11 research outputs found
Brief Announcement: Almost-Tight Approximation Distributed Algorithm for Minimum Cut
In this short paper, we present an improved algorithm for approximating the
minimum cut on distributed (CONGEST) networks. Let be the minimum
cut. Our algorithm can compute exactly in
\tilde{O}((\sqrt{n}+D)\poly(\lambda)) time, where is the number of nodes
(processors) in the network, is the network diameter, and hides
\poly\log n. By a standard reduction, we can convert this algorithm into a
-approximation \tilde{O}((\sqrt{n}+D)/\poly(\epsilon))-time
algorithm. The latter result improves over the previous
-approximation \tilde{O}((\sqrt{n}+D)/\poly(\epsilon))-time
algorithm of Ghaffari and Kuhn [DISC 2013]. Due to the lower bound of
by Das Sarma et al. [SICOMP 2013], this running
time is {\em tight} up to a \poly\log n factor. Our algorithm is an extremely
simple combination of Thorup's tree packing theorem [Combinatorica 2007],
Kutten and Peleg's tree partitioning algorithm [J. Algorithms 1998], and
Karger's dynamic programming [JACM 2000].Comment: To appear as a brief announcement at PODC 201
A Simple Deterministic Distributed MST Algorithm, with Near-Optimal Time and Message Complexities
Distributed minimum spanning tree (MST) problem is one of the most central
and fundamental problems in distributed graph algorithms. Garay et al.
\cite{GKP98,KP98} devised an algorithm with running time , where is the hop-diameter of the input -vertex -edge
graph, and with message complexity . Peleg and Rubinovich
\cite{PR99} showed that the running time of the algorithm of \cite{KP98} is
essentially tight, and asked if one can achieve near-optimal running time
**together with near-optimal message complexity**.
In a recent breakthrough, Pandurangan et al. \cite{PRS16} answered this
question in the affirmative, and devised a **randomized** algorithm with time
and message complexity . They asked if
such a simultaneous time- and message-optimality can be achieved by a
**deterministic** algorithm.
In this paper, building upon the work of \cite{PRS16}, we answer this
question in the affirmative, and devise a **deterministic** algorithm that
computes MST in time , using messages. The polylogarithmic factors in the time
and message complexities of our algorithm are significantly smaller than the
respective factors in the result of \cite{PRS16}. Also, our algorithm and its
analysis are very **simple** and self-contained, as opposed to rather
complicated previous sublinear-time algorithms \cite{GKP98,KP98,E04b,PRS16}
A Faster Distributed Single-Source Shortest Paths Algorithm
We devise new algorithms for the single-source shortest paths (SSSP) problem
with non-negative edge weights in the CONGEST model of distributed computing.
While close-to-optimal solutions, in terms of the number of rounds spent by the
algorithm, have recently been developed for computing SSSP approximately, the
fastest known exact algorithms are still far away from matching the lower bound
of rounds by Peleg and Rubinovich [SIAM
Journal on Computing 2000], where is the number of nodes in the network
and is its diameter. The state of the art is Elkin's randomized algorithm
[STOC 2017] that performs rounds. We
significantly improve upon this upper bound with our two new randomized
algorithms for polynomially bounded integer edge weights, the first performing
rounds and the second performing rounds. Our bounds also compare favorably to the
independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a
-approximation -round algorithm for directed SSSP and a new work/depth trade-off for exact
SSSP on directed graphs in the PRAM model.Comment: Presented at the the 59th Annual IEEE Symposium on Foundations of
Computer Science (FOCS 2018
On the Distributed Complexity of Large-Scale Graph Computations
Motivated by the increasing need to understand the distributed algorithmic
foundations of large-scale graph computations, we study some fundamental graph
problems in a message-passing model for distributed computing where
machines jointly perform computations on graphs with nodes (typically, ). The input graph is assumed to be initially randomly partitioned among
the machines, a common implementation in many real-world systems.
Communication is point-to-point, and the goal is to minimize the number of
communication {\em rounds} of the computation.
Our main contribution is the {\em General Lower Bound Theorem}, a theorem
that can be used to show non-trivial lower bounds on the round complexity of
distributed large-scale data computations. The General Lower Bound Theorem is
established via an information-theoretic approach that relates the round
complexity to the minimal amount of information required by machines to solve
the problem. Our approach is generic and this theorem can be used in a
"cookbook" fashion to show distributed lower bounds in the context of several
problems, including non-graph problems. We present two applications by showing
(almost) tight lower bounds for the round complexity of two fundamental graph
problems, namely {\em PageRank computation} and {\em triangle enumeration}. Our
approach, as demonstrated in the case of PageRank, can yield tight lower bounds
for problems (including, and especially, under a stochastic partition of the
input) where communication complexity techniques are not obvious.
Our approach, as demonstrated in the case of triangle enumeration, can yield
stronger round lower bounds as well as message-round tradeoffs compared to
approaches that use communication complexity techniques
Distributed Edge Connectivity in Sublinear Time
We present the first sublinear-time algorithm for a distributed
message-passing network sto compute its edge connectivity exactly in
the CONGEST model, as long as there are no parallel edges. Our algorithm takes
time to compute and a
cut of cardinality with high probability, where and are the
number of nodes and the diameter of the network, respectively, and
hides polylogarithmic factors. This running time is sublinear in (i.e.
) whenever is. Previous sublinear-time
distributed algorithms can solve this problem either (i) exactly only when
[Thurimella PODC'95; Pritchard, Thurimella, ACM
Trans. Algorithms'11; Nanongkai, Su, DISC'14] or (ii) approximately [Ghaffari,
Kuhn, DISC'13; Nanongkai, Su, DISC'14].
To achieve this we develop and combine several new techniques. First, we
design the first distributed algorithm that can compute a -edge connectivity
certificate for any in time .
Second, we show that by combining the recent distributed expander decomposition
technique of [Chang, Pettie, Zhang, SODA'19] with techniques from the
sequential deterministic edge connectivity algorithm of [Kawarabayashi, Thorup,
STOC'15], we can decompose the network into a sublinear number of clusters with
small average diameter and without any mincut separating a cluster (except the
`trivial' ones). Finally, by extending the tree packing technique from [Karger
STOC'96], we can find the minimum cut in time proportional to the number of
components. As a byproduct of this technique, we obtain an -time
algorithm for computing exact minimum cut for weighted graphs.Comment: Accepted at 51st ACM Symposium on Theory of Computing (STOC 2019
Distributed Weighted Min-Cut in Nearly-Optimal Time
Minimum-weight cut (min-cut) is a basic measure of a network's connectivity
strength. While the min-cut can be computed efficiently in the sequential
setting [Karger STOC'96], there was no efficient way for a distributed network
to compute its own min-cut without limiting the input structure or dropping the
output quality: In the standard CONGEST model, existing algorithms with
nearly-optimal time (e.g. [Ghaffari, Kuhn, DISC'13; Nanongkai, Su, DISC'14])
can guarantee a solution that is -approximation at best while the
exact -time algorithm [Ghaffari, Nowicki,
Thorup, SODA'20] works only on *simple* networks (no weights and no parallel
edges). Here and denote the network's number of vertices and
hop-diameter, respectively. For the weighted case, the best bound was [Daga, Henzinger, Nanongkai, Saranurak, STOC'19].
In this paper, we provide an *exact* -time algorithm
for computing min-cut on *weighted* networks. Our result improves even the
previous algorithm that works only on simple networks. Its time complexity
matches the known lower bound up to polylogarithmic factors. At the heart of
our algorithm are a clever routing trick and two structural lemmas regarding
the structure of a minimum cut of a graph. These two structural lemmas
considerably strengthen and generalize the framework of Mukhopadhyay-Nanongkai
[STOC'20] and can be of independent interest.Comment: Major changes: (i) The fragment decomposition technique is
simplified, (ii) Introduction and technical overview have been redone, and
(iii) The technical sections have been made simpler for better readabilit