SIAM Journal on Computing, Volume 53, Issue 2, Page 524-571, April 2024. Abstract. Let [math] be an [math]-point metric space. We assume that [math] is given in the distance oracle model, that is, [math] and for every pair of points [math] from [math] we can query their distance [math] in constant time. A [math]-nearest neighbor ([math]-NN) graph for [math] is a directed graph [math] that has an edge to each of [math]’s [math] nearest neighbors. We use [math] to denote the sum of edge weights of [math]. In this paper, we study the problem of approximating [math] in sublinear time when we are given oracle access to the metric space [math] that defines [math]. Our goal is to develop an algorithm that solves this problem faster than the time required to compute [math]. We first present an algorithm that in [math] time with probability at least [math] approximates [math] to within a factor of [math]. Next, we present a more elaborate sublinear algorithm that in time [math] computes an estimate [math] of [math] that satisfies with probability at least [math] [math], where [math] denotes the cost of the minimum spanning tree of [math]. Further, we complement these results with near matching lower bounds. We show that any algorithm that for a given metric space [math] of size [math], with probability at least [math], estimates [math] to within a [math] factor requires [math] time. Similarly, any algorithm that with probability at least [math] estimates [math] to within an additive error term [math] requires [math] time.
{"title":"Sublinear Time Approximation of the Cost of a Metric [math]-Nearest Neighbor Graph","authors":"Artur Czumaj, Christian Sohler","doi":"10.1137/22m1544105","DOIUrl":"https://doi.org/10.1137/22m1544105","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 2, Page 524-571, April 2024. <br/> Abstract. Let [math] be an [math]-point metric space. We assume that [math] is given in the distance oracle model, that is, [math] and for every pair of points [math] from [math] we can query their distance [math] in constant time. A [math]-nearest neighbor ([math]-NN) graph for [math] is a directed graph [math] that has an edge to each of [math]’s [math] nearest neighbors. We use [math] to denote the sum of edge weights of [math]. In this paper, we study the problem of approximating [math] in sublinear time when we are given oracle access to the metric space [math] that defines [math]. Our goal is to develop an algorithm that solves this problem faster than the time required to compute [math]. We first present an algorithm that in [math] time with probability at least [math] approximates [math] to within a factor of [math]. Next, we present a more elaborate sublinear algorithm that in time [math] computes an estimate [math] of [math] that satisfies with probability at least [math] [math], where [math] denotes the cost of the minimum spanning tree of [math]. Further, we complement these results with near matching lower bounds. We show that any algorithm that for a given metric space [math] of size [math], with probability at least [math], estimates [math] to within a [math] factor requires [math] time. Similarly, any algorithm that with probability at least [math] estimates [math] to within an additive error term [math] requires [math] time.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140615382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michael A. Bender, Alex Conway, Martín Farach-Colton, Hanna Komlós, William Kuszmaul, Nicole Wein
SIAM Journal on Computing, Ahead of Print. Abstract. The online list-labeling problem is an algorithmic primitive with a large literature of upper bounds, lower bounds, and applications. The goal is to store a dynamically changing set of [math] items in an array of [math] slots, while maintaining the invariant that the items appear in sorted order and while minimizing the relabeling cost, defined to be the number of items that are moved per insertion/deletion. For the linear regime, where [math], an upper bound of [math] on the relabeling cost has been known since 1981. A lower bound of [math] is known for deterministic algorithms and for so-called smooth algorithms, but the best general lower bound remains [math]. The central open question in the field is whether [math] is optimal for all algorithms. In this paper, we give a randomized data structure that achieves an expected relabeling cost of [math] per operation. More generally, if [math] for [math], the expected relabeling cost becomes [math]. Our solution is history independent, meaning that the state of the data structure is independent of the order in which items are inserted/deleted. For history-independent data structures, we also prove a matching lower bound: for all [math] between [math] and some sufficiently small positive constant, the optimal expected cost for history-independent list-labeling solutions is [math].
{"title":"Online List Labeling: Breaking the [math] Barrier","authors":"Michael A. Bender, Alex Conway, Martín Farach-Colton, Hanna Komlós, William Kuszmaul, Nicole Wein","doi":"10.1137/22m1534468","DOIUrl":"https://doi.org/10.1137/22m1534468","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. The online list-labeling problem is an algorithmic primitive with a large literature of upper bounds, lower bounds, and applications. The goal is to store a dynamically changing set of [math] items in an array of [math] slots, while maintaining the invariant that the items appear in sorted order and while minimizing the relabeling cost, defined to be the number of items that are moved per insertion/deletion. For the linear regime, where [math], an upper bound of [math] on the relabeling cost has been known since 1981. A lower bound of [math] is known for deterministic algorithms and for so-called smooth algorithms, but the best general lower bound remains [math]. The central open question in the field is whether [math] is optimal for all algorithms. In this paper, we give a randomized data structure that achieves an expected relabeling cost of [math] per operation. More generally, if [math] for [math], the expected relabeling cost becomes [math]. Our solution is history independent, meaning that the state of the data structure is independent of the order in which items are inserted/deleted. For history-independent data structures, we also prove a matching lower bound: for all [math] between [math] and some sufficiently small positive constant, the optimal expected cost for history-independent list-labeling solutions is [math].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"56 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140585453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Computing, Ahead of Print. Abstract. We give an almost quadratic [math] lower bound on the space usage of any [math]-pass streaming algorithm solving the (directed) [math]-[math] reachability problem. This means that any such algorithm must essentially store the entire graph. As corollaries, we obtain almost quadratic space lower bounds for additional fundamental problems, including maximum matching, shortest path, matrix rank, and linear programming. Our main technical contribution is the definition and construction of set hiding graphs, that may be of independent interest: we give a general way of encoding a set [math] as a directed graph with [math] vertices, such that deciding whether [math] boils down to deciding if [math] is reachable from [math], for a specific pair of vertices [math] in the graph. Furthermore, we prove that our graph “hides” [math], in the sense that no low-space streaming algorithm with a small number of passes can learn (almost) anything about [math].
{"title":"Almost Optimal SuperConstant-Pass Streaming Lower Bounds for Reachability","authors":"Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, Huacheng Yu","doi":"10.1137/21m1417740","DOIUrl":"https://doi.org/10.1137/21m1417740","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. We give an almost quadratic [math] lower bound on the space usage of any [math]-pass streaming algorithm solving the (directed) [math]-[math] reachability problem. This means that any such algorithm must essentially store the entire graph. As corollaries, we obtain almost quadratic space lower bounds for additional fundamental problems, including maximum matching, shortest path, matrix rank, and linear programming. Our main technical contribution is the definition and construction of set hiding graphs, that may be of independent interest: we give a general way of encoding a set [math] as a directed graph with [math] vertices, such that deciding whether [math] boils down to deciding if [math] is reachable from [math], for a specific pair of vertices [math] in the graph. Furthermore, we prove that our graph “hides” [math], in the sense that no low-space streaming algorithm with a small number of passes can learn (almost) anything about [math].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"43 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140584995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Amey Bhangale, Prahladh Harsha, Orr Paradise, Avishay Tal
SIAM Journal on Computing, Volume 53, Issue 2, Page 480-523, April 2024. Abstract. We introduce a variant of Probabilistically Checkable Proofs (PCPs) that we refer to as rectangular PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the row of each query and the other determining the column. We construct PCPs that are efficient, short, smooth, and (almost) rectangular. As a key application, we show that proofs for hard languages in NTIME[math], when viewed as matrices, are rigid infinitely often. This strengthens and simplifies a recent result of Alman and Chen [FOCS, 2019] constructing explicit rigid matrices in FNP. Namely, we prove the following theorem: There is a constant [math] such that there is an FNP-machine that, for infinitely many [math], on input [math] outputs [math] matrices with entries in [math] that are [math]-far (in Hamming distance) from matrices of rank at most [math]. Our construction of rectangular PCPs starts with an analysis of how randomness yields queries in the Reed–Muller-based outer PCP of Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan [SIAM J. Comput., 36 (2006), pp. 889–974; CCC, 2005]. We then show how to preserve rectangularity under PCP composition and a smoothness-inducing transformation. This warrants refined and stronger notions of rectangularity, which we prove for the outer PCP and its transforms.
{"title":"Rigid Matrices from Rectangular PCPs","authors":"Amey Bhangale, Prahladh Harsha, Orr Paradise, Avishay Tal","doi":"10.1137/22m1495597","DOIUrl":"https://doi.org/10.1137/22m1495597","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 2, Page 480-523, April 2024. <br/> Abstract. We introduce a variant of Probabilistically Checkable Proofs (PCPs) that we refer to as rectangular PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the row of each query and the other determining the column. We construct PCPs that are efficient, short, smooth, and (almost) rectangular. As a key application, we show that proofs for hard languages in NTIME[math], when viewed as matrices, are rigid infinitely often. This strengthens and simplifies a recent result of Alman and Chen [FOCS, 2019] constructing explicit rigid matrices in FNP. Namely, we prove the following theorem: There is a constant [math] such that there is an FNP-machine that, for infinitely many [math], on input [math] outputs [math] matrices with entries in [math] that are [math]-far (in Hamming distance) from matrices of rank at most [math]. Our construction of rectangular PCPs starts with an analysis of how randomness yields queries in the Reed–Muller-based outer PCP of Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan [SIAM J. Comput., 36 (2006), pp. 889–974; CCC, 2005]. We then show how to preserve rectangularity under PCP composition and a smoothness-inducing transformation. This warrants refined and stronger notions of rectangularity, which we prove for the outer PCP and its transforms.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"10 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140585120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Computing, Ahead of Print. Abstract. We consider the problem of finding a maximal independent set (MIS) in the shared blackboard communication model with vertex-partitioned inputs. There are [math] players corresponding to vertices of an undirected graph, and each player sees the edges incident on its vertex; this way, each edge is known by both its endpoints and is thus shared by two players. The players communicate in simultaneous rounds by posting their messages on a shared blackboard visible to all players, with the goal of computing an MIS of the graph. While the MIS problem is well studied in other distributed models and while shared blackboard is, perhaps, the simplest broadcast model, lower bounds for our problem were only known against one-round protocols. We present a lower bound on the round-communication tradeoff for computing an MIS in this model. Specifically, we show that, when [math] rounds of interaction are allowed, at least one player needs to communicate [math] bits. In particular, with logarithmic bandwidth, finding an MIS requires [math] rounds. This lower bound can be compared with the algorithm of Ghaffari et al. [Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, 2018, pp. 129–138] that solves the MIS in [math] rounds but with a logarithmic bandwidth for an average player. Additionally, our lower bound further extends to the closely related problem of maximal bipartite matching. The presence of edge-sharing gives the algorithms in our model a surprising power, and numerous algorithmic results exploiting this power are known. For a similar reason, proving lower bounds in this model is much more challenging because this sharing in the players’ inputs prohibits the use of standard number-in-hand communication complexity arguments. Thus, to prove our results, we devise a new round elimination framework, which we call partial-input embedding, that may also be useful in future work for proving round-sensitive lower bounds in the presence of shared inputs. Finally, we discuss several implications of our results to multiround (adaptive) distributed sketching algorithms, broadcast congested clique, and the welfare maximization problem in two-sided matching markets.
SIAM 计算期刊》，提前印刷。 摘要我们考虑的是在顶点分区输入的共享黑板通信模型中寻找最大独立集（MIS）的问题。无向图的顶点对应着 [math] 玩家，每个玩家都能看到自己顶点上的边，这样，每条边的端点都是已知的，因此由两个玩家共享。棋手在同时进行的几轮比赛中，通过在所有棋手都能看到的共享黑板上发布信息进行交流，目的是计算图的 MIS。虽然 MIS 问题在其他分布式模型中得到了很好的研究，而且共享黑板也许是最简单的广播模型，但我们的问题的下界只针对一轮协议。我们提出了在该模型中计算 MIS 的回合-通信权衡下限。具体来说，我们证明了当允许 [math] 轮交互时，至少有一个玩家需要通信 [math] 比特。特别是在带宽为对数的情况下，找到一个 MIS 需要 [math] 轮。这个下界可以与 Ghaffari 等人的算法[Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, 2018, pp.此外，我们的下界还进一步扩展到了密切相关的最大双方格匹配问题。边缘共享的存在给我们模型中的算法带来了惊人的威力，利用这种威力的算法结果不胜枚举。出于类似的原因，在这个模型中证明下界更具挑战性，因为玩家输入的共享禁止使用标准的手数通信复杂度论证。因此，为了证明我们的结果，我们设计了一个新的回合消除框架，我们称之为部分输入嵌入（partial-input embedding）。最后，我们讨论了我们的结果对多轮（自适应）分布式草图算法、广播拥塞小集团以及双面匹配市场中的福利最大化问题的若干影响。
{"title":"Rounds vs. Communication Tradeoffs for Maximal Independent Sets","authors":"Sepehr Assadi, Gillat Kol, Zhijun Zhang","doi":"10.1137/22m1536972","DOIUrl":"https://doi.org/10.1137/22m1536972","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. We consider the problem of finding a maximal independent set (MIS) in the shared blackboard communication model with vertex-partitioned inputs. There are [math] players corresponding to vertices of an undirected graph, and each player sees the edges incident on its vertex; this way, each edge is known by both its endpoints and is thus shared by two players. The players communicate in simultaneous rounds by posting their messages on a shared blackboard visible to all players, with the goal of computing an MIS of the graph. While the MIS problem is well studied in other distributed models and while shared blackboard is, perhaps, the simplest broadcast model, lower bounds for our problem were only known against one-round protocols. We present a lower bound on the round-communication tradeoff for computing an MIS in this model. Specifically, we show that, when [math] rounds of interaction are allowed, at least one player needs to communicate [math] bits. In particular, with logarithmic bandwidth, finding an MIS requires [math] rounds. This lower bound can be compared with the algorithm of Ghaffari et al. [Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, 2018, pp. 129–138] that solves the MIS in [math] rounds but with a logarithmic bandwidth for an average player. Additionally, our lower bound further extends to the closely related problem of maximal bipartite matching. The presence of edge-sharing gives the algorithms in our model a surprising power, and numerous algorithmic results exploiting this power are known. For a similar reason, proving lower bounds in this model is much more challenging because this sharing in the players’ inputs prohibits the use of standard number-in-hand communication complexity arguments. Thus, to prove our results, we devise a new round elimination framework, which we call partial-input embedding, that may also be useful in future work for proving round-sensitive lower bounds in the presence of shared inputs. Finally, we discuss several implications of our results to multiround (adaptive) distributed sketching algorithms, broadcast congested clique, and the welfare maximization problem in two-sided matching markets.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"33 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140302407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Clément Legrand-Duchesne, Ashutosh Rai, Martin Tancer
SIAM Journal on Computing, Volume 53, Issue 2, Page 431-479, April 2024. Abstract. Deciding whether a diagram of a knot can be untangled with a given number of moves (as a part of the input) is known to be NP-complete. In this paper we determine the parameterized complexity of this problem with respect to a natural parameter called defect. Roughly speaking, it measures the efficiency of the moves used in the shortest untangling sequence of Reidemeister moves. We show that in a shortest untangling sequence the [math] moves, that is, the moves removing two adjacent crossings, can be essentially performed greedily. Using that, we show that this problem belongs to W[P] when parameterized by the defect. We also show that this problem is W[P]-hard by a reduction from Minimum axiom set.
{"title":"Parameterized Complexity of Untangling Knots","authors":"Clément Legrand-Duchesne, Ashutosh Rai, Martin Tancer","doi":"10.1137/22m1501969","DOIUrl":"https://doi.org/10.1137/22m1501969","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 2, Page 431-479, April 2024. <br/> Abstract. Deciding whether a diagram of a knot can be untangled with a given number of moves (as a part of the input) is known to be NP-complete. In this paper we determine the parameterized complexity of this problem with respect to a natural parameter called defect. Roughly speaking, it measures the efficiency of the moves used in the shortest untangling sequence of Reidemeister moves. We show that in a shortest untangling sequence the [math] moves, that is, the moves removing two adjacent crossings, can be essentially performed greedily. Using that, we show that this problem belongs to W[P] when parameterized by the defect. We also show that this problem is W[P]-hard by a reduction from Minimum axiom set.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"165 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140197632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zeyu Guo, Ray Li, Chong Shangguan, Itzhak Tamo, Mary Wootters
SIAM Journal on Computing, Volume 53, Issue 2, Page 389-430, April 2024. Abstract. This paper shows that there exist Reed–Solomon (RS) codes, over exponentially large finite fields in the code length, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving the list-decoding capacity. In particular, we show that for any [math] there exist RS codes with rate [math] that are list-decodable from radius of [math]. We generalize this result to list-recovery, showing that there exist [math]-list-recoverable RS codes with rate [math]. Along the way we use our techniques to give a new proof of a result of Blackburn on optimal linear perfect hash matrices, and strengthen it to obtain a construction of strongly perfect hash matrices. To derive the results in this paper we show a surprising connection of the above problems to graph theory, and in particular to the tree packing theorem of Nash-Williams and Tutte. We also state a new conjecture that generalizes the tree packing theorem to hypergraphs and show that if this conjecture holds, then there would exist RS codes that are optimally (nonasymptotically) list-decodable.
{"title":"Improved List-Decodability and List-Recoverability of Reed–Solomon Codes via Tree Packings","authors":"Zeyu Guo, Ray Li, Chong Shangguan, Itzhak Tamo, Mary Wootters","doi":"10.1137/21m1463707","DOIUrl":"https://doi.org/10.1137/21m1463707","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 2, Page 389-430, April 2024. <br/> Abstract. This paper shows that there exist Reed–Solomon (RS) codes, over exponentially large finite fields in the code length, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving the list-decoding capacity. In particular, we show that for any [math] there exist RS codes with rate [math] that are list-decodable from radius of [math]. We generalize this result to list-recovery, showing that there exist [math]-list-recoverable RS codes with rate [math]. Along the way we use our techniques to give a new proof of a result of Blackburn on optimal linear perfect hash matrices, and strengthen it to obtain a construction of strongly perfect hash matrices. To derive the results in this paper we show a surprising connection of the above problems to graph theory, and in particular to the tree packing theorem of Nash-Williams and Tutte. We also state a new conjecture that generalizes the tree packing theorem to hypergraphs and show that if this conjecture holds, then there would exist RS codes that are optimally (nonasymptotically) list-decodable.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"36 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140170229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Arnaud Casteigts, Michael Raskin, Malte Renken, Viktor Zamaraev
SIAM Journal on Computing, Volume 53, Issue 2, Page 346-388, April 2024. Abstract. A graph whose edges only appear at certain points in time is called a temporal graph (among other names). Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological order (i.e., a temporal path). In this paper, we consider a simple model of random temporal graph, obtained from an Erdős–Rényi random graph, [math], by considering a random permutation [math] of the edges and interpreting the ranks in [math] as presence times. We give a thorough study of the temporal connectivity of such graphs and derive implications for the existence of several kinds of sparse spanners. It turns out that temporal reachability in this model exhibits a surprisingly regular sequence of thresholds. In particular, we show that at [math], any fixed pair of vertices can asymptotically almost surely (a.a.s.) reach each other; at [math], at least one vertex (and, in fact, any fixed vertex) can a.a.s. reach all others; and at [math], all the vertices can a.a.s. reach each other; i.e., the graph is temporally connected. Furthermore, the graph admits a temporal spanner of size [math] as soon as it becomes temporally connected, which is nearly optimal, as [math] is a lower bound. This result is quite significant because temporal graphs do not admit spanners of size [math] in general [Kempe, Kleinberg, and Kumar, J. Comput. System Sci., 64 (2002), pp. 820–842]. In fact, they do not even always admit spanners of size [math] [Axiotis and Fotakis, On the size and the approximability of minimum temporally connected subgraphs, 2016, pp. 149:1–149:14]. Thus, our result implies that the obstructions found in these works—and more generally any non-negligible obstruction—are statistically insignificant: nearly optimal spanners always exist in random temporal graphs. All the above thresholds are sharp. Carrying the study of temporal spanners a step further, we show that pivotal spanners—i.e., spanners of size [math] composed of two spanning trees glued at a single vertex (one descending in time, the other ascending subsequently)—exist a.a.s. at [math], this threshold being also sharp. Finally, we show that optimal spanners (of size [math]) also exist a.a.s. at [math]. Whether this value is a sharp threshold is open; we conjecture that it is. For completeness, we compare the above results to existing results in related areas, including edge-ordered graphs, gossip theory, and population protocols, showing that our results can be interpreted in these settings as well and that in some cases they improve known results therein. Finally, we discuss an intriguing connection between our results and Janson’s celebrated results on percolation in weighted graphs.
SIAM 计算期刊》，第 53 卷第 2 期，第 346-388 页，2024 年 4 月。 摘要一个图的边只出现在特定的时间点上，这个图被称为时间图（还有其他名称）。如果每个有序顶点对由一条按时间顺序遍历边的路径（即时间路径）连接，那么这样的图就是时间连接图。在本文中，我们考虑了一个简单的随机时间图模型，它是从厄尔多斯-雷尼随机图[math]中通过考虑边的随机排列[math]并将[math]中的等级解释为存在时间而得到的。我们对这种图的时间连通性进行了深入研究，并推导出几种稀疏跨度图存在的意义。结果发现，在这个模型中，时间可达性呈现出令人惊讶的规则阈值序列。特别是，我们证明在[math]处，任何一对固定顶点都可以渐近地几乎肯定（a.a.s.）到达对方；在[math]处，至少有一个顶点（事实上，任何一个固定顶点）可以a.a.s.到达所有其他顶点；在[math]处，所有顶点都可以a.a.s.到达对方；也就是说，这个图是时间连通的。此外，一旦该图在时间上连通，就会产生一个大小为 [math] 的时空扳手，这几乎是最优的，因为 [math] 是一个下限。这个结果非常重要，因为一般情况下，时态图并不接受大小为 [math] 的跨度[Kempe, Kleinberg, and Kumar, J. Comput. System Sci.，64 (2002)，pp.］事实上，它们甚至并不总是允许大小为[math]的跨度[Axiotis and Fotakis, On the size and the approximability of minimum temporally connected subgraphs, 2016, pp.149:1-149:14]。因此，我们的结果意味着，这些著作中发现的障碍--更一般地说，任何不可忽略的障碍--在统计上都是不重要的：在随机时空图中，总是存在近乎最优的扳手。上述所有阈值都很尖锐。为了进一步研究时空跨域图，我们证明了枢轴跨域图--即大小为[math]的跨域图，由粘在一个顶点上的两棵生成树（一棵在时间上递减，另一棵随后递增）组成--在[math]时a.a.s.存在，这个阈值也是尖锐的。最后，我们证明了最优跨接树（大小为 [math]）在 [math] 时也是 a.a.s. 存在的。这个值是否是一个尖锐的临界值还没有定论；我们猜想它是。为了完整起见，我们将上述结果与相关领域的现有结果进行了比较，包括边序图、流言理论和种群协议，表明我们的结果也可以在这些环境中得到解释，而且在某些情况下，它们改进了其中的已知结果。最后，我们讨论了我们的结果与詹森关于加权图中渗流的著名结果之间的有趣联系。
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Sjoerd Dirksen, Shahar Mendelson, Alexander Stollenwerk
SIAM Journal on Computing, Volume 53, Issue 2, Page 315-345, April 2024. Abstract. We consider the problem of embedding a subset of [math] into a low-dimensional Hamming cube in an almost isometric way. We construct a simple, data-oblivious, and computationally efficient map that achieves this task with high probability; we first apply a specific structured random matrix, which we call the double circulant matrix; using that a matrix requires linear storage and matrix-vector multiplication that can be performed in near-linear time. We then binarize each vector by comparing each of its entries to a random threshold, selected uniformly at random from a well-chosen interval. We estimate the number of bits required for this encoding scheme in terms of two natural geometric complexity parameters of the set: its Euclidean covering numbers and its localized Gaussian complexity. The estimate we derive turns out to be the best that one can hope for, up to logarithmic terms. The key to the proof is a phenomenon of independent interest: we show that the double circulant matrix mimics the behavior of the Gaussian matrix in two important ways. First, it maps an arbitrary set in [math] into a set of well-spread vectors. Second, it yields a fast near-isometric embedding of any finite subset of [math] into [math]. This embedding achieves the same dimension reduction as the Gaussian matrix in near-linear time, under an optimal condition—up to logarithmic factors—on the number of points to be embedded. This improves a well-known construction due to Ailon and Chazelle.
{"title":"Fast Metric Embedding into the Hamming Cube","authors":"Sjoerd Dirksen, Shahar Mendelson, Alexander Stollenwerk","doi":"10.1137/22m1520220","DOIUrl":"https://doi.org/10.1137/22m1520220","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 2, Page 315-345, April 2024. <br/> Abstract. We consider the problem of embedding a subset of [math] into a low-dimensional Hamming cube in an almost isometric way. We construct a simple, data-oblivious, and computationally efficient map that achieves this task with high probability; we first apply a specific structured random matrix, which we call the double circulant matrix; using that a matrix requires linear storage and matrix-vector multiplication that can be performed in near-linear time. We then binarize each vector by comparing each of its entries to a random threshold, selected uniformly at random from a well-chosen interval. We estimate the number of bits required for this encoding scheme in terms of two natural geometric complexity parameters of the set: its Euclidean covering numbers and its localized Gaussian complexity. The estimate we derive turns out to be the best that one can hope for, up to logarithmic terms. The key to the proof is a phenomenon of independent interest: we show that the double circulant matrix mimics the behavior of the Gaussian matrix in two important ways. First, it maps an arbitrary set in [math] into a set of well-spread vectors. Second, it yields a fast near-isometric embedding of any finite subset of [math] into [math]. This embedding achieves the same dimension reduction as the Gaussian matrix in near-linear time, under an optimal condition—up to logarithmic factors—on the number of points to be embedded. This improves a well-known construction due to Ailon and Chazelle.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"21 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Computing, Volume 53, Issue 2, Page 287-314, April 2024. Abstract. The Kneser graph [math] is defined for integers [math] and [math] with [math] as the graph whose vertices are all the [math]-subsets of [math] where two such sets are adjacent if they are disjoint. The Schrijver graph [math] is defined as the subgraph of [math] induced by the collection of all [math]-subsets of [math] that do not include two consecutive elements modulo [math]. It is known that the chromatic number of both [math] and [math] is [math]. In the computational Kneser and Schrijver problems, we are given access to a coloring with [math] colors of the vertices of [math] and [math], respectively, and the goal is to find a monochromatic edge. We prove that the problems admit randomized algorithms with running time [math], hence they are fixed-parameter tractable with respect to the parameter [math]. The analysis involves structural results on intersecting families and on induced subgraphs of Kneser and Schrijver graphs. We also study the Agreeable-Set problem of assigning a small subset of a set of [math] items to a group of [math] agents, so that all agents value the subset at least as much as its complement. As an application of our algorithm for the Kneser problem, we obtain a randomized polynomial-time algorithm for the Agreeable-Set problem for instances with [math]. We further show that the Agreeable-Set problem is at least as hard as a variant of the Kneser problem with extended access to the input coloring.
{"title":"Fixed-Parameter Algorithms for the Kneser and Schrijver Problems","authors":"Ishay Haviv","doi":"10.1137/23m1557076","DOIUrl":"https://doi.org/10.1137/23m1557076","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 2, Page 287-314, April 2024. <br/> Abstract. The Kneser graph [math] is defined for integers [math] and [math] with [math] as the graph whose vertices are all the [math]-subsets of [math] where two such sets are adjacent if they are disjoint. The Schrijver graph [math] is defined as the subgraph of [math] induced by the collection of all [math]-subsets of [math] that do not include two consecutive elements modulo [math]. It is known that the chromatic number of both [math] and [math] is [math]. In the computational Kneser and Schrijver problems, we are given access to a coloring with [math] colors of the vertices of [math] and [math], respectively, and the goal is to find a monochromatic edge. We prove that the problems admit randomized algorithms with running time [math], hence they are fixed-parameter tractable with respect to the parameter [math]. The analysis involves structural results on intersecting families and on induced subgraphs of Kneser and Schrijver graphs. We also study the Agreeable-Set problem of assigning a small subset of a set of [math] items to a group of [math] agents, so that all agents value the subset at least as much as its complement. As an application of our algorithm for the Kneser problem, we obtain a randomized polynomial-time algorithm for the Agreeable-Set problem for instances with [math]. We further show that the Agreeable-Set problem is at least as hard as a variant of the Kneser problem with extended access to the input coloring.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"45 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140126989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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