Compact Self-Stabilizing Leader Election for Arbitrary Networks

TL;DR

This paper introduces a novel self-stabilizing leader election algorithm for arbitrary networks that significantly reduces space complexity to sub-logarithmic levels, surpassing previous bounds for silent algorithms.

Contribution

It presents the first self-stabilizing leader election algorithm for arbitrary networks with sub-logarithmic space complexity, breaking the $ ext{O}( ext{log } n)$ barrier.

Findings

Achieves space complexity of $O( ext{max}igrace ext{log } riangle, ext{log log } n igrace)$ bits per node.

First to break the $ ext{O}( ext{log } n)$ space barrier for silent algorithms in arbitrary networks.

Uses sophisticated tools like distance-2 coloring to encode spanning trees compactly.

Abstract

We present a self-stabilizing leader election algorithm for arbitrary networks, with space-complexity $O(\max\{\log \Delta, \log \log n\})$ bits per node in $n$-node networks with maximum degree~$\Delta$. This space complexity is sub-logarithmic in $n$ as long as $\Delta = n^{o(1)}$. The best space-complexity known so far for arbitrary networks was $O(\log n)$ bits per node, and algorithms with sub-logarithmic space-complexities were known for the ring only. To our knowledge, our algorithm is the first algorithm for self-stabilizing leader election to break the $\Omega(\log n)$ bound for silent algorithms in arbitrary networks. Breaking this bound was obtained via the design of a (non-silent) self-stabilizing algorithm using sophisticated tools such as solving the distance-2 coloring problem in a silent self-stabilizing manner, with space-complexity $O(\max\{\log \Delta, \log \log n\})$ bits per node. Solving this latter coloring problem allows us to implement a sub-logarithmic encoding of spanning trees --- storing the IDs of the neighbors requires $\Omega(\log n)$ bits per node, while we encode spanning trees using $O(\max\{\log \Delta, \log \log n\})$ bits per node. Moreover, we show how to construct such compactly encoded spanning trees without relying on variables encoding distances or number of nodes, as these two types of variables would also require $\Omega(\log n)$ bits per node.