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 $t \in V \set minus \{s\}$. We show how to solve this problem in near-linear $O(n \log^3 n)$ time. Previously, nothing better was known than running a single-source single-sink max flow algorithm $n-1$ times, giving a total time bound of $O(n^2 \log n)$ with the algorithm of Borradaile and Klein. An important implication is that all-pairs max $st$-flow values in $G$ can be computed in near-quadratic time. This is close to optimal as the output size is $\Theta(n^2)$. We give a quadratic lower bound on the number of distinct max flow values and an $\Omega(n^3)$ 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 flow values is $O(n)$. Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of $\Theta(n^2)$ max-flow 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 $O(n^{1.5} \log^2 n)$ preprocessing time, it can report the set of arcs $C$ crossing a min $st$-cut in $O(|C|)$ time.