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

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 log⁵ n) preprocessing time and O(n log n) space. We use a Gomory-Hu tree to represent all the pairwise min st-cuts implicitly. Previously, no subquadratic time algorithm was known for this problem. Our oracle can be extended to report the min st-cuts in time proportional to their size. Since all-pairs min si-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 log⁵ n) time and O(n log n) space and an explicit representation with additional O(C) time and space where G is the size of the basis. To obtain our results, we require that shortest paths be unique; this assumption can be removed deterministically with an additional O(log² n) running-time factor.