13 research outputs found
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
Fast Partial Distance Estimation and Applications
We study approximate distributed solutions to the weighted {\it
all-pairs-shortest-paths} (APSP) problem in the CONGEST model. We obtain the
following results.
A deterministic -approximation to APSP in
rounds. This improves over the best previously known algorithm, by both
derandomizing it and by reducing the running time by a factor.
In many cases, routing schemes involve relabeling, i.e., assigning new names
to nodes and require that these names are used in distance and routing queries.
It is known that relabeling is necessary to achieve running times of . In the relabeling model, we obtain the following results.
A randomized -approximation to APSP, for any integer ,
running in rounds, where is the hop diameter of
the network. This algorithm simplifies the best previously known result and
reduces its approximation ratio from to . Also, the new
algorithm uses uses labels of asymptotically optimal size, namely
bits.
A randomized -approximation to APSP, for any integer ,
running in time and producing {\it
compact routing tables} of size . The node lables consist
of bits. This improves on the approximation ratio of
for tables of that size achieved by the best previously known algorithm, which
terminates faster, in rounds
Approximation of Distances and Shortest Paths in the Broadcast Congest Clique
We study the broadcast version of the CONGEST-CLIQUE model of distributed computing. This model operates in synchronized rounds; in each round, any node in a network of size n can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC\u2714] a randomized (2+o(1))-approximation algorithm to compute all pairs shortest paths (APSP) in time ~{O}(sqrt{n}) on weighted graphs. We complement this result by proving that any randomized (2-o(1))-approximation of APSP and (2-o(1))-approximation of the diameter of a graph takes ~Omega(n) time in the worst case. This demonstrates that getting a negligible improvement in the approximation factor requires significantly more time. Furthermore this bound implies that already computing a (2-o(1))-approximation of all pairs shortest paths is among the hardest graph-problems in the broadcast-version of the CONGEST-CLIQUE model, as any graph-problem where each node receives a linear amount of input can be solved trivially in linear time in this model. This contrasts a recent (1+o(1))-approximation for APSP that runs in time O(n^{0.15715}) and an exact algorithm for APSP that runs in time ~O(n^{1/3}) in the unicast version of the CONGEST-CLIQUE model, a more powerful variant of the broadcast version.
This lower bound in the broadcast CONGEST-CLIQUE model is derived by first establishing a new lower bound for (2-o(1))-approximating the diameter in weighted graphs in the CONGEST model, which is of independent interest. This lower bound is then transferred to the CONGEST-CLIQUE model.
On the positive side we provide a deterministic version of Nanongkai\u27s (2+o(1))-approximation algorithm for APSP. To do so we present a fast deterministic construction of small hitting sets. We also show how to replace another randomized part within Nanongkai\u27s algorithm with a deterministic source-detection algorithm designed for the CONGEST model
Distributed Exact Shortest Paths in Sublinear Time
The distributed single-source shortest paths problem is one of the most
fundamental and central problems in the message-passing distributed computing.
Classical Bellman-Ford algorithm solves it in time, where is the
number of vertices in the input graph . Peleg and Rubinovich (FOCS'99)
showed a lower bound of for this problem, where
is the hop-diameter of .
Whether or not this problem can be solved in time when is
relatively small is a major notorious open question. Despite intensive research
\cite{LP13,N14,HKN15,EN16,BKKL16} that yielded near-optimal algorithms for the
approximate variant of this problem, no progress was reported for the original
problem.
In this paper we answer this question in the affirmative. We devise an
algorithm that requires time, for , and time, for larger . This
running time is sublinear in in almost the entire range of parameters,
specifically, for . For the all-pairs shortest paths
problem, our algorithm requires time, regardless of
the value of .
We also devise the first algorithm with non-trivial complexity guarantees for
computing exact shortest paths in the multipass semi-streaming model of
computation.
From the technical viewpoint, our algorithm computes a hopset of a
skeleton graph of without first computing itself. We then conduct
a Bellman-Ford exploration in , while computing the required edges
of on the fly. As a result, our algorithm computes exactly those edges of
that it really needs, rather than computing approximately the entire