Deterministic Low-Diameter Decompositions for Weighted Graphs and Distributed and Parallel Applications
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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 [Rozhoň, Ghaffari STOC 2020] 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 ℓ_1-embedding into polylogarithmic dimensional space with polylogarithmic distortion. The best prior deterministic algorithms for ℓ_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 [Chang, Ghaffari PODC 21] to O(log^4 n).
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