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We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time. When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sub-linear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilon-approximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d-1)/(2d-1)}) poly log n time algorithm for computing epsilon-approximate bottleneck matching in d-dimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G'(V',E') of size |V'|^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HK-Algorithm.
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GraphRay: Distributed pathfinder network scaling
Pathfinder network scaling is a graph sparsification technique that has been popularly used due to its efficacy of extracting the “important” structure of a graph. However, existing algorithms to compute the pathfinder network (PFNET) of a graph have prohibitively expensive time complexity for large graphs: O(n 3 ) for the general case and O(n 2 log n) for a specific parameter setting, PFNET(r = ∞, q = n - 1), which is considered in many applications. In this paper, we introduce the first distributed technique to compute the pathfinder network with the specific parameters (r = ∞ and q = n - 1) of a large graph with millions of edges. The results of our experiments show our technique is scalable; it efficiently utilizes a parallel distributed computing environment, reducing the running times as more processing units are added.
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- Award ID(s):
- 1741536
- PAR ID:
- 10078462
- Date Published:
- Journal Name:
- IEEE 7th Symposium on Large Data Analysis and Visualization (LDAV 2017)
- Page Range / eLocation ID:
- 74 to 83
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/LDAV.2017.8231853
