Single Source -- All Sinks Max Flows in Planar Digraphs

Let G = (V, E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t ϵ V \ {s}. We show how to solve this problem in near-linear O(n log³ n) time. Previously, nothing better was known than running a single-source singlesink max How algorithm n-1 times, giving a total time bound of O(n² log n) with the algorithm of Borradaile and Klein. An important implication is that all-pairs max st-How values in G can be computed in near-quadratic time. This is close to optimal as the output size is 8(n²). We give a quadratic lower bound on the number of distinct max How values and an Ω(n³) lower bound for the total size of all min cut-sets. This distinguishes the problem from the undirected case where the number of distinct max How values is O(n). Previous to our result, no algorithm which could solve the all-pairs max How values problem faster than the time of 8(n²) max-How computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed s and all t, after O(n^1.5 log² n) preprocessing time, it can report the set of arcs C crossing a min st-cut in O(|C|) time.