Randomized algorithms
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SIAM Journal on Computing
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FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Perfect matchings via uniform sampling in regular bipartite graphs
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Introduction to Algorithms, Third Edition
Introduction to Algorithms, Third Edition
Load balanced Birkhoff-von Neumann switches, part I: one-stage buffering
Computer Communications
An O(n log n) algorithm for a load balancing problem on paths
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Efficient algorithms for maximum weight matchings in general graphs with small edge weights
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Finding all maximally-matchable edges in a bipartite graph
Theoretical Computer Science
A near-linear time ε-approximation algorithm for geometric bipartite matching
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Integrating microsecond circuit switching into the data center
Proceedings of the ACM SIGCOMM 2013 conference on SIGCOMM
A new approach to computing maximum flows using electrical flows
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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In this paper we consider the well-studied problem of finding a perfect matching in a d-regular bipartite graph on 2n nodes with m=nd edges. The best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes time O(m√n). In regular bipartite graphs, however, a matching is known to be computable in O(m) time (due to Cole, Ost, and Schirra). In a recent line of work by Goel, Kapralov, and Khanna the O(m) time bound was improved first to ~ O(min m, n2.5/d) and then to ~O(min {m, n2/d\}). In this paper, we give a randomized algorithm that finds a perfect matching in a d-regular graph and runs in O(n log n) time (both in expectation and with high probability). The algorithm performs an appropriately truncated alternating random walk to successively find augmenting paths. Our algorithm may be viewed as using adaptive uniform sampling, and is thus able to bypass the limitations of (non-adaptive) uniform sampling established in earlier work. Our techniques also give an algorithm that successively finds a matching in the support of a doubly stochastic matrix in expected time O(n log2 n), with O(m) pre-processing time; this gives a simple O(m+mn log2 n) time algorithm for finding the Birkhoff-von Neumann decomposition of a doubly stochastic matrix. We show that randomization is crucial for obtaining o(nd) time algorithms by establishing an Ω(nd) lower bound for deterministic algorithms. We also show that there does not exist a randomized algorithm that finds a matching in a regular bipartite multigraph and takes o(n log n) time with high probability.