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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.

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Synchronizing clocks in distributed systems is well-understood, both in terms of faulttolerance in fully connected systems and the dependence of local and global worst-case skews (i.e., maximum clock difference between neighbors and arbitrary pairs of nodes, respectively) on the diameter of fault-free systems. However, so far nothing non-trivial is known about the local skew that can be achieved in topologies that are not fully connected even under a single Byzantine fault. Put simply, in this work we show that the most powerful known techniques for fault-tolerant and gradient clock synchronization are compatible, in the sense that the best of both worlds can be achieved simultaneously.Concretely, we combine the Lynch-Welch algorithm [17] for synchronizing a clique of n nodes despite up to f < n/3 Byzantine faults with the gradient clock synchronization (GCS) algorithm by Lenzen et al. [13] in order to render the latter resilient to faults. As this is not possible on general graphs, we augment an input graph G by replacing each node by 3f + 1 fully connected copies, which execute an instance of the Lynch-Welch algorithm. We then interpret these clusters as supernodes executing the GCS algorithm, where for each cluster its correct nodes' Lynch-Welch clocks provide estimates of the logical clock of the supernode in the GCS algorithm. By connecting clusters corresponding to neighbors in G in a fully bipartite manner, supernodes can inform each other about (estimates of) their logical clock values. This way, we achieve asymptotically optimal local skew, granted that no cluster contains more than f faulty nodes, at factor O(f ) and O(f 2 ) overheads in terms of nodes and edges, respectively. Note that tolerating f faulty neighbors trivially requires degree larger than f , so this is asymptotically optimal as well.

The Gale-Shapley algorithm for the Stable Marriage Problem is known to take Θ(n 2 ) steps to find a stable marriage in the worst case, but only Θ(n log n) steps in the average case (with n women and n men). In 1976, Knuth asked whether the worst-case running time can be improved in a model of computation that does not require sequential access to the whole input. A partial negative answer was given by Ng and Hirschberg, who showed that Θ(n 2 ) queries are required in a model that allows certain natural random-access queries to the participants' preferences. A significantly more general -albeit slightly weaker -lower bound follows from Segal's general analysis of communication complexity, namely that Ω(n 2 ) Boolean queries are required in order to find a stable marriage, regardless of the set of allowed Boolean queries.Using a reduction to the communication complexity of the disjointness problem, we give a far simpler, yet significantly more powerful argument showing that Ω(n 2 ) Boolean queries of any type are indeed required for finding a stable -or even an approximately stable -marriage. Notably, unlike Segal's lower bound, our lower bound generalizes also to (A) randomized algorithms, (B) allowing arbitrary separate preprocessing of the women's preferences profile and of the men's preferences profile, (C) several variants of the basic problem, such as whether a given pair is married in every/some stable marriage, and (D) determining whether a proposed marriage is stable or far from stable. In order to analyze "approximately stable" marriages, we introduce the notion of "distance to stability" and provide an efficient algorithm for its computation.Our proof of Theorem 1.1 comes from a reduction to the well-known lower bounds for the disjointness problem [18,28] in Yao's [36] model of two-party communication complexity (see [20] for a survey). We consider a scenario in which Alice holds the preferences of the n women and Bob holds the preferences of the n men, and show that each of the problems from Theorem 1.1 requires the exchange of Ω(n 2 ) bits of communication between Alice and Bob.We note that Segal [33] shows by a general argument that any deterministic or nondeterministic 5 communication protocol among all 2n participants for finding a stable marriage requires Ω(n 2 ) bits of communication. Our argument for Theorem 1.1(a), in addition to being significantly simpler, generalizes Segal's result to account for randomized algorithms, 6 and even when considering only two-party communication between Alice and Bob (essentially allowing arbitrary communication within the set of women and within the set of men without cost). Furthermore, our lower bound holds even for merely determining whether a given marriage is stable or far from stable (Theorem 1.1(b)), as well as for the additional related problems described in Theorem 1.1(c,d). These results immediately imply the same lower bounds for any type of Boolean queries in the original computation model, as Boolean queries can be simulated by a communication proto...

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In their seminal work on the Stable Marriage Problem, Gale and Shapley [4] describe an algorithm which finds a stable matching in O(n 2 ) communication rounds. Their algorithm has a natural interpretation as a distributed algorithm where each player is represented by a single processor. In this distributed model, Floréen, Kaski, Polishchuk, and Suomela [3] recently showed that for bounded preference lists, terminating the Gale-Shapley algorithm after a constant number of rounds results in an almost stable matching. In this paper, we describe a new deterministic distributed algorithm which finds an almost stable matching in O(log 5 n) communication rounds for arbitrary preferences. We also present a faster randomized variant which requires O(log 2 n) rounds. This run-time can be improved to O(1) rounds for "almost regular" (and in particular complete) preferences. To our knowledge, these are the first sub-polynomial round distributed algorithms for any variant of the stable marriage problem with unbounded preferences.

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