Computing Least Fixed Points with Overwrite Semantics in Parallel and Distributed Systems

TL;DR

This paper develops new methods for computing least fixed points in parallel and distributed systems using overwrite semantics, providing convergence guarantees under relaxed synchronization models, applicable to various algorithms.

Contribution

It introduces the first exact convergence guarantees for overwrite-based fixed point computations in parallel and distributed settings without requiring join operations or contraction assumptions.

Findings

Proves convergence under interleaving semantics with fair scheduling.

Establishes convergence with update-only-on-change semantics.

Demonstrates bounded staleness and locality guarantees in distributed execution.

Abstract

We present methods to compute least fixed points of multiple monotone inflationary functions in parallel and distributed settings. While the classic Knaster-Tarski theorem addresses a single function with sequential iteration, modern computing systems require parallel execution with overwrite semantics, non-atomic updates, and stale reads. We prove three convergence theorems under progressively relaxed synchronization: (1) Interleaving semantics with fair scheduling, (2) Parallel execution with update-only-on-change semantics (processes write only on those coordinates whose values change), and (3) Distributed execution with bounded staleness (updates propagate within $T$ rounds) and $i$-locality (each process modifies only its own component). Our approach differs from prior work in fundamental ways: Cousot-Cousot's chaotic iteration uses join-based merges that preserve information. Instead, we use coordinate-wise overwriting. Bertsekas's asynchronous methods assume contractions. We use coordinate-wise overwriting with structural constraints (locality, bounded staleness) instead. Applications include parallel and distributed algorithms for the transitive closure, stable marriage, shortest paths, and fair division with subsidy problems. Our results provide the first exact least-fixed-point convergence guarantees for overwrite-based parallel updates without join operations or contraction assumptions.