Congested Clique Algorithms for Graph Spanners

TL;DR

This paper presents faster algorithms for constructing graph spanners in the congested clique model, improving round complexity for both randomized and deterministic methods, with new derandomization techniques.

Contribution

It introduces new randomized and deterministic algorithms for graph spanner construction in the congested clique model, reducing round complexity and employing a novel derandomization theorem.

Findings

Randomized $(2k-1)$-spanner construction in $O( ext{log }k)$ rounds.

Deterministic $(2k-1)$-spanner construction in $O( ext{log }k + ( ext{log} ext{log }n)^3)$ rounds.

Deterministic $O(k)$-spanner construction in $O( ext{log }k)$ rounds.

Abstract

Graph spanners are sparse subgraphs that faithfully preserve the distances in the original graph up to small stretch. Spanner have been studied extensively as they have a wide range of applications ranging from distance oracles, labeling schemes and routing to solving linear systems and spectral sparsification. A $k$-spanner maintains pairwise distances up to multiplicative factor of $k$. It is a folklore that for every $n$-vertex graph $G$, one can construct a $(2k-1)$ spanner with $O(n^{1+1/k})$ edges. In a distributed setting, such spanners can be constructed in the standard CONGEST model using $O(k^2)$ rounds, when randomization is allowed. In this work, we consider spanner constructions in the congested clique model, and show: (1) A randomized construction of a $(2k-1)$-spanner with $\widetilde{O}(n^{1+1/k})$ edges in $O(\log k)$ rounds. The previous best algorithm runs in $O(k)$ rounds. (2) A deterministic construction of a $(2k-1)$-spanner with $\widetilde{O}(n^{1+1/k})$ edges in $O(\log k +(\log\log n)^3)$ rounds. The previous best algorithm runs in $O(k\log n)$ rounds. This improvement is achieved by a new derandomization theorem for hitting sets which might be of independent interest. (3) A deterministic construction of a $O(k)$-spanner with $O(k \cdot n^{1+1/k})$ edges in $O(\log k)$ rounds.