{"title":"Time Dependent Biased Random Walks","authors":"J. Haslegrave, Thomas Sauerwald, John Sylvester","doi":"10.1145/3498848","DOIUrl":"https://doi.org/10.1145/3498848","url":null,"abstract":"We study the biased random walk where at each step of a random walk a “controller” can, with a certain small probability, move the walk to an arbitrary neighbour. This model was introduced by Azar et al. [STOC’1992]; we extend their work to the time dependent setting and consider cover times of this walk. We obtain new bounds on the cover and hitting times. Azar et al. conjectured that the controller can increase the stationary probability of a vertex from p to p1-ε; while this conjecture is not true in full generality, we propose a best-possible amended version of this conjecture and confirm it for a broad class of graphs. We also consider the problem of computing an optimal strategy for the controller to minimise the cover time and show that for directed graphs determining the cover time is PSPACE-complete.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130944794","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}
E. Chlamtáč, M. Dinitz, G. Kortsarz, Bundit Laekhanukit
{"title":"Approximating Spanners and Directed Steiner Forest","authors":"E. Chlamtáč, M. Dinitz, G. Kortsarz, Bundit Laekhanukit","doi":"10.1145/3381451","DOIUrl":"https://doi.org/10.1145/3381451","url":null,"abstract":"It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Bodwin SODA’16 8 J.ACM’17, Bodwin-Williams SODA’16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances, we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an O(n3/5 + ε)-approximation for distance preservers and pairwise spanners (for arbitrary constant ε > 0). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an O(log n)-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an O(1)-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an O(n3/5 + ε)-approximation for the Directed Steiner Forest problem (for arbitrary constant ε > 0) when all edges have uniform costs, improving the previous best O(n2/3 + ε)-approximation due to Berman et al. [ICALP’11] (which holds for general edge costs).","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131842495","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}
{"title":"Online Algorithms for Weighted Paging with Predictions","authors":"Zhihao Jiang, Debmalya Panigrahi, Kevin Sun","doi":"10.1145/3548774","DOIUrl":"https://doi.org/10.1145/3548774","url":null,"abstract":"In this article, we initiate the study of the weighted paging problem with predictions. This continues the recent line of work in online algorithms with predictions, particularly that of Lykouris and Vassilvitski (ICML 2018) and Rohatgi (SODA 2020) on unweighted paging with predictions. We show that unlike unweighted paging, neither a fixed lookahead nor a knowledge of the next request for every page is sufficient information for an algorithm to overcome the existing lower bounds in weighted paging. However, a combination of the two, which we call strong per request prediction (SPRP), suffices to give a 2-competitive algorithm. We also explore the question of gracefully degrading algorithms with increasing prediction error, and give both upper and lower bounds for a set of natural measures of prediction error.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122294785","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}
André Berger, László Kozma, Matthias Mnich, Roland Vincze
{"title":"Time- and Space-optimal Algorithm for the Many-visits TSP","authors":"André Berger, László Kozma, Matthias Mnich, Roland Vincze","doi":"10.1145/3382038","DOIUrl":"https://doi.org/10.1145/3382038","url":null,"abstract":"The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of n cities that visits each city c a prescribed number kc of times. Travel costs may be asymmetric, and visiting a city twice in a row may incur a non-zero cost. The MV-TSP problem finds applications in scheduling, geometric approximation, and Hamiltonicity of certain graph families. The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou (SICOMP, 1984). It runs in time nO(n) + O(n3 log ∑ c kc) and requires nᶿ(n) space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its logarithmic dependence on the total length ∑ckc of the tour, allowing the algorithm to handle instances with very long tours. The superexponential dependence on the number of cities in both the time and space complexity, however, renders the algorithm impractical for all but the narrowest range of this parameter. In this article, we improve upon the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time 2O(n), i.e., single-exponential in the number of cities, using polynomial space. The space requirement of our algorithm is (essentially) the size of the output, and assuming the Exponential-Time Hypothesis (ETH), the problem cannot be solved in time 2o(n). Our algorithm is deterministic, and arguably both simpler and easier to analyze than the original approach of Cosmadakis and Papadimitriou. It involves an optimization over directed spanning trees and a recursive, centroid-based decomposition of trees.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"358 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126690719","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}
{"title":"Clustering in Hypergraphs to Minimize Average Edge Service Time","authors":"Ori Rottenstreich, Haim Kaplan, A. Hassidim","doi":"10.1145/3386121","DOIUrl":"https://doi.org/10.1145/3386121","url":null,"abstract":"We study the problem of clustering the vertices of a weighted hypergraph such that on average the vertices of each edge can be covered by a small number of clusters. This problem has many applications, such as for designing medical tests, clustering files on disk servers, and placing network services on servers. The edges of the hypergraph model groups of items that are likely to be needed together, and the optimization criteria that we use can be interpreted as the average delay (or cost) to serve the items of a typical edge. We describe and analyze algorithms for this problem for the case in which the clusters have to be disjoint and for the case where clusters can overlap. The analysis is often subtle and reveals interesting structure and invariants that one can utilize.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"141 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114343744","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}
{"title":"Finding a Shortest Odd Hole","authors":"M. Chudnovsky, A. Scott, P. Seymour","doi":"10.1145/3447869","DOIUrl":"https://doi.org/10.1145/3447869","url":null,"abstract":"An odd hole in a graph is an induced cycle with odd length greater than 3. In an earlier paper (with Sophie Spirkl), solving a longstanding open problem, we gave a polynomial-time algorithm to test if a graph has an odd hole. We subsequently showed that, for every t, there is a polynomial-time algorithm to test whether a graph contains an odd hole of length at least t. In this article, we give an algorithm that finds a shortest odd hole, if one exists.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124694629","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}
{"title":"Dynamic Distribution-Sensitive Point Location","authors":"Siu-Wing Cheng, Man-Kit Lau","doi":"10.4230/LIPIcs.SoCG.2020.30","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2020.30","url":null,"abstract":"We propose a dynamic data structure for the distribution-sensitive point location problem in the plane. Suppose that there is a fixed query distribution within a convex subdivision S, and we are given an oracle that can return in O(1) time the probability of a query point falling into a polygonal region of constant complexity. We can maintain S such that each query is answered in Oopt(S)) expected time, where opt (S) is the expected time of the best linear decision tree for answering point location queries in S. The space and construction time are O(nlog2n), where n is the number of vertices of S. An update of S as a mixed sequence of k edge insertions and deletions takes O(klog4 n) amortized time. As a corollary, the randomized incremental construction of the Voronoi diagram of n sites can be performed in O(nlog4 n) expected time so that, during the incremental construction, a nearest neighbor query at any time can be answered optimally with respect to the intermediate Voronoi diagram at that time.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123893354","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}
Kai Jin, Siu-Wing Cheng, Man-Kwun Chiu, Man Ting Wong
{"title":"A Generalization of Self-Improving Algorithms","authors":"Kai Jin, Siu-Wing Cheng, Man-Kwun Chiu, Man Ting Wong","doi":"10.1145/3531227","DOIUrl":"https://doi.org/10.1145/3531227","url":null,"abstract":"Ailon et al. [SICOMP’11] proposed self-improving algorithms for sorting and Delaunay triangulation (DT) when the input instances x1, ... , xn follow some unknown product distribution. That is, xi is drawn independently from a fixed unknown distribution 𝒟i. After spending O(n1+ε) time in a learning phase, the subsequent expected running time is O((n + H)/ε), where H ∊ {HS,HDT}, and HS and HDT are the entropies of the distributions of the sorting and DT output, respectively. In this article, we allow dependence among the xi’s under the group product distribution. There is a hidden partition of [1, n] into groups; the xi’s in the kth group are fixed unknown functions of the same hidden variable uk; and the uk’s are drawn from an unknown product distribution. We describe self-improving algorithms for sorting and DT under this model when the functions that map uk to xi’s are well-behaved. After an O(poly(n))-time training phase, we achieve O(n + HS) and O(nα (n) + HDT) expected running times for sorting and DT, respectively, where α (⋅) is the inverse Ackermann function.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122136816","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}
{"title":"Submatrix Maximum Queries in Monge and Partial Monge Matrices Are Equivalent to Predecessor Search","authors":"Paweł Gawrychowski, S. Mozes, O. Weimann","doi":"10.1145/3381416","DOIUrl":"https://doi.org/10.1145/3381416","url":null,"abstract":"We present an optimal data structure for submatrix maximum queries in n× n Monge matrices. Our result is a two-way reduction showing that the problem is equivalent to the classical predecessor problem in a universe of polynomial size. This gives a data structure of O(n) space that answers submatrix maximum queries in O(log log n) time, as well as a matching lower bound, showing that O(log log n) query-time is optimal for any data structure of size O(npolylog(n)). Our result settles the problem, improving on the O(log2 n) query time in SODA’12, and on the O(log n) query-time in ICALP’14. In addition, we show that partial Monge matrices can be handled in the same bounds as full Monge matrices. In both previous results, partial Monge matrices incurred additional inverse-Ackermann factors.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"248 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133362457","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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