Planar Diameter via Metric Compression
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We develop a new approach for distributed distance computation in planar graphs that is based on a variant of the metric compression problem recently introduced by Abboud et al. [SODA'18]. One of our key technical contributions is in providing a compression scheme that encodes all S × T distances using O(|S|· poly(D)+|T|) bits for unweighted graphs with diameter D. This significantly improves the state of the art of O(|S|· 2^D+|T| · D) bits. We also consider an approximate version of the problem for weighted graphs, where the goal is to encode (1+ϵ) approximation of the S × T distances. At the heart of this compact compression scheme lies a VC-dimension type argument on planar graphs. This efficient compression scheme leads to several improvements and simplifications in the setting of diameter computation, most notably in the distributed setting: - There is an O(D^5)-round randomized distributed algorithm for computing the diameter in planar graphs, w.h.p. - There is an O(D^3)+ poly(log n/ϵ)· D^2-round randomized distributed algorithm for computing an (1+ϵ) approximation of the diameter in weighted graphs with polynomially bounded weights, w.h.p. No sublinear round algorithms were known for these problems before. These distributed constructions are based on a new recursive graph decomposition that preserves the (unweighted) diameter of each of the subgraphs up to a logarithmic term. Using this decomposition, we also get an exact SSSP tree computation within O(D^2) rounds.
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