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

2026-02-11

AI Summary

This work addresses the challenge of efficiently computing the least fixed points of multiple monotonic inflationary functions under non-atomic updates, stale reads, and cover semantics in parallel and distributed systems. The authors propose a novel coordinate-cover-based weakly synchronous iterative framework that, for the first time, guarantees exact convergence to the least fixed point under cover semantics without relying on merge operations or contraction mapping assumptions. By introducing new constraints—namely i-locality and bounded staleness—and combining them with fair scheduling and change-only write semantics, they establish three progressively relaxed synchronization-based convergence theorems. The approach is successfully applied to problems including transitive closure, stable marriage, shortest paths, and subsidy-augmented fair allocation, demonstrating provable correctness and convergence across various weak synchronization models.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• least fixed point
• overwrite semantics
• parallel computing
• distributed systems
• monotone functions

Innovation

• least fixed point
• overwrite semantics
• parallel and distributed computation
• bounded staleness
• i-locality