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 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 $O(\log^{4}n)$.