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arXiv 2602.10486cs.DCcs.DS
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在并行和分布式系统中利用覆盖语义计算最小固定点

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

University of Texas at Austin(德克萨斯大学奥斯汀分校);Purdue University(普渡大学)

Vijay K. Garg, Rohan Garg

AI总结:

本文提出在并行和分布式系统中利用覆盖语义计算最小固定点的方法,解决了传统方法无法处理的非原子更新和过期读取问题。

摘要:

我们提出了在并行和分布式环境中计算多个单调膨胀函数的最小固定点的方法。虽然经典的Knaster-Tarski定理处理单一函数的顺序迭代,但现代计算系统需要并行执行具有覆盖语义、非原子更新和过期读取。我们证明了三个在逐步放松同步条件下收敛的定理:(1) 交错语义与公平调度;(2) 并行执行仅在值变化时更新语义(进程只在值变化的坐标上写入);(3) 分布式执行具有受限过期(更新在T轮内传播)和i-局部性(每个进程只修改自身组件)。我们的方法与先前工作在根本上有所不同:Cousot-Cousot的混沌迭代使用基于连接的合并以保留信息。相反,我们使用坐标-wise覆盖。Bertsekas的异步方法假设收缩。我们使用基于结构约束(局部性、受限过期)的坐标-wise覆盖。应用包括并行和分布式算法中的传递闭包、稳定婚姻、最短路径和带有补贴的公平分配问题。我们的结果为没有连接操作或收缩假设的覆盖基于并行更新提供了首次精确的最小固定点收敛保证。

英文摘要:

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.

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