SIAM Journal on Computing最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"title":"Sharp Thresholds in Random Simple Temporal Graphs","authors":"Arnaud Casteigts, Michael Raskin, Malte Renken, Viktor Zamaraev","doi":"10.1137/22m1511916","DOIUrl":"https://doi.org/10.1137/22m1511916","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 2, Page 346-388, April 2024. <br/> 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.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"3 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140173081","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}

{"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}

{"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}