Brief Announcement: Fully Lattice Linear Algorithms

Abstract

Lattice-linear systems allow nodes to execute asynchronously. We introduce eventually lattice-linear algorithms, where lattices are induced only among the states in a subset of the state space. The algorithm guarantees that the system transitions to a state in one of the lattices. Then, the algorithm behaves lattice linearly while traversing to an optimal state through that lattice.We present a lattice-linear self-stabilizing algorithm for service demand based minimal dominating set (SDMDS) problem. Using this as an example, we elaborate the working of, and define, eventually lattice-linear algorithms. Then, we present eventually lattice-linear self-stabilizing algorithms for minimal vertex cover (MVC), maximal independent set (MIS), graph colouring (GC) and 2-dominating set problems (2DS).Algorithms for SDMDS, MVC and MIS converge in 1 round plus n moves (within 2n moves), GC in n + 4m moves, and 2DS in 1 round plus 2n moves (within 3n moves). These results are an improvement over the existing literature. We also present experimental results to show performance gain demonstrating the benefit of lattice-linearity.

Multiplication and Modulo are Lattice Linear

“…For example, the fully lattice linear algorithms [8] follow the trace strictly according to the required specification for any system that initializes in an arbitrary feasible state, even if the nodes run asynchronously.…”

“…These algorithms induce multiple disjoint lattices in a subset of the state space. Fully lattice linear algorithms were developed for the minimal dominating set [8]; these algorithms induce multiple lattices in the entire state space. On the other hand, lattice linear problems form a single lattice among the states.…”