4,817 research outputs found
All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs
For an undirected -vertex graph with non-negative edge-weights, we
consider the following type of query: given two vertices and in ,
what is the weight of a minimum -cut in ? We solve this problem in
preprocessing time for graphs of bounded genus, giving the first
sub-quadratic time algorithm for this class of graphs. Our result also improves
by a logarithmic factor a previous algorithm by Borradaile, Sankowski and
Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm
constructs a Gomory-Hu tree for the given graph, providing a data structure
with space that can answer minimum-cut queries in constant time. The
dependence on the genus of the input graph in our preprocessing time is
Planar Ultrametric Rounding for Image Segmentation
We study the problem of hierarchical clustering on planar graphs. We
formulate this in terms of an LP relaxation of ultrametric rounding. To solve
this LP efficiently we introduce a dual cutting plane scheme that uses minimum
cost perfect matching as a subroutine in order to efficiently explore the space
of planar partitions. We apply our algorithm to the problem of hierarchical
image segmentation
Max -Flow Oracles and Negative Cycle Detection in Planar Digraphs
We study the maximum -flow oracle problem on planar directed graphs
where the goal is to design a data structure answering max -flow value (or
equivalently, min -cut value) queries for arbitrary source-target pairs
. For the case of polynomially bounded integer edge capacities, we
describe an exact max -flow oracle with truly subquadratic space and
preprocessing, and sublinear query time. Moreover, if
-approximate answers are acceptable, we obtain a static oracle
with near-linear preprocessing and query time and a
dynamic oracle supporting edge capacity updates and queries in
worst-case time.
To the best of our knowledge, for directed planar graphs, no (approximate)
max -flow oracles have been described even in the unweighted case, and
only trivial tradeoffs involving either no preprocessing or precomputing all
the possible answers have been known.
One key technical tool we develop on the way is a sublinear (in the number of
edges) algorithm for finding a negative cycle in so-called dense distance
graphs. By plugging it in earlier frameworks, we obtain improved bounds for
other fundamental problems on planar digraphs. In particular, we show: (1) a
deterministic time algorithm for negatively-weighted SSSP in
planar digraphs with integer edge weights at least . This improves upon the
previously known bounds in the important case of weights polynomial in , and
(2) an improved bound on finding a perfect matching in a
bipartite planar graph.Comment: Extended abstract to appear in SODA 202
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