Making Self-Stabilizing any Locally Greedy Problem

TL;DR

This paper introduces a method to convert synchronous distributed algorithms for locally greedy and mendable problems into self-stabilizing algorithms in anonymous networks, enabling robust solutions with new algorithms for ruling sets and graph coloring.

Contribution

It presents the first explicit self-stabilizing algorithm for computing a $(k,k-1)$-ruling set and demonstrates how to use this to achieve distance-$K$ coloring, facilitating simulation of local algorithms.

Findings

First explicit self-stabilizing ruling set algorithm

Distance-$K$ coloring of graphs achieved

Simulation of local algorithms in anonymous networks

Abstract

We propose a way to transform synchronous distributed algorithms solving locally greedy and mendable problems into self-stabilizing algorithms in anonymous networks. Mendable problems are a generalization of greedy problems where any partial solution may be transformed -- instead of completed -- into a global solution: every time we extend the partial solution we are allowed to change the previous partial solution up to a given distance. Locally here means that to extend a solution for a node, we need to look at a constant distance from it. In order to do this, we propose the first explicit self-stabilizing algorithm computing a $(k,k-1)$-ruling set (i.e. a "maximal independent set at distance $k$"). By combining multiple time this technique, we compute a distance-$K$ coloring of the graph. With this coloring we can finally simulate \local~model algorithms running in a constant number of rounds, using the colors as unique identifiers. Our algorithms work under the Gouda daemon, which is similar to the probabilistic daemon: if an event should eventually happen, it will occur under this daemon.