1,134 research outputs found
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
A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs
We consider the problem of computing all-pairs shortest paths in a directed
graph with real weights assigned to vertices.
For an 0-1 matrix let be the complete weighted graph
on the rows of where the weight of an edge between two rows is equal to
their Hamming distance. Let be the weight of a minimum weight spanning
tree of
We show that the all-pairs shortest path problem for a directed graph on
vertices with nonnegative real weights and adjacency matrix can be
solved by a combinatorial randomized algorithm in time
As a corollary, we conclude that the transitive closure of a directed graph
can be computed by a combinatorial randomized algorithm in the
aforementioned time.
We also conclude that the all-pairs shortest path problem for uniform disk
graphs, with nonnegative real vertex weights, induced by point sets of bounded
density within a unit square can be solved in time
Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision
The quantum query complexity of Boolean matrix multiplication is typically
studied as a function of the matrix dimension, n, as well as the number of 1s
in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values
of \ell. This is an improvement over previous algorithms for all values of
\ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps
n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing
that our algorithm is essentially tight.
We first reduce Boolean matrix multiplication to several instances of graph
collision. We then provide an algorithm that takes advantage of the fact that
the underlying graph in all of our instances is very dense to find all graph
collisions efficiently
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
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
Quantum Algorithms for Matrix Products over Semirings
In this paper we construct quantum algorithms for matrix products over
several algebraic structures called semirings, including the (max,min)-matrix
product, the distance matrix product and the Boolean matrix product. In
particular, we obtain the following results.
We construct a quantum algorithm computing the product of two n x n matrices
over the (max,min) semiring with time complexity O(n^{2.473}). In comparison,
the best known classical algorithm for the same problem, by Duan and Pettie,
has complexity O(n^{2.687}). As an application, we obtain a O(n^{2.473})-time
quantum algorithm for computing the all-pairs bottleneck paths of a graph with
n vertices, while classically the best upper bound for this task is
O(n^{2.687}), again by Duan and Pettie.
We construct a quantum algorithm computing the L most significant bits of
each entry of the distance product of two n x n matrices in time O(2^{0.64L}
n^{2.46}). In comparison, prior to the present work, the best known classical
algorithm for the same problem, by Vassilevska and Williams and Yuster, had
complexity O(2^{L}n^{2.69}). Our techniques lead to further improvements for
classical algorithms as well, reducing the classical complexity to
O(2^{0.96L}n^{2.69}), which gives a sublinear dependency on 2^L.
The above two algorithms are the first quantum algorithms that perform better
than the -time straightforward quantum algorithm based on
quantum search for matrix multiplication over these semirings. We also consider
the Boolean semiring, and construct a quantum algorithm computing the product
of two n x n Boolean matrices that outperforms the best known classical
algorithms for sparse matrices. For instance, if the input matrices have
O(n^{1.686...}) non-zero entries, then our algorithm has time complexity
O(n^{2.277}), while the best classical algorithm has complexity O(n^{2.373}).Comment: 19 page
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