Planar Diameter via Metric Compression

TL;DR

This paper introduces a new metric compression scheme for planar graphs that improves distributed diameter computation, enabling sublinear round algorithms for exact and approximate diameter and SSSP problems.

Contribution

It presents a novel compression method for all-pairs distances in planar graphs, leading to the first sublinear distributed algorithms for diameter and SSSP computations.

Findings

Compression scheme uses fewer bits than previous methods.

Distributed algorithms achieve sublinear rounds for diameter and SSSP.

Approximate diameter computation in weighted graphs with polynomial weights.

Abstract

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 $S \times T$ distances using $\widetilde{O}(|S|\cdot poly(D)+|T|)$ bits for unweighted graphs with diameter $D$. This significantly improves the state of the art of $\widetilde{O}(|S|\cdot 2^{D}+|T| \cdot D)$ bits. We also consider an approximate version of the problem for \emph{weighted} graphs, where the goal is to encode $(1+\epsilon)$ approximation of the $S \times T$ 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 $\widetilde{O}(D^5)$-round randomized distributed algorithm for computing the diameter in planar graphs, w.h.p. - There is an $\widetilde{O}(D^3)+ poly(\log n/\epsilon)\cdot D^2$-round randomized distributed algorithm for computing an $(1+\epsilon)$ 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 \emph{exact} SSSP tree computation within $\widetilde{O}(D^2)$ rounds.