Exponentially Faster Shortest Paths in the Congested Clique
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Published:2022-08-31
Issue:4
Volume:69
Page:1-42
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ISSN:0004-5411
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Container-title:Journal of the ACM
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language:en
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Short-container-title:J. ACM
Author:
Dory Michal1ORCID,
Parter Merav2ORCID
Affiliation:
1. ETH Zurich, Zürich, Switzerland
2. Weizmann Institute, Rehovot, Israel
Abstract
We present improved deterministic algorithms for approximating shortest paths in the
Congested Clique
model of distributed computing. We obtain poly(log log
n
)-round algorithms for the following problems in unweighted undirected
n
-vertex graphs:
( 1 +
ϵ
)-approximation of multi-source shortest paths (MSSP) from
O
(√
n
) sources.
(2 +
ϵ
)-approximation of all pairs shortest paths (APSP).
(1 +
ϵ
, β)-approximation of APSP where
β
=
O
(log log
n
/
ϵ
)
log log
n
.
These bounds improve
exponentially
over the state-of-the-art poly-logarithmic bounds due to [Censor-Hillel et al., PODC19]. It also provides the first nearly-additive bounds for the APSP problem in sub-polynomial time. Our approach is based on distinguishing between short and long distances based on some distance threshold
t = O
(
β
/
ϵ
) where
β
=
O
(log log
n
/
ϵ
)
log log
n
. Handling the long distances is done by devising a new algorithm for computing a sparse (1 +
ϵ
,
β
) emulator with
O
(
n
log log
n
) edges. For the short distances, we provide
distance-sensitive
variants for the distance tool-kit of [Censor-Hillel et al., PODC19]. By exploiting the fact that this tool-kit should be applied only on local balls of radius
t
, their round complexities get improved from poly (log
n
) to poly (log
t
).
Finally, our deterministic solutions for these problems are based on a derandomization scheme of a novel variant of the hitting set problem, which might be of independent interest.
Funder
Israel Science Foundation
European Union’s Horizon 2020 Research and Innovation Program
The Minerva
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Reference35 articles.
1. A Hierarchy of Lower Bounds for Sublinear Additive Spanners
2. Ruben Becker, Andreas Karrenbauer, Sebastian Krinninger, and Christoph Lenzen. 2017. Near-optimal approximate shortest paths and transshipment in distributed and streaming models. In 31st International Symposium on Distributed Computing.
3. Fast Approximate Shortest Paths in the Congested Clique
4. Algebraic methods in the congested clique
5. Sparse matrix multiplication and triangle listing in the congested clique model;Censor-Hillel Keren;Theoretical Computer Science,2019