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 l₁- into polylogarithmic dimensional space with polylogarithmic distortion. The best prior deterministic algorithms for l₁-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(log²n) in an unweighted graph, our new clustering algorithm still leads to an improvement in round complexity from O(log¹⁰n) rounds [CG21] to O( log⁴n).