Space-Time Tradeoffs for Distributed Verification

TL;DR

This paper introduces a new framework called $t$-PLS for distributed network verification that explores the tradeoffs between verification time, label size, message length, and space, providing optimal bounds and new techniques.

Contribution

It defines $t$-PLS, constructs a universal $t$-PLS, and establishes optimal space-time tradeoffs for verifying acyclicity in distributed networks.

Findings

Universal $t$-PLS matches known communication bounds.

Optimal label size and space complexity for acyclicity testing.

Recursive verifier operates efficiently without prior run-time knowledge.

Abstract

Verifying that a network configuration satisfies a given boolean predicate is a fundamental problem in distributed computing. Many variations of this problem have been studied, for example, in the context of proof labeling schemes (PLS), locally checkable proofs (LCP), and non-deterministic local decision (NLD). In all of these contexts, verification time is assumed to be constant. Korman, Kutten and Masuzawa [PODC 2011] presented a proof-labeling scheme for MST, with poly-logarithmic verification time, and logarithmic memory at each vertex. In this paper we introduce the notion of a $t$-PLS, which allows the verification procedure to run for super-constant time. Our work analyzes the tradeoffs of $t$-PLS between time, label size, message length, and computation space. We construct a universal $t$-PLS and prove that it uses the same amount of total communication as a known one-round universal PLS, and $t$ factor smaller labels. In addition, we provide a general technique to prove lower bounds for space-time tradeoffs of $t$-PLS. We use this technique to show an optimal tradeoff for testing that a network is acyclic (cycle free). Our optimal $t$-PLS for acyclicity uses label size and computation space $O((\log n)/t)$. We further describe a recursive $O(\log^* n)$ space verifier for acyclicity which does not assume previous knowledge of the run-time $t$.