3,961 research outputs found
Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths
Computing all-pairs shortest paths is a fundamental and much-studied problem
with many applications. Unfortunately, despite intense study, there are still
no significantly faster algorithms for it than the time
algorithm due to Floyd and Warshall (1962). Somewhat faster algorithms exist
for the vertex-weighted version if fast matrix multiplication may be used.
Yuster (SODA 2009) gave an algorithm running in time ,
but no combinatorial, truly subcubic algorithm is known.
Motivated by the recent framework of efficient parameterized algorithms (or
"FPT in P"), we investigate the influence of the graph parameters clique-width
() and modular-width () on the running times of algorithms for solving
All-Pairs Shortest Paths. We obtain efficient (and combinatorial) parameterized
algorithms on non-negative vertex-weighted graphs of times
, resp. . If fast matrix
multiplication is allowed then the latter can be improved to
using the algorithm of Yuster as a black box.
The algorithm relative to modular-width is adaptive, meaning that the running
time matches the best unparameterized algorithm for parameter value equal
to , and they outperform them already for for any
Speeding up shortest path algorithms
Given an arbitrary, non-negatively weighted, directed graph we
present an algorithm that computes all pairs shortest paths in time
, where is the number of
different edges contained in shortest paths and is a running
time of an algorithm to solve a single-source shortest path problem (SSSP).
This is a substantial improvement over a trivial times application of
that runs in . In our algorithm we use
as a black box and hence any improvement on results also in improvement
of our algorithm.
Furthermore, a combination of our method, Johnson's reweighting technique and
topological sorting results in an all-pairs
shortest path algorithm for arbitrarily-weighted directed acyclic graphs.
In addition, we also point out a connection between the complexity of a
certain sorting problem defined on shortest paths and SSSP.Comment: 10 page
Quantum and approximation algorithms for maximum witnesses of Boolean matrix products
The problem of finding maximum (or minimum) witnesses of the Boolean product
of two Boolean matrices (MW for short) has a number of important applications,
in particular the all-pairs lowest common ancestor (LCA) problem in directed
acyclic graphs (dags). The best known upper time-bound on the MW problem for
n\times n Boolean matrices of the form O(n^{2.575}) has not been substantially
improved since 2006. In order to obtain faster algorithms for this problem, we
study quantum algorithms for MW and approximation algorithms for MW (in the
standard computational model). Some of our quantum algorithms are input or
output sensitive. Our fastest quantum algorithm for the MW problem, and
consequently for the related problems, runs in time
\tilde{O}(n^{2+\lambda/2})=\tilde{O}(n^{2.434}), where \lambda satisfies the
equation \omega(1, \lambda, 1) = 1 + 1.5 \, \lambda and \omega(1, \lambda, 1)
is the exponent of the multiplication of an n \times n^{\lambda}$ matrix by an
n^{\lambda} \times n matrix. Next, we consider a relaxed version of the MW
problem (in the standard model) asking for reporting a witness of bounded rank
(the maximum witness has rank 1) for each non-zero entry of the matrix product.
First, by adapting the fastest known algorithm for maximum witnesses, we obtain
an algorithm for the relaxed problem that reports for each non-zero entry of
the product matrix a witness of rank at most \ell in time
\tilde{O}((n/\ell)n^{\omega(1,\log_n \ell,1)}). Then, by reducing the relaxed
problem to the so called k-witness problem, we provide an algorithm that
reports for each non-zero entry C[i,j] of the product matrix C a witness of
rank O(\lceil W_C(i,j)/k\rceil ), where W_C(i,j) is the number of witnesses for
C[i,j], with high probability. The algorithm runs in
\tilde{O}(n^{\omega}k^{0.4653} +n^2k) time, where \omega=\omega(1,1,1).Comment: 14 pages, 3 figure
Combining All Pairs Shortest Paths and All Pairs Bottleneck Paths Problems
We introduce a new problem that combines the well known All Pairs Shortest
Paths (APSP) problem and the All Pairs Bottleneck Paths (APBP) problem to
compute the shortest paths for all pairs of vertices for all possible flow
amounts. We call this new problem the All Pairs Shortest Paths for All Flows
(APSP-AF) problem. We firstly solve the APSP-AF problem on directed graphs with
unit edge costs and real edge capacities in
time,
where is the number of vertices, is the number of distinct edge
capacities (flow amounts) and is the time taken
to multiply two -by- matrices over a ring. Secondly we extend the problem
to graphs with positive integer edge costs and present an algorithm with
worst case time complexity, where is
the upper bound on edge costs
The graph bottleneck identity
A matrix is said to determine a
\emph{transitional measure} for a digraph on vertices if for all
the \emph{transition inequality} holds and reduces to the equality (called the \emph{graph
bottleneck identity}) if and only if every path in from to contains
. We show that every positive transitional measure produces a distance by
means of a logarithmic transformation. Moreover, the resulting distance
is \emph{graph-geodetic}, that is,
holds if and only if every path in connecting and contains .
Five types of matrices that determine transitional measures for a digraph are
considered, namely, the matrices of path weights, connection reliabilities,
route weights, and the weights of in-forests and out-forests. The results
obtained have undirected counterparts. In [P. Chebotarev, A class of
graph-geodetic distances generalizing the shortest-path and the resistance
distances, Discrete Appl. Math., URL
http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to
fill the gap between the shortest path distance and the resistance distance.Comment: 12 pages, 18 references. Advances in Applied Mathematic
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
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