Space-Time Tradeoffs for Distributed Verification

Abstract

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 nondeterministic local decision (NLD). In all of these contexts, verification time is assumed to be constant. Korman, Kutten and Masuzawa [15] 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.

Redundancy in distributed proofs

“…The certificate of each node consists of the n × n adjacency matrix of the graph, an array of n entries each equals to the k-bit label at the corresponding node, and an array of n entries listing the identities of the n nodes. It was proved in [41] that the universal proof-labeling scheme can be scaled by a factor t. Theorem 3 significantly improves that result, by showing that the universal proof-labeling scheme can actually be scaled by a factor b(t), which can be exponential in t.…”

“…In this paper, we aim at designing deterministic and generic ways for reducing the certificate size of prooflabeling schemes. This is achieved by following the guidelines of [41], that is, trading time for space by exploiting the inherent redundancy in distributed proofs. We focus only on questions regarding the existence of proof-labeling schemes, and give upper and lower bounds on their sizes.…”

“…Our work focuses on the second model, of locally checkable proofs. However, for consistency with previous literature on scaling [41], we stick to the name proof-labeling schemes. Note that the main difference is the use of node identifiers, and when the certificates are of size Ω(log n), the prover can add the idbandwidth restrictionentifiers to the certificates, making the models equivalent.…”

“…Our work was inspired by [41], which aims at reducing the size of the certificates by trading time for space, i.e., allowing the verification procedure to take t rounds, for a non-constant t, in order to reduce the certificate size. They show a tradeoff of this kind for example for proving the acyclicity of the input graph.…”

“…For more references to work related to distributed verification, or distributed decision in general, see the survey [18]. To our knowledge, the aforementioned works [36,41] are the only prior works studying tradeoffs between time and certificate size.…”