Deterministic Low-Diameter Decompositions for Weighted Graphs and Distributed and Parallel Applications

This paper presents new deterministic and distributed low-diameter decomposition algorithms for weighted graphs. In particular, we show that if one can efficiently compute approximate distances in a parallel or a distributed setting, one can also efficiently compute low-diameter decompositions. This...

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Bibliographic Details
Published in:Proceedings / annual Symposium on Foundations of Computer Science pp. 1114 - 1121
Main Authors: Rozhon, Vaclav, Elkin, Michael, Grunau, Christoph, Haeupler, Bernhard
Format: Conference Proceeding
Language:English
Published: IEEE 01.10.2022
Subjects:
Approximation algorithms
Clustering algorithms
Complexity theory
Computational modeling
Computer science
Distortion
Parallel algorithms
ISSN:2575-8454
Online Access:Get full text
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Summary:This paper presents new deterministic and distributed low-diameter decomposition algorithms for weighted graphs. In particular, we show that if one can efficiently compute approximate distances in a parallel or a distributed setting, one can also efficiently compute low-diameter decompositions. This consequently implies solutions to many fundamental distance based problems using a polylogarithmic number of approximate distance computations.Our low-diameter decomposition generalizes and extends the line of work starting from [RG20] to weighted graphs in a very model-independent manner. Moreover, our clustering results have additional useful properties, including strong-diameter guarantees, separation properties, restricting cluster centers to specified terminals, and more. Applications include:-The first near-linear work and polylogarithmic depth randomized and deterministic parallel algorithm for low-stretch spanning trees (LSST) with polylogarithmic stretch. Previously, the best parallel LSST algorithm required m.n^{o(1)} work and n^{o(1)} depth and was inherently randomized. No deterministic LSST algorithm with truly sub-quadratic work and sub-linear depth was known.-The first near-linear work and polylogarithmic depth deterministic algorithm for computing an \ell_{1}-embedding into polylogarithmic dimensional space with polylogarithmic distortion. The best prior deterministic algorithms for \ell_{1}-embeddings either require large polynomial work or are inherently sequential.Even when we apply our techniques to the classical problem of computing a ball-carving with strong-diameter O(\log^{2}n) in an unweighted graph, our new clustering algorithm still leads to an improvement in round complexity from O(\log^{10}n) rounds [CG21] to O(\log^{4}n).
ISSN:2575-8454
DOI:10.1109/FOCS54457.2022.00107