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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Graph spanners are sparse subgraphs which approximately preserve all pairwise shortest-path distances in an input graph. The notion of approximation can be additive, multiplicative, or both, and many variants of this problem have been extensively studied. We study the problem of computing a graph spanner when the edges of the input graph are distributed across two or more sites in an arbitrary, possibly worst-case partition, and the goal is for the sites to minimize the communication used to output a spanner. We assume the message-passing model of communication, for which there is a point-to-point link between all pairs of sites as well as a coordinator who is responsible for producing the output. We stress that the subset of edges that each site has is not related to the network topology, which is fixed to be point-to-point. While this model has been extensively studied for related problems such as graph connectivity, it has not been systematically studied for graph spanners. We present the first tradeoffs for total communication versus the quality of the spanners computed, for two or more sites, as well as for additive and multiplicative notions of distortion. We show separations in the communication complexity when edges are allowed to occur on multiple sites, versus when each edge occurs on at most one site. We obtain nearly tight bounds (up to polylog factors) for the communication of additive 2-spanners in both the with and without duplication models, multiplicative (2k-1)-spanners in the with duplication model, and multiplicative 3 and 5-spanners in the without duplication model. Our lower bound for multiplicative 3-spanners employs biregular bipartite graphs rather than the usual Erdős girth conjecture graphs and may be of wider interest.

Manuel Fernández V, David P. Woodruff, and Taisuke Yasuda. Graph Spanners in the Message-Passing Model. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 77:1-77:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fernandezv_et_al:LIPIcs.ITCS.2020.77, author = {Fern\'{a}ndez V, Manuel and Woodruff, David P. and Yasuda, Taisuke}, title = {{Graph Spanners in the Message-Passing Model}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {77:1--77:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.77}, URN = {urn:nbn:de:0030-drops-117620}, doi = {10.4230/LIPIcs.ITCS.2020.77}, annote = {Keywords: Graph spanners, Message-passing model, Communication complexity} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Consider a variant of the Mastermind game in which queries are l_p distances, rather than the usual Hamming distance. That is, a codemaker chooses a hidden vector y in {-k,-k+1,...,k-1,k}^n and answers to queries of the form ||y-x||_p where x in {-k,-k+1,...,k-1,k}^n. The goal is to minimize the number of queries made in order to correctly guess y.
In this work, we show an upper bound of O(min{n,(n log k)/(log n)}) queries for any real 1<=p<infty and O(n) queries for p=infty. To prove this result, we in fact develop a nonadaptive polynomial time algorithm that works for a natural class of separable distance measures, i.e., coordinate-wise sums of functions of the absolute value. We also show matching lower bounds up to constant factors, even for adaptive algorithms for the approximation version of the problem, in which the problem is to output y' such that ||y'-y||_p <= R for any R <= k^{1-epsilon}n^{1/p} for constant epsilon>0. Thus, essentially any approximation of this problem is as hard as finding the hidden vector exactly, up to constant factors. Finally, we show that for the noisy version of the problem, i.e., the setting when the codemaker answers queries with any q = (1 +/- epsilon)||y-x||_p, there is no query efficient algorithm.

Manuel Fernández V, David P. Woodruff, and Taisuke Yasuda. The Query Complexity of Mastermind with l_p Distances. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 1:1-1:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fernandezv_et_al:LIPIcs.APPROX-RANDOM.2019.1, author = {Fern\'{a}ndez V, Manuel and Woodruff, David P. and Yasuda, Taisuke}, title = {{The Query Complexity of Mastermind with l\underlinep Distances}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {1:1--1:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.1}, URN = {urn:nbn:de:0030-drops-112165}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.1}, annote = {Keywords: Mastermind, Query Complexity, l\underlinep Distance} }

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