2 research outputs found
Maximum st-flow in directed planar graphs via shortest paths
Minimum cuts have been closely related to shortest paths in planar graphs via
planar duality - so long as the graphs are undirected. Even maximum flows are
closely related to shortest paths for the same reason - so long as the source
and the sink are on a common face. In this paper, we give a correspondence
between maximum flows and shortest paths via duality in directed planar graphs
with no constraints on the source and sink. We believe this a promising avenue
for developing algorithms that are more practical than the current
asymptotically best algorithms for maximum st-flow.Comment: 20 pages, 4 figures. Short version to be published in proceedings of
IWOCA'1
Faster Shortest Paths in Dense Distance Graphs, with Applications
We show how to combine two techniques for efficiently computing shortest
paths in directed planar graphs. The first is the linear-time shortest-path
algorithm of Henzinger, Klein, Subramanian, and Rao [STOC'94]. The second is
Fakcharoenphol and Rao's algorithm [FOCS'01] for emulating Dijkstra's algorithm
on the dense distance graph (DDG). A DDG is defined for a decomposition of a
planar graph into regions of at most vertices each, for some parameter
. The vertex set of the DDG is the set of vertices
of that belong to more than one region (boundary vertices). The DDG has
arcs, such that distances in the DDG are equal to the distances in
. Fakcharoenphol and Rao's implementation of Dijkstra's algorithm on the DDG
(nicknamed FR-Dijkstra) runs in time, and is a
key component in many state-of-the-art planar graph algorithms for shortest
paths, minimum cuts, and maximum flows. By combining these two techniques we
remove the dependency in the running time of the shortest-path
algorithm, making it .
This work is part of a research agenda that aims to develop new techniques
that would lead to faster, possibly linear-time, algorithms for problems such
as minimum-cut, maximum-flow, and shortest paths with negative arc lengths. As
immediate applications, we show how to compute maximum flow in directed
weighted planar graphs in time, where is the minimum number
of edges on any path from the source to the sink. We also show how to compute
any part of the DDG that corresponds to a region with vertices and
boundary vertices in time, which is faster than has been
previously known for small values of