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 $O(n\log^4n)$ 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 $O(n\log^4n)$ 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 $\log^2 n$ factor in the preprocessing times.