Computer Science – Data Structures and Algorithms
Scientific paper
2009-02-10
Computer Science
Data Structures and Algorithms
Added analysis of the Hopcroft-Karp algorithm on the subsampled graph
Scientific paper
We consider the well-studied problem of finding a perfect matching in $d$-regular bipartite graphs with $2n$ vertices and $m = nd$ edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes $O(m \sqrt{n})$ time, in regular bipartite graphs, a perfect matching is known to be computable in $O(m)$ time. Very recently, the $O(m)$ bound was improved to $O(\min\{m, \frac{n^{2.5}\ln n}{d}\})$ expected time, an expression that is bounded by $\tilde{O}(n^{1.75})$. In this paper, we further improve this result by giving an $O(\min\{m, \frac{n^2\ln^3 n}{d}\})$ expected time algorithm for finding a perfect matching in regular bipartite graphs; as a function of $n$ alone, the algorithm takes expected time $O((n\ln n)^{1.5})$. To obtain this result, we design and analyze a two-stage sampling scheme that reduces the problem of finding a perfect matching in a regular bipartite graph to the same problem on a subsampled bipartite graph with $O(n\ln n)$ edges that has a perfect matching with high probability. The matching is then recovered using the Hopcroft-Karp algorithm. While the standard analysis of Hopcroft-Karp gives us an $\tilde{O}(n^{1.5})$ running time, we present a tighter analysis for our special case that results in the stronger $\tilde{O}(\min\{m, \frac{n^2}{d} \})$ time mentioned earlier. Our proof of correctness of this sampling scheme uses a new correspondence theorem between cuts and Hall's theorem ``witnesses'' for a perfect matching in a bipartite graph that we prove. We believe this theorem may be of independent interest; as another example application, we show that a perfect matching in the support of an $n \times n$ doubly stochastic matrix with $m$ non-zero entries can be found in expected time $\tilde{O}(m + n^{1.5})$.
Goel Ashish
Kapralov Michael
Khanna Sanjeev
No associations
LandOfFree
Perfect Matchings in Õ(n^{1.5}) Time in Regular Bipartite Graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Perfect Matchings in Õ(n^{1.5}) Time in Regular Bipartite Graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Perfect Matchings in Õ(n^{1.5}) Time in Regular Bipartite Graphs will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-19838