Fault Tolerant Coloring of the Asynchronous Cycle

TL;DR

This paper introduces a wait-free algorithm for proper coloring of asynchronous cycles that is optimal in both round complexity and color range, applicable to crash-prone nodes with unique identifiers.

Contribution

It provides a novel, optimal, wait-free coloring algorithm for asynchronous cycles with proven minimal rounds and color range, independent of cycle size.

Findings

Algorithm runs in O(log* n) rounds, which is optimal.

Uses only 5 colors, matching known lower bounds.

Applicable to crash-prone nodes with unique IDs.

Abstract

We present a wait-free algorithm for proper coloring the n nodes of the asynchronous cycle $C_n$, where each crash-prone node starts with its (unique) identifier as input. The algorithm is independent of $n \geq 3$, and runs in $\mathrm{O}(\log^* n)$ rounds in $C_n$. This round-complexity is optimal thanks to a known matching lower bound, which applies even to synchronous (failure-free) executions. The range of colors used by our algorithm, namely $\{0, ..., 4\}$, is optimal too, thanks to a known lower bound on the minimum number of names for which renaming is solvable wait-free in shared-memory systems, whenever $n$ is a power of a prime. Indeed, our model coincides with the shared-memory model whenever $n = 3$, and the minimum number of names for which renaming is possible in 3-process shared-memory systems is 5.