Distributed Minimum Cut Approximation
نویسندگان
چکیده
We study the problem of computing approximate minimum edge cuts by distributed algorithms. We present two randomized approximation algorithms that both run in a standard synchronous message passing model where in each round, O(log n) bits can be transmitted over every edge (a.k.a. the CONGEST model). The first algorithm is based on a simple and new approach for analyzing random edge sampling, which we call random layering technique. For any any weighted graph and any ∈ (0, 1), the algorithm finds a cut of size at most O( −1λ) in O(D) + Õ(n ) rounds, where λ is the minimum-cut size and the Õ-notation hides poly-logarithmic factors in n. In addition, using the outline of a centralized algorithm due to Matula [SODA ’93], we present a randomized algorithm to compute a cut of size at most (2 + )λ in Õ((D+ √ n)/ ) rounds for any > 0. The time complexities of our algorithms almost match the Ω̃(D + √ n) lower bound of Das Sarma et al. [STOC ’11], thus leading to an answer to an open question raised by Elkin [SIGACT-News ’04] and Das Sarma et al. [STOC ’11]. To complement our upper bound results, we also strengthen the Ω̃(D + √ n) lower bound of Das Sarma et al. by extending it to unweighted graphs. We show that the same lower bound also holds for unweighted multigraphs (or equivalently for weighted graphs in which O(w log n) bits can be transmitted in each round over an edge of weightw). For unweighted simple graphs, we show that computing an α-approximate minimum cut requires time at least Ω̃(D + √ n/α).
منابع مشابه
A Distributed Minimum Cut Approximation Scheme
In this paper, we study the problem of approximating the minimum cut in a distributed message-passing model, the CONGEST model. The minimum cut problem has been well-studied in the context of centralized algorithms. However, there were no known non-trivial algorithms in the distributed model until the recent work of Ghaffari and Kuhn. They gave algorithms for finding cuts of size O(ǫλ) and (2 +...
متن کاملAlmost-Tight Distributed Minimum Cut Algorithms
We study the problem of computing the minimum cut in a weighted distributed messagepassing networks (the CONGEST model). Let λ be the minimum cut, n be the number of nodes (processors) in the network, and D be the network diameter. Our algorithm can compute λ exactly in O(( √ n log∗ n+D)λ log n) time. To the best of our knowledge, this is the first paper that explicitly studies computing the ex...
متن کاملA Static 2-Approximation Algorithm for Vertex Connectivity and Incremental Approximation Algorithms for Edge and Vertex Connectivity
This paper presents insertions-only algorithms for maintaining the exact and/or approximate size of the minimum edge cut and the minimum vertex cut of a graph. The algorithms output the approximate or exact size k in time O(1) and a cut of size k in time linear in its size. For the minimum edge cut problem and for any 0 < 1, the amortized time per insertion is O(1= ) for a (2 + )-approximation,...
متن کاملA Static 2-Approximation Algorithm for Vertex Connectivity and Imcremental Approximation Algorithms for Edge and Vertex Connectivity
This paper presents insertions-only algorithms for maintaining the exact and/or approximate size of the minimum edge cut and the minimum vertex cut of a graph. The algorithms output the approximate or exact size k in time O(1) and a cut of size k in time linear in its size. For the minimum edge cut problem and for any 0 < 1, the amortized time per insertion is O(1== 2) for a (2 +)-approximation...
متن کاملDC-Based Approximations on Graphs
We now return to the one basic design paradigm we skipped in our discussion of approximation algorithms for NP-hard optimization problems, namely divide-and-conquer (DC), and apply it to problems on graphs. The basic idea is as follows: Find a balanced cut of small capacity, i.e., a cut in which both sides contain roughly the same number of vertices and such that few edges cross the cut. Ignore...
متن کامل