arXiv subjects

Will Rosenbaum

Publications and source records attributed to Will Rosenbaum.

23 records · Page 2Linked to original sources

arXiv2017

Space-Time Tradeoffs for Distributed Verification

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 $t$.

cs.DC↗

arXiv2017

On Sampling Edges Almost Uniformly

We consider the problem of sampling an edge almost uniformly from an unknown graph, $G = (V, E)$. Access to the graph is provided via queries of the following types: (1) uniform vertex queries, (2) degree queries, and (3) neighbor queries. We describe an algorithm that returns a random edge $e \in E$ using $\tilde{O}(n / \sqrt{\varepsilon m})$ queries in expectation, where $n = |V|$ is the number of vertices, and $m = |E|$ is the number of edges, such that each edge $e$ is sampled with probability $(1 \pm \varepsilon)/m$. We prove that our algorithm is optimal in the sense that any algorithm that samples an edge from an almost-uniform distribution must perform $Ω(n / \sqrt{m})$ queries.

cs.CC↗

arXiv2015

Fast distributed almost stable marriages

In their seminal work on the Stable Marriage Problem, Gale and Shapley 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, Floreen, Kaski, Polishchuk, and Suomela 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.

cs.GT↗

arXiv2014

It's Not Easy Being Three: The Approximability of Three-Dimensional Stable Matching Problems

In 1976, Knuth asked if the stable marriage problem (SMP) can be generalized to marriages consisting of 3 genders. In 1988, Alkan showed that the natural generalization of SMP to 3 genders ($3$GSM) need not admit a stable marriage. Three years later, Ng and Hirschberg proved that it is NP-complete to determine if given preferences admit a stable marriage. They further prove an analogous result for the $3$ person stable assignment ($3$PSA) problem. In light of Ng and Hirschberg's NP-hardness result for $3$GSM and $3$PSA, we initiate the study of approximate versions of these problems. In particular, we describe two optimization variants of $3$GSM and $3$PSA: maximally stable marriage/matching (MSM) and maximum stable submarriage/submatching (MSS). We show that both variants are NP-hard to approximate within some fixed constant factor. Conversely, we describe a simple polynomial time algorithm which computes constant factor approximations for the maximally stable marriage and matching problems. Thus both variants of MSM are APX-complete.

cs.CC↗

arXiv2014

On The Communication Complexity of Finding an (Approximate) Stable Marriage

In this paper, we consider the communication complexity of protocols that compute stable matchings. We work within the context of Gale and Shapley's original stable marriage problem\cite{GS62}: $n$ men and $n$ women each privately hold a total and strict ordering on all of the members of the opposite gender. They wish to collaborate in order to find a stable matching---a pairing of the men and women such that no unmatched pair mutually prefer each other to their assigned partners in the matching. We show that any communication protocol (deterministic, nondeterministic, or randomized) that correctly ouputs a stable matching requires $Ω(n^2)$ bits of communication. Thus, the original algorithm of Gale and Shapley is communication-optimal up to a logarithmic factor. We then introduce a "divorce metric" on the set of all matchings, which allows us to consider approximately stable matchings. We describe an efficient algorithm to compute the "distance to stability" of a given matching. We then show that even under the relaxed requirement that a protocol only yield an approximate stable matching, the $Ω(n^2)$ communication lower bound still holds.

cs.CC↗