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Min st-Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time

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https://ir.library.oregonstate.edu/concern/articles/5m60qt57r

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  • For an undirected n-vertex planar graph G with non-negative edge-weights, we consider the following type of query: given two vertices s and t in G, what is the weight of a min st-cut in G? We show how to answer such queries in constant time with O(n log⁴ n) preprocessing time and O(n log n) space. We use a Gomory-Hu tree to represent all the pairwise min cuts implicitly. Previously, no subquadratic time algorithm was known for this problem. Since all-pairs min cut and the minimum cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum cycle basis in O(n log⁴ n) time and O(n log n) space. Additionally, an explicit representation can be obtained in O(C) time and space where C is the size of the basis. These results require that shortest paths are unique. This can be guaranteed either by using randomization without overhead, or deterministically with an additional log² n factor in the preprocessing times.
  • © ACM, 2015. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in ACM Transactions on Algorithms (TALG), 11(3), Article 16, (January 2015)} http://doi.acm.org/10.1145/2684068
  • Keywords: Minimum cut, Minimum cycle basis, Planar graphs
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  • Borradaile, G., Sankowski, P., & Wulff-Nilsen, C. (2015). Min st-Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time. ACM Transactions on Algorithms (TALG), 11(3), 16. doi:10.1145/2684068
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  • 11
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  • 3
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  • This material is based on work supported by the National Science Foundation, under Grant No. CCF-0964037; by the Polish Ministry of Science, under Grant No. 206 355636; and by the European Research Council, under Project PAAl no. 259515
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