2,648 research outputs found
Max flow vitality in general and -planar graphs
The \emph{vitality} of an arc/node of a graph with respect to the maximum
flow between two fixed nodes and is defined as the reduction of the
maximum flow caused by the removal of that arc/node. In this paper we address
the issue of determining the vitality of arcs and/or nodes for the maximum flow
problem. We show how to compute the vitality of all arcs in a general
undirected graph by solving only max flow instances and, In
-planar graphs (directed or undirected) we show how to compute the vitality
of all arcs and all nodes in worst-case time. Moreover, after
determining the vitality of arcs and/or nodes, and given a planar embedding of
the graph, we can determine the vitality of a `contiguous' set of arcs/nodes in
time proportional to the size of the set.Comment: 12 pages, 3 figure
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\{s}. We show
how to solve this problem in near-linear O(n log^3 n) time. Previously, no
better solution 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^{3/2} log^{3/2} n) preprocessing time, it can
report the set of arcs C crossing a min st-cut in time roughly proportional to
the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201
Minimum Cycle Basis and All-Pairs Min Cut of a Planar Graph in Subquadratic Time
A minimum cycle basis of a weighted undirected graph is a basis of the
cycle space of such that the total weight of the cycles in this basis is
minimized. If is a planar graph with non-negative edge weights, such a
basis can be found in time and space, where is the size of . We
show that this is optimal if an explicit representation of the basis is
required. We then present an time and space
algorithm that computes a minimum cycle basis \emph{implicitly}. From this
result, we obtain an output-sensitive algorithm that explicitly computes a
minimum cycle basis in time and space,
where is the total size (number of edges and vertices) of the cycles in the
basis. These bounds reduce to and ,
respectively, when is unweighted. We get similar results for the all-pairs
min cut problem since it is dual equivalent to the minimum cycle basis problem
for planar graphs. We also obtain time and
space algorithms for finding, respectively, the weight vector and a Gomory-Hu
tree of . The previous best time and space bound for these two problems was
quadratic. From our Gomory-Hu tree algorithm, we obtain the following result:
with time and space for preprocessing, the
weight of a min cut between any two given vertices of can be reported in
constant time. Previously, such an oracle required quadratic time and space for
preprocessing. The oracle can also be extended to report the actual cut in time
proportional to its size
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