Fully Adaptive Self-Stabilizing Transformer for LCL Problems

TL;DR

This paper introduces a universal self-stabilizing transformer for local graph problems, enabling quick, adaptive recovery from faults with minimal message overhead, applicable to infinite graphs and producing anonymous solutions.

Contribution

It presents the first generic self-stabilizing transformer for LCL problems that is fully adaptive, size-uniform, and applicable to infinite graphs, improving fault recovery efficiency.

Findings

Transformer stabilizes in expected time proportional to log(k + Δ)

Produces anonymous, size-uniform self-stabilizing algorithms

Applicable to infinite graphs with degree bound Δ

Abstract

The first generic self-stabilizing transformer for local problems in a constrained bandwidth model is introduced. This transformer can be applied to a wide class of locally checkable labeling (LCL) problems, converting a given fault free synchronous algorithm that satisfies certain conditions into a self-stabilizing synchronous algorithm for the same problem. The resulting self-stabilizing algorithms are anonymous, size-uniform, and \emph{fully adaptive} in the sense that their time complexity is bounded as a function of the number $k$ of nodes that suffered faults (possibly at different times) since the last legal configuration. Specifically, for graphs whose degrees are up-bounded by $\Delta$, the algorithms produced by the transformer stabilize in time proportional to $\log (k + \Delta)$ in expectation, independently of the number of nodes in the graph. As such, the transformer is applicable also for infinite graphs (with degree bound $\Delta$). Another appealing feature of the transformer is its small message size overhead. The transformer is applied to known algorithms (or simple variants thereof) for some classic LCL problems, producing the first anonymous size-uniform self-stabilizing algorithms for these problems that are provably fully adaptive. From a technical point of view, the transformer's key design feature is a novel probabilistic tool that allows different nodes to act in synchrony even though their clocks may have been adversarially manipulated.