This article studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within n1-ε for any ε > 0, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring or alternatively, study restricted families of graphs. Toward understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm for C-coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. Optimizing the number of recolorings, sometimes referred to as the recourse bound, is important for various practical applications. In WADS ’17, Barba et al. devised two complementary algorithms: for any β > 0, the first (respectively, second) maintains an O(Cβn1/β) (respectively, O(Cβ)-coloring while recoloring O(β) (respectively, O(βn1/β)) vertices per update. Barba et al. also showed that the second trade-off appears to exhibit the right behavior, at least for β = O(1): any algorithm that maintains a C-coloring of an n-vertex dynamic forest must recolor Ω (n2C(C-1)) vertices per update, for any constant C ≥ 2. Our contribution is twofold: • We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: for any β > 0, we get a Ô (Cβlog2 n)-coloring with O(β) recolorings per update, where the Ô notation suppresses polyloglog(n) factors. In particular, for β = O(1), we get constant recolorings with polylog(n) colors; not only is this an exponential improvement over the previous bound but also it unveils a rather surprising phenomenon: the trade-off between the number of colors and recolorings is highly non-symmetric. • For uniformly sparse graphs, we use low out-degree orientations to strengthen the preceding result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest. From this data structure, we get a deterministic algorithm for graphs of arboricity ɑ that maintains an O(ɑ log2 n)-coloring in amortized O(1) time.
{"title":"Improved Dynamic Graph Coloring","authors":"Shay Solomon, Nicole Wein","doi":"10.1145/3392724","DOIUrl":"https://doi.org/10.1145/3392724","url":null,"abstract":"This article studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within n1-ε for any ε > 0, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring or alternatively, study restricted families of graphs. Toward understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm for C-coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. Optimizing the number of recolorings, sometimes referred to as the recourse bound, is important for various practical applications. In WADS ’17, Barba et al. devised two complementary algorithms: for any β > 0, the first (respectively, second) maintains an O(Cβn1/β) (respectively, O(Cβ)-coloring while recoloring O(β) (respectively, O(βn1/β)) vertices per update. Barba et al. also showed that the second trade-off appears to exhibit the right behavior, at least for β = O(1): any algorithm that maintains a C-coloring of an n-vertex dynamic forest must recolor Ω (n2C(C-1)) vertices per update, for any constant C ≥ 2. Our contribution is twofold: • We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: for any β > 0, we get a Ô (Cβlog2 n)-coloring with O(β) recolorings per update, where the Ô notation suppresses polyloglog(n) factors. In particular, for β = O(1), we get constant recolorings with polylog(n) colors; not only is this an exponential improvement over the previous bound but also it unveils a rather surprising phenomenon: the trade-off between the number of colors and recolorings is highly non-symmetric. • For uniformly sparse graphs, we use low out-degree orientations to strengthen the preceding result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest. From this data structure, we get a deterministic algorithm for graphs of arboricity ɑ that maintains an O(ɑ log2 n)-coloring in amortized O(1) time.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131029685","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 present fully polynomial-time (deterministic or randomised) approximation schemes for Holant problems, defined by a non-negative constraint function satisfying a generalised second-order recurrence modulo in a couple of exceptional cases. As a consequence, any non-negative Holant problem on cubic graphs has an efficient approximation algorithm unless the problem is equivalent to approximately counting perfect matchings, a central open problem in the area. This is in sharp contrast to the computational phase transition shown by two-state spin systems on cubic graphs. Our main technique is the recently established connection between zeros of graph polynomials and approximate counting.
{"title":"Zeros of Holant Problems","authors":"Heng Guo, Chao Liao, P. Lu, Chihao Zhang","doi":"10.1145/3418056","DOIUrl":"https://doi.org/10.1145/3418056","url":null,"abstract":"We present fully polynomial-time (deterministic or randomised) approximation schemes for Holant problems, defined by a non-negative constraint function satisfying a generalised second-order recurrence modulo in a couple of exceptional cases. As a consequence, any non-negative Holant problem on cubic graphs has an efficient approximation algorithm unless the problem is equivalent to approximately counting perfect matchings, a central open problem in the area. This is in sharp contrast to the computational phase transition shown by two-state spin systems on cubic graphs. Our main technique is the recently established connection between zeros of graph polynomials and approximate counting.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129579819","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}
F. Fomin, P. Golovach, D. Lokshtanov, Fahad Panolan, Saket Saurabh
We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constraints. The new constrained clustering problem generalizes a number of problems and by solving it, we obtain the first linear time-approximation schemes for a number of well-studied fundamental problems concerning clustering of binary vectors and low-rank approximation of binary matrices. Among the problems solvable by our approach are Low GF(2)-Rank Approximation, Low Boolean-Rank Approximation, and various versions of Binary Clustering. For example, for Low GF(2)-Rank Approximation problem, where for an m× n binary matrix A and integer r> 0, we seek for a binary matrix B of GF(2) rank at most r such that the ℓ0-norm of matrix A−B is minimum, our algorithm, for any ε > 0 in time f(r,ε)⋅ n⋅ m, where f is some computable function, outputs a (1+ε)-approximate solution with probability at least (1−1e). This is the first linear time approximation scheme for these problems. We also give (deterministic) PTASes for these problems running in time nf(r)1ε2log 1ε, where f is some function depending on the problem. Our algorithm for the constrained clustering problem is based on a novel sampling lemma, which is interesting on its own.
{"title":"Approximation Schemes for Low-rank Binary Matrix Approximation Problems","authors":"F. Fomin, P. Golovach, D. Lokshtanov, Fahad Panolan, Saket Saurabh","doi":"10.1145/3365653","DOIUrl":"https://doi.org/10.1145/3365653","url":null,"abstract":"We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constraints. The new constrained clustering problem generalizes a number of problems and by solving it, we obtain the first linear time-approximation schemes for a number of well-studied fundamental problems concerning clustering of binary vectors and low-rank approximation of binary matrices. Among the problems solvable by our approach are Low GF(2)-Rank Approximation, Low Boolean-Rank Approximation, and various versions of Binary Clustering. For example, for Low GF(2)-Rank Approximation problem, where for an m× n binary matrix A and integer r> 0, we seek for a binary matrix B of GF(2) rank at most r such that the ℓ0-norm of matrix A−B is minimum, our algorithm, for any ε > 0 in time f(r,ε)⋅ n⋅ m, where f is some computable function, outputs a (1+ε)-approximate solution with probability at least (1−1e). This is the first linear time approximation scheme for these problems. We also give (deterministic) PTASes for these problems running in time nf(r)1ε2log 1ε, where f is some function depending on the problem. Our algorithm for the constrained clustering problem is based on a novel sampling lemma, which is interesting on its own.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130634402","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 exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex u, a target vertex v and a set X of k failed vertices, such an oracle returns the length of a shortest u-to-v path that avoids all vertices in X. We propose oracles that can handle any number k of failures. We show several tradeoffs between space, query time, and preprocessing time. In particular, for a directed weighted planar graph with n vertices and any constant k, we show an Õ(n)-size, Õ(√ n)-query-time oracle.1 We then present a space vs. query time tradeoff: for any q ε [ 1,√ n ], we propose an oracle of size nk+1+o(1)/q2k that answers queries in Õ(q) time. For single vertex failures (k = 1), our n2+o(1)/q2-size, Õ(q)-query-time oracle improves over the previously best known tradeoff of Baswana et al. SODA 2012 by polynomial factors for q ≥ nt, for any t ∈ (0,1/2]. For multiple failures, no planarity exploiting results were previously known.
{"title":"Exact Distance Oracles for Planar Graphs with Failing Vertices","authors":"P. Charalampopoulos, S. Mozes, Benjamin Tebeka","doi":"10.1145/3511541","DOIUrl":"https://doi.org/10.1145/3511541","url":null,"abstract":"We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex u, a target vertex v and a set X of k failed vertices, such an oracle returns the length of a shortest u-to-v path that avoids all vertices in X. We propose oracles that can handle any number k of failures. We show several tradeoffs between space, query time, and preprocessing time. In particular, for a directed weighted planar graph with n vertices and any constant k, we show an Õ(n)-size, Õ(√ n)-query-time oracle.1 We then present a space vs. query time tradeoff: for any q ε [ 1,√ n ], we propose an oracle of size nk+1+o(1)/q2k that answers queries in Õ(q) time. For single vertex failures (k = 1), our n2+o(1)/q2-size, Õ(q)-query-time oracle improves over the previously best known tradeoff of Baswana et al. SODA 2012 by polynomial factors for q ≥ nt, for any t ∈ (0,1/2]. For multiple failures, no planarity exploiting results were previously known.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131176701","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}
Ali Bibak, Charles Carlson, Karthekeyan Chandrasekaran
Finding locally optimal solutions for MAX-CUT and MAX-k-CUT are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin (ACM Transactions on Algorithms, 2017) showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei (STOC, 2017) showed that the smoothed complexity of FLIP for max-cut in complete graphs is (OΦ5n15.1), where Φ is an upper bound on the random edge-weight density and Φ is the number of vertices in the input graph. While Angel, Bubeck, Peres, and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress toward improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O(Φ n7.83). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for MAX-3-CUT in complete graphs is polynomial and for MAX-k-CUT in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest toward showing smoothed polynomial complexity of FLIP for MAX-k-CUT in complete graphs for larger constants k.
寻找MAX-CUT和MAX-k-CUT的局部最优解是众所周知的pls完全问题。寻找这种局部最优解的一种本能的方法是FLIP方法。尽管FLIP在最坏情况下需要指数级的时间，但在实际情况下，它往往会很快终止。为了解释这种差异，我们在平滑复杂度框架下研究了FLIP的运行时间。Etscheid和Röglin (ACM Transactions on Algorithms, 2017)表明，任意图中最大切的FLIP平滑复杂度为拟多项式。Angel, Bubeck, Peres, and Wei (STOC, 2017)表明，完全图中最大切的FLIP平滑复杂度为(OΦ5n15.1)，其中Φ为随机边权密度的上界，Φ为输入图中的顶点数。虽然Angel, Bubeck, Peres和Wei的结果显示了第一个多项式平滑复杂性，但他们也推测他们的运行时间界限远非最佳。在这项工作中，我们在改进运行时界限方面取得了实质性进展。证明了完全图中最大切的FLIP的平滑复杂度为0 (Φ n7.83)。我们的结果基于一个精心选择的矩阵，该矩阵的秩捕获了方法的运行时间，以及该矩阵的改进秩界和基于该矩阵的改进联合界。此外，我们的技术为在平滑框架中分析FLIP提供了一个通用框架。我们通过证明完全图中MAX-3-CUT的FLIP平滑复杂度是多项式，任意图中MAX-k-CUT的FLIP平滑复杂度是拟多项式来说明这个一般框架。我们相信，我们的技术也应该对在较大常数k的完全图中显示MAX-k-CUT的FLIP的光滑多项式复杂度感兴趣。
{"title":"Improving the Smoothed Complexity of FLIP for Max Cut Problems","authors":"Ali Bibak, Charles Carlson, Karthekeyan Chandrasekaran","doi":"10.1145/3454125","DOIUrl":"https://doi.org/10.1145/3454125","url":null,"abstract":"Finding locally optimal solutions for MAX-CUT and MAX-k-CUT are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin (ACM Transactions on Algorithms, 2017) showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei (STOC, 2017) showed that the smoothed complexity of FLIP for max-cut in complete graphs is (OΦ5n15.1), where Φ is an upper bound on the random edge-weight density and Φ is the number of vertices in the input graph. While Angel, Bubeck, Peres, and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress toward improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O(Φ n7.83). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for MAX-3-CUT in complete graphs is polynomial and for MAX-k-CUT in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest toward showing smoothed polynomial complexity of FLIP for MAX-k-CUT in complete graphs for larger constants k.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"2003 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127331921","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 introduce the zip tree,1 a form of randomized binary search tree that integrates previous ideas into one practical, performant, and pleasant-to-implement package. A zip tree is a binary search tree in which each node has a numeric rank and the tree is (max)-heap-ordered with respect to ranks, with rank ties broken in favor of smaller keys. Zip trees are essentially treaps [8], except that ranks are drawn from a geometric distribution instead of a uniform distribution, and we allow rank ties. These changes enable us to use fewer random bits per node. We perform insertions and deletions by unmerging and merging paths (unzipping and zipping) rather than by doing rotations, which avoids some pointer changes and improves efficiency. The methods of zipping and unzipping take inspiration from previous top-down approaches to insertion and deletion by Stephenson [10], Martínez and Roura [5], and Sprugnoli [9]. From a theoretical standpoint, this work provides two main results. First, zip trees require only O(log log n) bits (with high probability) to represent the largest rank in an n-node binary search tree; previous data structures require O(log n) bits for the largest rank. Second, zip trees are naturally isomorphic to skip lists [7], and simplify Dean and Jones’ mapping between skip lists
{"title":"Zip Trees","authors":"R. Tarjan, Caleb C. Levy, Stephen Timmel","doi":"10.1145/3476830","DOIUrl":"https://doi.org/10.1145/3476830","url":null,"abstract":"We introduce the zip tree,1 a form of randomized binary search tree that integrates previous ideas into one practical, performant, and pleasant-to-implement package. A zip tree is a binary search tree in which each node has a numeric rank and the tree is (max)-heap-ordered with respect to ranks, with rank ties broken in favor of smaller keys. Zip trees are essentially treaps [8], except that ranks are drawn from a geometric distribution instead of a uniform distribution, and we allow rank ties. These changes enable us to use fewer random bits per node. We perform insertions and deletions by unmerging and merging paths (unzipping and zipping) rather than by doing rotations, which avoids some pointer changes and improves efficiency. The methods of zipping and unzipping take inspiration from previous top-down approaches to insertion and deletion by Stephenson [10], Martínez and Roura [5], and Sprugnoli [9]. From a theoretical standpoint, this work provides two main results. First, zip trees require only O(log log n) bits (with high probability) to represent the largest rank in an n-node binary search tree; previous data structures require O(log n) bits for the largest rank. Second, zip trees are naturally isomorphic to skip lists [7], and simplify Dean and Jones’ mapping between skip lists","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"67 1-3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123469098","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}
Let V be a set of n points in Rd, which we call voters. A point p ∈ Rd is preferred over another point p′ ∈ Rd by a voter υ ∈ V if dist(υ, p) < dist(υ, p′). A point p is called a plurality point if it is preferred by at least as many voters as any other point p′. We present an algorithm that decides in O(nlogn) time whether V admits a plurality point in the L2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute a minimum-cost subset W ⊂ V such that VW admits a plurality point, and to compute a so-called minimum-radius plurality ball. Finally, we consider the problem in the personalized L1 norm, where each point υ ∈ V has a preference vector ⟨w1(υ),…,wd(υ)⟩ and the distance from υ to any point p ∈ Rd is given by ∑i=1d wi(υ)· |xi(υ)−xi(p)|. For this case we can compute in O(nd−1) time the set of all plurality points of V. When all preference vectors are equal, the running time improves to O(n).
设V是Rd中n个点的集合，我们称之为投票人。如果dist(υ， p) < dist(υ， p ')，选民υ∈V更喜欢点p∈Rd而不是另一个点p '∈Rd。如果一个点p得到至少和其他点p一样多的选民的支持，就称为“多数点”。我们提出了一种算法，该算法在O(nlogn)时间内决定V在L2范数中是否有复数点，如果有，则找到(唯一的)复数点。我们还给出了计算最小代价子集W∧V使得VW允许一个复数点的有效算法，以及计算一个所谓的最小半径复数球的有效算法。最后，我们考虑个性化L1范数中的问题，其中每个点υ∈V具有⟨w1(υ)，…，wd(υ)⟩的偏好向量，并且从υ到任何点p∈Rd的距离由∑i=1d wi(υ)·|xi(υ)−xi(p)|给出。对于这种情况，我们可以在O(nd−1)时间内计算出v的所有复数点的集合。当所有偏好向量相等时，运行时间提高到O(n)。
{"title":"Faster Algorithms for Computing Plurality Points","authors":"M. D. Berg, Joachim Gudmundsson, M. Mehr","doi":"10.1145/3186990","DOIUrl":"https://doi.org/10.1145/3186990","url":null,"abstract":"Let V be a set of n points in Rd, which we call voters. A point p ∈ Rd is preferred over another point p′ ∈ Rd by a voter υ ∈ V if dist(υ, p) < dist(υ, p′). A point p is called a plurality point if it is preferred by at least as many voters as any other point p′. We present an algorithm that decides in O(nlogn) time whether V admits a plurality point in the L2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute a minimum-cost subset W ⊂ V such that VW admits a plurality point, and to compute a so-called minimum-radius plurality ball. Finally, we consider the problem in the personalized L1 norm, where each point υ ∈ V has a preference vector ⟨w1(υ),…,wd(υ)⟩ and the distance from υ to any point p ∈ Rd is given by ∑i=1d wi(υ)· |xi(υ)−xi(p)|. For this case we can compute in O(nd−1) time the set of all plurality points of V. When all preference vectors are equal, the running time improves to O(n).","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130050464","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 a new construction of locality-sensitive hash functions for Hamming space that is covering in the sense that is it guaranteed to produce a collision for every pair of vectors within a given radius r. The construction is efficient in the sense that the expected number of hash collisions between vectors at distance cr, for a given c>1, comes close to that of the best possible data independent LSH without the covering guarantee, namely, the seminal LSH construction of Indyk and Motwani (STOC’98). The efficiency of the new construction essentially matches their bound when the search radius is not too large—e.g., when cr = o(log (n)/ log log n), where n is the number of points in the dataset, and when cr = log (n)/k, where k is an integer constant. In general, it differs by at most a factor ln (4) in the exponent of the time bounds. As a consequence, LSH-based similarity search in Hamming space can avoid the problem of false negatives at little or no cost in efficiency.
{"title":"CoveringLSH","authors":"R. Pagh","doi":"10.1145/3155300","DOIUrl":"https://doi.org/10.1145/3155300","url":null,"abstract":"We consider a new construction of locality-sensitive hash functions for Hamming space that is covering in the sense that is it guaranteed to produce a collision for every pair of vectors within a given radius r. The construction is efficient in the sense that the expected number of hash collisions between vectors at distance cr, for a given c>1, comes close to that of the best possible data independent LSH without the covering guarantee, namely, the seminal LSH construction of Indyk and Motwani (STOC’98). The efficiency of the new construction essentially matches their bound when the search radius is not too large—e.g., when cr = o(log (n)/ log log n), where n is the number of points in the dataset, and when cr = log (n)/k, where k is an integer constant. In general, it differs by at most a factor ln (4) in the exponent of the time bounds. As a consequence, LSH-based similarity search in Hamming space can avoid the problem of false negatives at little or no cost in efficiency.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121993802","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 study the problem of two-dimensional orthogonal range counting with additive error. Given a set P of n points drawn from an n× n grid and an error parameter ε, the goal is to build a data structure, such that for any orthogonal range R, it can return the number of points in P ∩ R with additive error ε n. A well-known solution for this problem is obtained by using ε-approximation, which is a subset A⊆ P that can estimate the number of points in P ∩ R with the number of points in A ∩ R. It is known that an ε-approximation of size O(1/ε log 2.5 1/ε) exists for any P with respect to orthogonal ranges, and the best lower bound is Ω(1/ε log 1/ε). The ε-approximation is a rather restricted data structure, as we are not allowed to store any information other than the coordinates of the points. In this article, we explore what can be achieved without any restriction on the data structure. We first describe a simple data structure that uses O(1/ε(log 21/ε + log n)) bits and answers queries with error ε n. We then prove a lower bound that any data structure that answers queries with error ε n will have to use Ω(1/ε (log 21/ε + log n)) bits. Our lower bound is information-theoretic: We show that there is a collection of 2Ω(nlog n) point sets with large union combinatorial discrepancy and thus are hard to distinguish unless we use Ω(nlog n) bits.
{"title":"Tight Space Bounds for Two-Dimensional Approximate Range Counting","authors":"Zhewei Wei, K. Yi","doi":"10.1145/3205454","DOIUrl":"https://doi.org/10.1145/3205454","url":null,"abstract":"We study the problem of two-dimensional orthogonal range counting with additive error. Given a set P of n points drawn from an n× n grid and an error parameter ε, the goal is to build a data structure, such that for any orthogonal range R, it can return the number of points in P ∩ R with additive error ε n. A well-known solution for this problem is obtained by using ε-approximation, which is a subset A⊆ P that can estimate the number of points in P ∩ R with the number of points in A ∩ R. It is known that an ε-approximation of size O(1/ε log 2.5 1/ε) exists for any P with respect to orthogonal ranges, and the best lower bound is Ω(1/ε log 1/ε). The ε-approximation is a rather restricted data structure, as we are not allowed to store any information other than the coordinates of the points. In this article, we explore what can be achieved without any restriction on the data structure. We first describe a simple data structure that uses O(1/ε(log 21/ε + log n)) bits and answers queries with error ε n. We then prove a lower bound that any data structure that answers queries with error ε n will have to use Ω(1/ε (log 21/ε + log n)) bits. Our lower bound is information-theoretic: We show that there is a collection of 2Ω(nlog n) point sets with large union combinatorial discrepancy and thus are hard to distinguish unless we use Ω(nlog n) bits.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125365273","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}
Marcin Bienkowski, T. Jurdzinski, M. Korzeniowski, D. Kowalski
We consider the problems of online and stochastic packet queueing in a distributed system of n nodes with queues, where the communication between the nodes is done via a multiple access channel. In the online setting, in each round, an arbitrary number of packets can be injected to nodes’ queues. Two measures of performance are considered: the total number of packets in all queues, called the total load, and the maximum queue size, called the maximum load. We develop a deterministic distributed algorithm that is asymptotically optimal with respect to both complexity measures, in a competitive way. More precisely, the total load of our algorithm is bigger than the total load of any other algorithm, including centralized online solutions, by only an additive term of O(n2), whereas the maximum queue size of our algorithm is at most n times bigger than the maximum queue size of any other algorithm, with an extra additive O(n). The optimality for both measures is justified by proving the corresponding lower bounds, which also separates nearly exponentially distributed solutions from the centralized ones. Next, we show that our algorithm is also stochastically stable for any expected injection rate smaller or equal to 1. This is the first solution to the stochastic queueing problem on a multiple access channel that achieves such stability for the (highest possible) rate equal to 1.
{"title":"Distributed Online and Stochastic Queueing on a Multiple Access Channel","authors":"Marcin Bienkowski, T. Jurdzinski, M. Korzeniowski, D. Kowalski","doi":"10.1145/3182396","DOIUrl":"https://doi.org/10.1145/3182396","url":null,"abstract":"We consider the problems of online and stochastic packet queueing in a distributed system of n nodes with queues, where the communication between the nodes is done via a multiple access channel. In the online setting, in each round, an arbitrary number of packets can be injected to nodes’ queues. Two measures of performance are considered: the total number of packets in all queues, called the total load, and the maximum queue size, called the maximum load. We develop a deterministic distributed algorithm that is asymptotically optimal with respect to both complexity measures, in a competitive way. More precisely, the total load of our algorithm is bigger than the total load of any other algorithm, including centralized online solutions, by only an additive term of O(n2), whereas the maximum queue size of our algorithm is at most n times bigger than the maximum queue size of any other algorithm, with an extra additive O(n). The optimality for both measures is justified by proving the corresponding lower bounds, which also separates nearly exponentially distributed solutions from the centralized ones. Next, we show that our algorithm is also stochastically stable for any expected injection rate smaller or equal to 1. This is the first solution to the stochastic queueing problem on a multiple access channel that achieves such stability for the (highest possible) rate equal to 1.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131034554","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：