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

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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(n⁴n) preprocessing time and O(n 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(n⁴n) time and O(n 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 ² n factor in the preprocessing times.