A reporting query returns the objects satisfying a predicate q from an input set. In prioritized reporting, each object carries a real-valued weight (which can be query dependent), and a query returns the objects that satisfy q and have weights at least a threshold τ. A top-k query finds, among all the objects satisfying q, the k ones of the largest weights; a max query is a special instance with k = 1. We want to design data structures of small space to support queries (and possibly updates) efficiently. Previous work has shown that a top-k structure can also support max and prioritized queries with no performance deterioration. This article explores the opposite direction: do prioritized queries, possibly combined with max queries, imply top-k search? Subject to mild conditions, we provide affirmative answers with two reduction techniques. The first converts a prioritized structure into a static top-k structure with the same space complexity and only a logarithmic blowup in query time. If a max structure is available in addition, our second reduction yields a top-k structure with no degradation in expected performance (this holds for the space, query, and update complexities). Our techniques significantly simplify the design of top-k structures because structures for max and prioritized queries are often easier to obtain. We demonstrate this by developing top-k structures for interval stabbing, 3D dominance, halfspace reporting, linear ranking, and L∞ nearest neighbor search in the RAM and the external memory computation models.
{"title":"Generic Techniques for Building Top-k Structures","authors":"S. Rahul, Yufei Tao","doi":"10.1145/3546074","DOIUrl":"https://doi.org/10.1145/3546074","url":null,"abstract":"A reporting query returns the objects satisfying a predicate q from an input set. In prioritized reporting, each object carries a real-valued weight (which can be query dependent), and a query returns the objects that satisfy q and have weights at least a threshold τ. A top-k query finds, among all the objects satisfying q, the k ones of the largest weights; a max query is a special instance with k = 1. We want to design data structures of small space to support queries (and possibly updates) efficiently. Previous work has shown that a top-k structure can also support max and prioritized queries with no performance deterioration. This article explores the opposite direction: do prioritized queries, possibly combined with max queries, imply top-k search? Subject to mild conditions, we provide affirmative answers with two reduction techniques. The first converts a prioritized structure into a static top-k structure with the same space complexity and only a logarithmic blowup in query time. If a max structure is available in addition, our second reduction yields a top-k structure with no degradation in expected performance (this holds for the space, query, and update complexities). Our techniques significantly simplify the design of top-k structures because structures for max and prioritized queries are often easier to obtain. We demonstrate this by developing top-k structures for interval stabbing, 3D dominance, halfspace reporting, linear ranking, and L∞ nearest neighbor search in the RAM and the external memory computation models.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131619578","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}
Leader election is a fundamental task in distributed computing. It is a symmetry breaking problem, calling for one node of the network to become the leader, and for all other nodes to become non-leaders. We consider leader election in anonymous radio networks modeled as simple undirected connected graphs. Nodes communicate in synchronous rounds. In each round, a node can either transmit a message to all its neighbours, or stay silent and listen. A node v hears a message from a neighbour w in a given round if v listens in this round and if w is its only neighbour transmitting in this round. If v listens in a round in which more than one neighbour transmits, then v hears noise that is different from any message and different from silence. We assume that nodes are identical (anonymous) and execute the same deterministic algorithm. Under this scenario, symmetry can be broken only in one way: by different wake-up times of the nodes. In which situations is it possible to break symmetry and elect a leader using time as symmetry breaker? In order to answer this question, we consider configurations. A configuration is the underlying graph with nodes tagged by non-negative integers with the following meaning. A node can either wake up spontaneously in the round shown on its tag, according to some global clock, or can be woken up hearing a message sent by one of its already awoken neighbours. The local clock of a node starts at its wakeup and nodes do not have access to the global clock determining their tags. A configuration is feasible if there exists a distributed algorithm that elects a leader for this configuration. Our main result is a complete algorithmic characterization of feasible configurations. More precisely, we design a centralized decision algorithm, working in polynomial time, whose input is a configuration and which decides if the configuration is feasible. Using this algorithm we also provide a dedicated deterministic distributed leader election algorithm for each feasible configuration that elects a leader for this configuration in time O(n2σ, where n is the number of nodes and σ is the difference between the largest and smallest tag of the configuration. We then ask the question whether there exists a universal deterministic distributed algorithm electing a leader for all feasible configurations. The answer turns out to be no, and we show that such a universal algorithm cannot exist even for the class of 4-node feasible configurations. We also prove that a distributed version of our decision algorithm cannot exist.
{"title":"Deterministic Leader Election in Anonymous Radio Networks","authors":"Avery Miller, A. Pelc, R. Yadav","doi":"10.1145/3527171","DOIUrl":"https://doi.org/10.1145/3527171","url":null,"abstract":"Leader election is a fundamental task in distributed computing. It is a symmetry breaking problem, calling for one node of the network to become the leader, and for all other nodes to become non-leaders. We consider leader election in anonymous radio networks modeled as simple undirected connected graphs. Nodes communicate in synchronous rounds. In each round, a node can either transmit a message to all its neighbours, or stay silent and listen. A node v hears a message from a neighbour w in a given round if v listens in this round and if w is its only neighbour transmitting in this round. If v listens in a round in which more than one neighbour transmits, then v hears noise that is different from any message and different from silence. We assume that nodes are identical (anonymous) and execute the same deterministic algorithm. Under this scenario, symmetry can be broken only in one way: by different wake-up times of the nodes. In which situations is it possible to break symmetry and elect a leader using time as symmetry breaker? In order to answer this question, we consider configurations. A configuration is the underlying graph with nodes tagged by non-negative integers with the following meaning. A node can either wake up spontaneously in the round shown on its tag, according to some global clock, or can be woken up hearing a message sent by one of its already awoken neighbours. The local clock of a node starts at its wakeup and nodes do not have access to the global clock determining their tags. A configuration is feasible if there exists a distributed algorithm that elects a leader for this configuration. Our main result is a complete algorithmic characterization of feasible configurations. More precisely, we design a centralized decision algorithm, working in polynomial time, whose input is a configuration and which decides if the configuration is feasible. Using this algorithm we also provide a dedicated deterministic distributed leader election algorithm for each feasible configuration that elects a leader for this configuration in time O(n2σ, where n is the number of nodes and σ is the difference between the largest and smallest tag of the configuration. We then ask the question whether there exists a universal deterministic distributed algorithm electing a leader for all feasible configurations. The answer turns out to be no, and we show that such a universal algorithm cannot exist even for the class of 4-node feasible configurations. We also prove that a distributed version of our decision algorithm cannot exist.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132302353","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 address the problem of designing, implementing, and experimenting with compressed data structures that support rank and select queries over a dictionary of integers. We shine a new light on this classical problem by showing a connection between the input integers and the geometry of a set of points in a Cartesian plane suitably derived from them. We then build upon some results in computational geometry to introduce the first compressed rank/select dictionary based on the idea of “learning” the distribution of such points via proper linear approximations (LA). We therefore call this novel data structure the la_vector. We prove time and space complexities of the la_vector in several scenarios: in the worst case, in the case of input distributions with finite mean and variance, and taking into account the kth order entropy of some of its building blocks. We also discuss improved hybrid data structures, namely, ones that suitably orchestrate known compressed rank/select dictionaries with the la_vector. We corroborate our theoretical results with a large set of experiments over datasets originating from a variety of applications (Web search, DNA sequencing, information retrieval, and natural language processing) and show that our approach provides new interesting space-time tradeoffs with respect to many well-established compressed rank/select dictionary implementations. In particular, we show that our select is the fastest, and our rank is on the space-time Pareto frontier.
{"title":"A Learned Approach to Design Compressed Rank/Select Data Structures","authors":"A. Boffa, P. Ferragina, Giorgio Vinciguerra","doi":"10.1145/3524060","DOIUrl":"https://doi.org/10.1145/3524060","url":null,"abstract":"We address the problem of designing, implementing, and experimenting with compressed data structures that support rank and select queries over a dictionary of integers. We shine a new light on this classical problem by showing a connection between the input integers and the geometry of a set of points in a Cartesian plane suitably derived from them. We then build upon some results in computational geometry to introduce the first compressed rank/select dictionary based on the idea of “learning” the distribution of such points via proper linear approximations (LA). We therefore call this novel data structure the la_vector. We prove time and space complexities of the la_vector in several scenarios: in the worst case, in the case of input distributions with finite mean and variance, and taking into account the kth order entropy of some of its building blocks. We also discuss improved hybrid data structures, namely, ones that suitably orchestrate known compressed rank/select dictionaries with the la_vector. We corroborate our theoretical results with a large set of experiments over datasets originating from a variety of applications (Web search, DNA sequencing, information retrieval, and natural language processing) and show that our approach provides new interesting space-time tradeoffs with respect to many well-established compressed rank/select dictionary implementations. In particular, we show that our select is the fastest, and our rank is on the space-time Pareto frontier.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126545265","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 𝒢 be a minor-closed graph class. We say that a graph G is a k-apex of 𝒢 if G contains a set S of at most k vertices such that GS belongs to 𝒢. We denote by 𝒜k (𝒢) the set of all graphs that are k-apices of 𝒢. In the first paper of this series, we obtained upper bounds on the size of the graphs in the minor-obstruction set of 𝒜k (𝒢), i.e., the minor-minimal set of graphs not belonging to 𝒜k (𝒢). In this article, we provide an algorithm that, given a graph G on n vertices, runs in time 2poly(k) ⋅ n3 and either returns a set S certifying that G ∈ 𝒜k (𝒢), or reports that G ∉ 𝒜k (𝒢). Here poly is a polynomial function whose degree depends on the maximum size of a minor-obstruction of 𝒢. In the special case where 𝒢 excludes some apex graph as a minor, we give an alternative algorithm running in 2poly(k) ⋅ n2-time.
{"title":"k-apices of Minor-closed Graph Classes. II. Parameterized Algorithms","authors":"Ignasi Sau, Giannos Stamoulis, D. Thilikos","doi":"10.1145/3519028","DOIUrl":"https://doi.org/10.1145/3519028","url":null,"abstract":"Let 𝒢 be a minor-closed graph class. We say that a graph G is a k-apex of 𝒢 if G contains a set S of at most k vertices such that GS belongs to 𝒢. We denote by 𝒜k (𝒢) the set of all graphs that are k-apices of 𝒢. In the first paper of this series, we obtained upper bounds on the size of the graphs in the minor-obstruction set of 𝒜k (𝒢), i.e., the minor-minimal set of graphs not belonging to 𝒜k (𝒢). In this article, we provide an algorithm that, given a graph G on n vertices, runs in time 2poly(k) ⋅ n3 and either returns a set S certifying that G ∈ 𝒜k (𝒢), or reports that G ∉ 𝒜k (𝒢). Here poly is a polynomial function whose degree depends on the maximum size of a minor-obstruction of 𝒢. In the special case where 𝒢 excludes some apex graph as a minor, we give an alternative algorithm running in 2poly(k) ⋅ n2-time.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127321977","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}
Sayan Bhattacharya, F. Grandoni, Janardhan Kulkarni, Quanquan C. Liu, Shay Solomon
The problem of (Δ +1)-vertex coloring a graph of maximum degree Δ has been extremely well studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA’18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized algorithm for maintaining a (Δ +1)-coloring with O(log Δ) expected amortized update time. In this article, we present an improved randomized algorithm for (Δ +1)-coloring that achieves O(1) amortized update time and show that this bound holds not only in expectation but also with high probability. Our starting point is the state-of-the-art randomized algorithm for maintaining a maximal matching (Solomon, FOCS’16). We carefully build on the approach of Solomon, but, due to inherent differences between the maximal matching and (Δ +1)-coloring problems, we need to deviate significantly from it in several crucial and highly nontrivial points.1
{"title":"Fully Dynamic (Δ +1)-Coloring in O(1) Update Time","authors":"Sayan Bhattacharya, F. Grandoni, Janardhan Kulkarni, Quanquan C. Liu, Shay Solomon","doi":"10.1145/3494539","DOIUrl":"https://doi.org/10.1145/3494539","url":null,"abstract":"The problem of (Δ +1)-vertex coloring a graph of maximum degree Δ has been extremely well studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA’18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized algorithm for maintaining a (Δ +1)-coloring with O(log Δ) expected amortized update time. In this article, we present an improved randomized algorithm for (Δ +1)-coloring that achieves O(1) amortized update time and show that this bound holds not only in expectation but also with high probability. Our starting point is the state-of-the-art randomized algorithm for maintaining a maximal matching (Solomon, FOCS’16). We carefully build on the approach of Solomon, but, due to inherent differences between the maximal matching and (Δ +1)-coloring problems, we need to deviate significantly from it in several crucial and highly nontrivial points.1","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116454697","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 show the existence of an exact mimicking network of kO(log k) edges for minimum multicuts over a set of terminals in an undirected graph, where k is the total capacity of the terminals, i.e., the sum of the degrees of the terminal vertices. Furthermore, using the best available approximation algorithm for Small Set Expansion, we show that a mimicking network of kO(log3 k) edges can be computed in randomized polynomial time. As a consequence, we show quasipolynomial kernels for several problems, including Edge Multiway Cut, Group Feedback Edge Set for an arbitrary group, and Edge Multicut parameterized by the solution size and the number of cut requests. The result combines the matroid-based irrelevant edge approach used in the kernel for s-Multiway Cut with a recursive decomposition and sparsification of the graph along sparse cuts. This is the first progress on the kernelization of Multiway Cut problems since the kernel for s-Multiway Cut for constant value of s (Kratsch and Wahlström, FOCS 2012).
我们证明了无向图中一组终端上的最小多截点的精确模拟网络存在kO(log k)条边，其中k是终端的总容量，即终端顶点的度之和。此外，利用小集展开的最佳逼近算法，我们证明了kO(log3k)条边的模拟网络可以在随机多项式时间内计算出来。因此，我们展示了几个问题的拟多项式核，包括边缘多路切割，任意组的组反馈边缘集，以及由解决方案大小和切割请求数量参数化的边缘多路切割。该结果结合了s-多路切割核中使用的基于矩阵的不相关边缘方法，以及沿着稀疏切割的图的递归分解和稀疏化。这是自s-Multiway Cut (s为定值)的核以来，多路切割问题核化的第一个进展(Kratsch and Wahlström, FOCS 2012)。
{"title":"Quasipolynomial Multicut-mimicking Networks and Kernels for Multiway Cut Problems","authors":"Magnus Wahlström","doi":"10.1145/3501304","DOIUrl":"https://doi.org/10.1145/3501304","url":null,"abstract":"We show the existence of an exact mimicking network of kO(log k) edges for minimum multicuts over a set of terminals in an undirected graph, where k is the total capacity of the terminals, i.e., the sum of the degrees of the terminal vertices. Furthermore, using the best available approximation algorithm for Small Set Expansion, we show that a mimicking network of kO(log3 k) edges can be computed in randomized polynomial time. As a consequence, we show quasipolynomial kernels for several problems, including Edge Multiway Cut, Group Feedback Edge Set for an arbitrary group, and Edge Multicut parameterized by the solution size and the number of cut requests. The result combines the matroid-based irrelevant edge approach used in the kernel for s-Multiway Cut with a recursive decomposition and sparsification of the graph along sparse cuts. This is the first progress on the kernelization of Multiway Cut problems since the kernel for s-Multiway Cut for constant value of s (Kratsch and Wahlström, FOCS 2012).","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121122859","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 give a fully dynamic (Las-Vegas style) algorithm with constant expected amortized time per update that maintains a proper (Δ +1)-vertex coloring of a graph with maximum degree at most Δ. This improves upon the previous O(log Δ)-time algorithm by Bhattacharya et al. (SODA 2018). Our algorithm uses an approach based on assigning random ranks to vertices and does not need to maintain a hierarchical graph decomposition. We show that our result does not only have optimal running time but is also optimal in the sense that already deciding whether a Δ-coloring exists in a dynamically changing graph with maximum degree at most Δ takes Ω (log n) time per operation.
{"title":"Constant-time Dynamic (Δ +1)-Coloring","authors":"M. Henzinger, Pan Peng","doi":"10.1145/3501403","DOIUrl":"https://doi.org/10.1145/3501403","url":null,"abstract":"We give a fully dynamic (Las-Vegas style) algorithm with constant expected amortized time per update that maintains a proper (Δ +1)-vertex coloring of a graph with maximum degree at most Δ. This improves upon the previous O(log Δ)-time algorithm by Bhattacharya et al. (SODA 2018). Our algorithm uses an approach based on assigning random ranks to vertices and does not need to maintain a hierarchical graph decomposition. We show that our result does not only have optimal running time but is also optimal in the sense that already deciding whether a Δ-coloring exists in a dynamically changing graph with maximum degree at most Δ takes Ω (log n) time per operation.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"212 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131407868","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 prove a general connection between the communication complexity of two-player games and the sample complexity of their multi-player locally private analogues. We use this connection to prove sample complexity lower bounds for locally differentially private protocols as straightforward corollaries of results from communication complexity. In particular, we (1) use a communication lower bound for the hidden layers problem to prove an exponential sample complexity separation between sequentially and fully interactive locally private protocols, and (2) use a communication lower bound for the pointer chasing problem to prove an exponential sample complexity separation between k-round and (k+1)-round sequentially interactive locally private protocols, for every k.
{"title":"Exponential Separations in Local Privacy","authors":"Matthew Joseph, Jieming Mao, Aaron Roth","doi":"10.1145/3459095","DOIUrl":"https://doi.org/10.1145/3459095","url":null,"abstract":"We prove a general connection between the communication complexity of two-player games and the sample complexity of their multi-player locally private analogues. We use this connection to prove sample complexity lower bounds for locally differentially private protocols as straightforward corollaries of results from communication complexity. In particular, we (1) use a communication lower bound for the hidden layers problem to prove an exponential sample complexity separation between sequentially and fully interactive locally private protocols, and (2) use a communication lower bound for the pointer chasing problem to prove an exponential sample complexity separation between k-round and (k+1)-round sequentially interactive locally private protocols, for every k.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123093610","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 articles that were presented at the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) 2019 in San Diego, California, on January 6–9, 2019. These articles, selected on the basis of their high ratings by the conference program committee, have been thoroughly reviewed according to the journal’s highest standards. In “Polynomial-time algorithm for Maximum Weight Independent Set on P6-free graphs”, Andrzej Grzesik, Tereza Klimošová, Marcin Pilipczuk, and Michał Pilipczuk study the classic problem of max-weight independent set. They provide the first polynomial-time algorithm to solve the problem for the class of P6-free graphs, i.e., all graphs that contain no path on six vertices as an induced subgraph. The approach is based on a careful enumeration of vertex subsets that inherently characterize all maximal independent sets. The article “I/O-Efficient Algorithms for Topological Sort and Related Problems” by Nairen Cao, Jeremy Fineman, Katina Russell, and Eugene Yang, introduces the first randomized algorithms for topological sorting and for identifying the strongly connected components of a directed graph in the I/O model. These algorithms have a runtime of O (sort (E) · poly(logV )), where sort (E) is the time needed for sorting E elements in the I/O-model. The algorithms are based on a recursive approach, which iteratively updates a random labeling until vertices in the same strongly connected component have the same label and the labeling represents a topological sorting of the strongly connected components. These new algorithms also imply efficient I/O-algorithms for various reachability and shortest paths problems in directed acyclic graphs. In the article “SETH-Based Lower Bounds for Subset Sum and Bicriteria Path”, by Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay, the authors introduce a reduction fromk-SAT to SUBSET SUM on dense instances. The new reduction proves that Bellman’s SUBSET SUM algorithm from 1969, which runs in O (Tn) time on n numbers and target T , cannot be improved to T 1−ε 2 (n) time for any constant ε > 0, unless the Strong Exponential Time Hypothesis (SETH) is false. The reduction is based on the results of Behrends (1946) about the existence of certain sequences of numbers, combined with clever partitioning and encoding. The authors also extend their techniques to prove SETH-based lower bounds for the BICRITERIA s, t-PATH problem. The article “Optimal Las Vegas Approximate Near Neighbors in p” by Alexander Wei presents Las Vegas data structures for solving approximate near neighbors in R under the p -norm. For 1 ≤ p ≤ 2, the author provides a data structure that matches the performance of optimal localitysensitive hashing. Moreover, using a locality-sensitive filter construction, the article gives the first
我们很高兴地发表《ACM算法汇刊》特刊，其中包含2019年1月6日至9日在加利福尼亚州圣地亚哥举行的第30届ACM- siam离散算法研讨会(SODA)上发表的六篇文章的完整版本。这些文章是由会议计划委员会根据其高评级选出的，并根据该杂志的最高标准进行了彻底的审查。Andrzej Grzesik、Tereza Klimošová、Marcin Pilipczuk和michaov Pilipczuk在“P6-free图上最大权独立集的多项式时间算法”中研究了最大权独立集的经典问题。他们提供了第一个多项式时间算法来解决P6-free图类的问题，即所有在六个顶点上不包含路径的图作为诱导子图。该方法基于顶点子集的仔细枚举，这些子集固有地表征所有最大独立集。Nairen Cao、Jeremy Fineman、Katina Russell和Eugene Yang撰写的文章“拓扑排序和相关问题的I/O高效算法”介绍了用于拓扑排序和识别I/O模型中有向图的强连接分量的第一个随机算法。这些算法的运行时间为O (sort (E)·poly(logV))，其中sort (E)是对I/O模型中的E个元素进行排序所需的时间。该算法基于递归方法，迭代更新随机标记，直到相同强连接组件中的顶点具有相同的标记，并且标记表示强连接组件的拓扑排序。这些新算法还隐含了有效的I/ o算法来解决有向无环图中的各种可达性和最短路径问题。在Amir Abboud, Karl Bringmann, Danny Hermelin和Dvir Shabtay的文章“基于seth的子集和双标准路径的下界”中，作者介绍了在密集实例上从k- sat到子集Sum的约简。新的约简证明了1969年的Bellman子集SUM算法，在n个数和目标T上运行O (Tn)时间，对于任意常数ε > 0，不能改进到t1−ε 2 (n)时间，除非强指数时间假设(SETH)是假的。这种约简是基于Behrends(1946)关于某些数列存在性的结果，并结合了巧妙的划分和编码。作者还扩展了他们的技术来证明基于seth的BICRITERIA s, t-PATH问题的下界。Alexander Wei的文章“Optimal Las Vegas Approximate Near Neighbors in p”提出了在p范数下求解R中的近似近邻的Las Vegas数据结构。对于1≤p≤2，作者提供了一种与最优位置敏感哈希性能相匹配的数据结构。此外，本文还采用了一种位置敏感滤波器结构，给出了第一种
{"title":"Introduction to the ACM-SIAM Symposium on Discrete Algorithms (SODA) 2019 Special Issue","authors":"M. Hoefer, T. Kopelowitz","doi":"10.1145/3508460","DOIUrl":"https://doi.org/10.1145/3508460","url":null,"abstract":"We are delighted to present a Special Issue of ACM Transactions on Algorithms, containing full versions of six articles that were presented at the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) 2019 in San Diego, California, on January 6–9, 2019. These articles, selected on the basis of their high ratings by the conference program committee, have been thoroughly reviewed according to the journal’s highest standards. In “Polynomial-time algorithm for Maximum Weight Independent Set on P6-free graphs”, Andrzej Grzesik, Tereza Klimošová, Marcin Pilipczuk, and Michał Pilipczuk study the classic problem of max-weight independent set. They provide the first polynomial-time algorithm to solve the problem for the class of P6-free graphs, i.e., all graphs that contain no path on six vertices as an induced subgraph. The approach is based on a careful enumeration of vertex subsets that inherently characterize all maximal independent sets. The article “I/O-Efficient Algorithms for Topological Sort and Related Problems” by Nairen Cao, Jeremy Fineman, Katina Russell, and Eugene Yang, introduces the first randomized algorithms for topological sorting and for identifying the strongly connected components of a directed graph in the I/O model. These algorithms have a runtime of O (sort (E) · poly(logV )), where sort (E) is the time needed for sorting E elements in the I/O-model. The algorithms are based on a recursive approach, which iteratively updates a random labeling until vertices in the same strongly connected component have the same label and the labeling represents a topological sorting of the strongly connected components. These new algorithms also imply efficient I/O-algorithms for various reachability and shortest paths problems in directed acyclic graphs. In the article “SETH-Based Lower Bounds for Subset Sum and Bicriteria Path”, by Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay, the authors introduce a reduction fromk-SAT to SUBSET SUM on dense instances. The new reduction proves that Bellman’s SUBSET SUM algorithm from 1969, which runs in O (Tn) time on n numbers and target T , cannot be improved to T 1−ε 2 (n) time for any constant ε > 0, unless the Strong Exponential Time Hypothesis (SETH) is false. The reduction is based on the results of Behrends (1946) about the existence of certain sequences of numbers, combined with clever partitioning and encoding. The authors also extend their techniques to prove SETH-based lower bounds for the BICRITERIA s, t-PATH problem. The article “Optimal Las Vegas Approximate Near Neighbors in p” by Alexander Wei presents Las Vegas data structures for solving approximate near neighbors in R under the p -norm. For 1 ≤ p ≤ 2, the author provides a data structure that matches the performance of optimal localitysensitive hashing. Moreover, using a locality-sensitive filter construction, the article gives the first","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116885118","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 show that approximate near neighbor search in high dimensions can be solved in a Las Vegas fashion (i.e., without false negatives) for ℓp (1≤ p≤ 2) while matching the performance of optimal locality-sensitive hashing. Specifically, we construct a data-independent Las Vegas data structure with query time O(dnρ) and space usage O(dn1+ρ) for (r, c r)-approximate near neighbors in Rd under the ℓp norm, where ρ = 1/cp + o(1). Furthermore, we give a Las Vegas locality-sensitive filter construction for the unit sphere that can be used with the data-dependent data structure of Andoni et al. (SODA 2017) to achieve optimal space-time tradeoffs in the data-dependent setting. For the symmetric case, this gives us a data-dependent Las Vegas data structure with query time O(dnρ) and space usage O(dn1+ρ) for (r, c r)-approximate near neighbors in Rd under the ℓp norm, where ρ = 1/(2cp - 1) + o(1). Our data-independent construction improves on the recent Las Vegas data structure of Ahle (FOCS 2017) for ℓp when 1 < p≤ 2. Our data-dependent construction performs even better for ℓp for all pε [1, 2] and is the first Las Vegas approximate near neighbors data structure to make use of data-dependent approaches. We also answer open questions of Indyk (SODA 2000), Pagh (SODA 2016), and Ahle by showing that for approximate near neighbors, Las Vegas data structures can match state-of-the-art Monte Carlo data structures in performance for both the data-independent and data-dependent settings and across space-time tradeoffs.
{"title":"Optimal Las Vegas Approximate Near Neighbors in ℓp","authors":"Alexander Wei","doi":"10.1145/3461777","DOIUrl":"https://doi.org/10.1145/3461777","url":null,"abstract":"We show that approximate near neighbor search in high dimensions can be solved in a Las Vegas fashion (i.e., without false negatives) for ℓp (1≤ p≤ 2) while matching the performance of optimal locality-sensitive hashing. Specifically, we construct a data-independent Las Vegas data structure with query time O(dnρ) and space usage O(dn1+ρ) for (r, c r)-approximate near neighbors in Rd under the ℓp norm, where ρ = 1/cp + o(1). Furthermore, we give a Las Vegas locality-sensitive filter construction for the unit sphere that can be used with the data-dependent data structure of Andoni et al. (SODA 2017) to achieve optimal space-time tradeoffs in the data-dependent setting. For the symmetric case, this gives us a data-dependent Las Vegas data structure with query time O(dnρ) and space usage O(dn1+ρ) for (r, c r)-approximate near neighbors in Rd under the ℓp norm, where ρ = 1/(2cp - 1) + o(1). Our data-independent construction improves on the recent Las Vegas data structure of Ahle (FOCS 2017) for ℓp when 1 < p≤ 2. Our data-dependent construction performs even better for ℓp for all pε [1, 2] and is the first Las Vegas approximate near neighbors data structure to make use of data-dependent approaches. We also answer open questions of Indyk (SODA 2000), Pagh (SODA 2016), and Ahle by showing that for approximate near neighbors, Las Vegas data structures can match state-of-the-art Monte Carlo data structures in performance for both the data-independent and data-dependent settings and across space-time tradeoffs.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130843553","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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