Fast Partial Distance Estimation and Applications

TL;DR

This paper presents new algorithms for approximate all-pairs shortest paths in distributed networks, achieving faster runtimes, better approximation ratios, and smaller labels, advancing the efficiency of distributed routing schemes.

Contribution

The paper introduces improved deterministic and randomized algorithms for approximate APSP in the CONGEST model, with faster runtimes, reduced approximation ratios, and optimal label sizes.

Findings

Deterministic $(1+o(1))$-approximation in $ ilde{O}(n)$ rounds.

Randomized $O(k)$-approximation in $ ilde{O}(n^{1/2+1/k}+D)$ rounds.

Compact routing tables with $O(k ext{log} n)$ bits and improved approximation ratios.

Abstract

We study approximate distributed solutions to the weighted {\it all-pairs-shortest-paths} (APSP) problem in the CONGEST model. We obtain the following results. $1.$ A deterministic $(1+o(1))$-approximation to APSP in $\tilde{O}(n)$ rounds. This improves over the best previously known algorithm, by both derandomizing it and by reducing the running time by a $\Theta(\log n)$ factor. In many cases, routing schemes involve relabeling, i.e., assigning new names to nodes and require that these names are used in distance and routing queries. It is known that relabeling is necessary to achieve running times of $o(n/\log n)$. In the relabeling model, we obtain the following results. $2.$ A randomized $O(k)$-approximation to APSP, for any integer $k>1$, running in $\tilde{O}(n^{1/2+1/k}+D)$ rounds, where $D$ is the hop diameter of the network. This algorithm simplifies the best previously known result and reduces its approximation ratio from $O(k\log k)$ to $O(k)$. Also, the new algorithm uses uses labels of asymptotically optimal size, namely $O(\log n)$ bits. $3.$ A randomized $O(k)$-approximation to APSP, for any integer $k>1$, running in time $\tilde{O}((nD)^{1/2}\cdot n^{1/k}+D)$ and producing {\it compact routing tables} of size $\tilde{O}(n^{1/k})$. The node lables consist of $O(k\log n)$ bits. This improves on the approximation ratio of $\Theta(k^2)$ for tables of that size achieved by the best previously known algorithm, which terminates faster, in $\tilde{O}(n^{1/2+1/k}+D)$ rounds.