however, holds also for nonpositive weights). One of the ingredients that we use is an algorithm to compute the distances from a set s of source nodes to a set T of target nodes in õ(Mnω+|S|ṡ |T|ṡ (Mn)14−ω) time. This improves on a result in Yuster,Zwick—FOCS’05. • We present the first distance sensitivity oracle that achieves simultaneously subcubic preprocessing time and sublinear query time. More precisely, for a given parameter α ∈ [0,1], our oracle has preprocessing time Õ(Mnω + 1 2+Mnω + α (4−ω)) and query time Õ(n1−&alpha). The previous best oracle for small integer weights has Õ(Mnω +1−α) preprocessing time and (superlinear) Õ(n1+α) query time [Weimann,Yuster-FOCS’10]. From a technical point of view, an interesting and novel aspect of our oracle is that it exploits as a subroutine our single-source replacement paths alg
{"title":"Faster Replacement Paths and Distance Sensitivity Oracles","authors":"F. Grandoni, V. V. Williams","doi":"10.1145/3365835","DOIUrl":"https://doi.org/10.1145/3365835","url":null,"abstract":"Shortest paths computation is one of the most fundamental problems in computer science. An important variant of the problem is when edges can fail, and one needs to compute shortest paths that avoid a (failing) edge. More formally, given a source node s, a target node t, and an edge e, a replacement path for the triple (s,t,e) is a shortest s-t path avoiding edge e. Replacement paths computation can be seen either as a static problem or as a data structure problem. In the static setting, a typical goal is to compute for fixed s and t, for every possible failed edge e, the length of the best replacement path around e (replacement paths problem). In the data structure setting, a typical goal is to design a data structure (distance sensitivity oracle) that, after some preprocessing, quickly answers queries of the form: What is the length of the replacement path for the triple (s,t,e)? In this article, we focus on n-node directed graphs with integer edge weights in [−M,M], and present improved replacement paths algorithms and distance sensitivity oracles based on fast matrix multiplication. In more detail, we obtain the following main results: • We describe a replacement paths algorithm with runtime Õ(Mnω), where ω < 2.373 is the fast matrix multiplication exponent. For a comparison, the previous fastest algorithms have runtime õ(Mn1+2ω /3) [Weimann,Yuster—FOCS’10] and, in the unweighted case, õ(n2.5) [Roditty, Zwick—ICALP’05]. Our result shows that, at least for small integer weights, the replacement paths problem in directed graphs may be easier than the related all-pairs shortest paths problem, as the current best runtime for the latter is õ(M14−ω n2+1 4−ω): this is Ω (n2.5) even if ω = 2. Our algorithm also implies that the k shortest simple s-t paths can be computed in õ(kMnω) time. • We consider the single-source generalization of the replacement paths problem, where only the source s is fixed. We show how to solve this problem in all-pairs shortest paths time, currently õ(M14−ω n2+14−ω). Our runtime reduces to õ(Mnω) for positive weights, hence matching our mentioned result for the simpler replacement paths case (that, however, holds also for nonpositive weights). One of the ingredients that we use is an algorithm to compute the distances from a set s of source nodes to a set T of target nodes in õ(Mnω+|S|ṡ |T|ṡ (Mn)14−ω) time. This improves on a result in Yuster,Zwick—FOCS’05. • We present the first distance sensitivity oracle that achieves simultaneously subcubic preprocessing time and sublinear query time. More precisely, for a given parameter α ∈ [0,1], our oracle has preprocessing time Õ(Mnω + 1 2+Mnω + α (4−ω)) and query time Õ(n1−&alpha). The previous best oracle for small integer weights has Õ(Mnω +1−α) preprocessing time and (superlinear) Õ(n1+α) query time [Weimann,Yuster-FOCS’10]. From a technical point of view, an interesting and novel aspect of our oracle is that it exploits as a subroutine our single-source replacement paths alg","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122773115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article presents an algorithm that solves the 3SUM problem for n real numbers in O((n2/ log2n)(log log n)O(1)) time, improving previous solutions by about a logarithmic factor. Our framework for shaving off two logarithmic factors can be applied to other problems, such as (median,+)-convolution/matrix multiplication and algebraic generalizations of 3SUM. This work also obtains the first subquadratic results on some 3SUM-hard problems in computational geometry, for example, deciding whether (the interiors of) a constant number of simple polygons have a common intersection.
{"title":"More Logarithmic-factor Speedups for 3SUM, (median,+)-convolution, and Some Geometric 3SUM-hard Problems","authors":"Timothy M. Chan","doi":"10.1145/3363541","DOIUrl":"https://doi.org/10.1145/3363541","url":null,"abstract":"This article presents an algorithm that solves the 3SUM problem for n real numbers in O((n2/ log2n)(log log n)O(1)) time, improving previous solutions by about a logarithmic factor. Our framework for shaving off two logarithmic factors can be applied to other problems, such as (median,+)-convolution/matrix multiplication and algebraic generalizations of 3SUM. This work also obtains the first subquadratic results on some 3SUM-hard problems in computational geometry, for example, deciding whether (the interiors of) a constant number of simple polygons have a common intersection.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129637974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Knuth assigned the following open problem a difficulty rating of 48/50 in The Art of Computer Programming Volume 4A: For odd n ≥ 3, can the permutations of { 1,2,… , n} be ordered in a cyclic list so that each permutation is transformed into the next by applying either the operation σ, a rotation to the left, or τ, a transposition of the first two symbols? The Sigma-Tau problem is equivalent to finding a Hamilton cycle in the directed Cayley graph generated by σ = (1 2 ⋅ n) and τ = (1 2). In this article, we solve the Sigma-Tau problem by providing a simple O(n)-time successor rule to generate successive permutations of a Hamilton cycle in the aforementioned Cayley graph.
{"title":"Solving the Sigma-Tau Problem","authors":"J. Sawada, A. Williams","doi":"10.1145/3359589","DOIUrl":"https://doi.org/10.1145/3359589","url":null,"abstract":"Knuth assigned the following open problem a difficulty rating of 48/50 in The Art of Computer Programming Volume 4A: For odd n ≥ 3, can the permutations of { 1,2,… , n} be ordered in a cyclic list so that each permutation is transformed into the next by applying either the operation σ, a rotation to the left, or τ, a transposition of the first two symbols? The Sigma-Tau problem is equivalent to finding a Hamilton cycle in the directed Cayley graph generated by σ = (1 2 ⋅ n) and τ = (1 2). In this article, we solve the Sigma-Tau problem by providing a simple O(n)-time successor rule to generate successive permutations of a Hamilton cycle in the aforementioned Cayley graph.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"273 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124420180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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