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

Vijay K. Garg, Rohan Garg

TL;DR

This work addresses computing the least fixed point of a family of monotone inflationary functions operating on a finite lattice under overwrite semantics in parallel and distributed environments. It introduces three convergence theorems that progressively relax synchronization: interleaving execution guarantees convergence for fair schedules; non-interleaving parallelism requires update-on-change and distributivity; distributed computation requires $i$-locality, strong fairness, and bounded staleness to preserve convergence to the least fixed point. Distinct from join-based chaotic iteration or contraction-based asynchronous methods, the approach demonstrates exact convergence guarantees for coordinate-wise overwrites, enabling scalable implementations of problems such as transitive closure, stable marriage, shortest paths, and fair division with subsidies. The results have practical significance for modern parallel and distributed systems where updates overwrite state, communications incur delays, and components evolve independently, providing a formal foundation for correct overwrite-based fixed-point computation with locality constraints.

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.

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

TL;DR

This work addresses computing the least fixed point of a family of monotone inflationary functions operating on a finite lattice under overwrite semantics in parallel and distributed environments. It introduces three convergence theorems that progressively relax synchronization: interleaving execution guarantees convergence for fair schedules; non-interleaving parallelism requires update-on-change and distributivity; distributed computation requires -locality, strong fairness, and bounded staleness to preserve convergence to the least fixed point. Distinct from join-based chaotic iteration or contraction-based asynchronous methods, the approach demonstrates exact convergence guarantees for coordinate-wise overwrites, enabling scalable implementations of problems such as transitive closure, stable marriage, shortest paths, and fair division with subsidies. The results have practical significance for modern parallel and distributed systems where updates overwrite state, communications incur delays, and components evolve independently, providing a formal foundation for correct overwrite-based fixed-point computation with locality constraints.

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 rounds) and -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.
Paper Structure (17 sections, 7 theorems, 25 equations, 3 tables, 10 algorithms)

This paper contains 17 sections, 7 theorems, 25 equations, 3 tables, 10 algorithms.

Table of Contents

  1. Introduction
  2. Fixed Points on Lattices Under Interleaving Execution
  3. Transitive Closure Example
  4. Fixed Point Theorem on Finite Distributive Lattices under Non--Interleaving Parallelism
  5. Transitive Closure Example
  6. Fixed Point Theorem on Finite Distributive Lattices under Distributed Computation
  7. Transitive Closure Example
  8. Greatest Common Fixed Points
  9. Applications
  10. Stable Marriage
  11. Shortest Paths
  12. Fair Division with Subsidy
  13. Related Work
  14. Conclusions and Future Work
  15. Appendix
  16. ...and 2 more sections

Key Result

Theorem 1

Let ${\cal F} = \{f_i : L \to L \mid i = 1,\dots,m\}$ be a set of inflationary and monotone functions. Consider an execution starting from $G_0=\bot$, where at each step $t$, a fair scheduler selects an index $i \in [m]$ and performs the atomic update $G_{t+1} = f_{i}(G_t)$. Then, the execution term

Theorems & Definitions (18)

  • Definition 1: Inflationary
  • Definition 2: Monotone
  • Theorem 1: Convergence to the least common fixed point under a fair schedule
  • proof
  • Theorem 2
  • Definition 3: Update-only-on-change
  • Theorem 3: Parallel convergence with non-interleaving semantics
  • Lemma 1: Monotone round progress
  • proof
  • proof
  • ...and 8 more