Single source--All sinks max flows in planar digraphs

J Lacki, Y Nussbaum, P Sankowski… - 2012 IEEE 53rd …, 2012 - ieeexplore.ieee.org
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, 2012ieeexplore.ieee.org
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∈V\setminus{s\}. We show how to solve this
problem in near-linear O(n\log^3n) 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\logn) 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 …
Let be a planar -vertex digraph. Consider the problem of computing max -flow values in from a fixed source to all sinks $t \in V \set minus \{s\}$. We show how to solve this problem in near-linear time. Previously, nothing better was known than running a single-source single-sink max flow algorithm times, giving a total time bound of with the algorithm of Borradaile and Klein. An important implication is that all-pairs max -flow values in can be computed in near-quadratic time. This is close to optimal as the output size is . We give a quadratic lower bound on the number of distinct max flow values and an 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 . Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of max-flow computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed and all , after preprocessing time, it can report the set of arcs crossing a min -cut in time.
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