10,535 research outputs found
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
Algebraic Methods in the Congested Clique
In this work, we use algebraic methods for studying distance computation and
subgraph detection tasks in the congested clique model. Specifically, we adapt
parallel matrix multiplication implementations to the congested clique,
obtaining an round matrix multiplication algorithm, where
is the exponent of matrix multiplication. In conjunction
with known techniques from centralised algorithmics, this gives significant
improvements over previous best upper bounds in the congested clique model. The
highlight results include:
-- triangle and 4-cycle counting in rounds, improving upon the
triangle detection algorithm of Dolev et al. [DISC 2012],
-- a -approximation of all-pairs shortest paths in
rounds, improving upon the -round -approximation algorithm of Nanongkai [STOC 2014], and
-- computing the girth in rounds, which is the first
non-trivial solution in this model.
In addition, we present a novel constant-round combinatorial algorithm for
detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266
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
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