Eventually Lattice-Linear Algorithms
Arya Tanmay Gupta, Sandeep S Kulkarni
TL;DR
The paper tackles the challenge of designing correct, asynchronous algorithms for classic graph problems by introducing eventually lattice-linear self-stabilizing algorithms (ELLSS), which induce lattices on subsets of the state space. Its core idea is to decompose rules into a first phase that reaches a lattice-admissible feasible state and a second phase that follows lattice-linearity to reach an optimal state, tolerating old reads under monotone AMR. The authors instantiate this framework for SDMDS and extend it to MVC, MIS, GC, and 2DS, achieving strong convergence guarantees (often 1 round plus a linear number of moves) and demonstrating practical performance benefits in experiments. The work provides a general, robust paradigm for asynchronous, self-stabilizing computation in distributed systems, with implications for backbone construction, scheduling, and resource allocation in networks.
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, MVCc 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.