Extending Lattice linearity for Self-Stabilizing Algorithms

TL;DR

This paper extends the concept of lattice linearity to self-stabilizing algorithms, enabling increased concurrency and correctness without synchronization, demonstrated through several graph problems.

Contribution

It introduces eventually lattice linear algorithms and applies this extension to multiple classical graph problems, improving convergence properties.

Findings

Converges in 2n moves for SDDS problem

Applicable to minimal vertex cover, maximal independent set, and graph coloring

Increases concurrency by eliminating synchronization needs

Abstract

In this article, we focus on extending the notion of lattice linearity to self-stabilizing programs. Lattice linearity allows a node to execute its actions with old information about the state of other nodes and still preserve correctness. It increases the concurrency of the program execution by eliminating the need for synchronization among its nodes. The extension -- denoted as eventually lattice linear algorithms -- is performed with an example of the service-demand based minimal dominating set (SDDS) problem, which is a generalization of the dominating set problem; it converges in $2n$ moves. Subsequently, we also show that the same approach could be used in various other problems including minimal vertex cover, maximal independent set and graph coloring.