1,611 research outputs found
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
An Improved Distributed Algorithm for Maximal Independent Set
The Maximal Independent Set (MIS) problem is one of the basics in the study
of locality in distributed graph algorithms. This paper presents an extremely
simple randomized algorithm providing a near-optimal local complexity for this
problem, which incidentally, when combined with some recent techniques, also
leads to a near-optimal global complexity.
Classical algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86]
provide the global complexity guarantee that, with high probability, all nodes
terminate after rounds. In contrast, our initial focus is on the
local complexity, and our main contribution is to provide a very simple
algorithm guaranteeing that each particular node terminates after rounds, with probability at least
. The guarantee holds even if the randomness outside -hops
neighborhood of is determined adversarially. This degree-dependency is
optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04].
Interestingly, this local complexity smoothly transitions to a global
complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider
[FOCS'12, arXiv: 1202.1983v3], we get a randomized MIS algorithm with a high
probability global complexity of ,
where denotes the maximum degree. This improves over the result of Barenboim et al., and gets close
to the lower bound of Kuhn et al.
Corollaries include improved algorithms for MIS in graphs of upper-bounded
arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local
Computation Algorithms (LCA) model, and a faster distributed algorithm for the
Lov\'asz Local Lemma
Towards a complexity theory for the congested clique
The congested clique model of distributed computing has been receiving
attention as a model for densely connected distributed systems. While there has
been significant progress on the side of upper bounds, we have very little in
terms of lower bounds for the congested clique; indeed, it is now know that
proving explicit congested clique lower bounds is as difficult as proving
circuit lower bounds.
In this work, we use various more traditional complexity-theoretic tools to
build a clearer picture of the complexity landscape of the congested clique:
-- Nondeterminism and beyond: We introduce the nondeterministic congested
clique model (analogous to NP) and show that there is a natural canonical
problem family that captures all problems solvable in constant time with
nondeterministic algorithms. We further generalise these notions by introducing
the constant-round decision hierarchy (analogous to the polynomial hierarchy).
-- Non-constructive lower bounds: We lift the prior non-uniform counting
arguments to a general technique for proving non-constructive uniform lower
bounds for the congested clique. In particular, we prove a time hierarchy
theorem for the congested clique, showing that there are decision problems of
essentially all complexities, both in the deterministic and nondeterministic
settings.
-- Fine-grained complexity: We map out relationships between various natural
problems in the congested clique model, arguing that a reduction-based
complexity theory currently gives us a fairly good picture of the complexity
landscape of the congested clique
Massively Parallel Algorithms for Distance Approximation and Spanners
Over the past decade, there has been increasing interest in
distributed/parallel algorithms for processing large-scale graphs. By now, we
have quite fast algorithms -- usually sublogarithmic-time and often
-time, or even faster -- for a number of fundamental graph
problems in the massively parallel computation (MPC) model. This model is a
widely-adopted theoretical abstraction of MapReduce style settings, where a
number of machines communicate in an all-to-all manner to process large-scale
data. Contributing to this line of work on MPC graph algorithms, we present
round MPC algorithms for computing
-spanners in the strongly sublinear regime of local memory. To
the best of our knowledge, these are the first sublogarithmic-time MPC
algorithms for spanner construction. As primary applications of our spanners,
we get two important implications, as follows:
-For the MPC setting, we get an -round algorithm for
approximation of all pairs shortest paths (APSP) in the
near-linear regime of local memory. To the best of our knowledge, this is the
first sublogarithmic-time MPC algorithm for distance approximations.
-Our result above also extends to the Congested Clique model of distributed
computing, with the same round complexity and approximation guarantee. This
gives the first sub-logarithmic algorithm for approximating APSP in weighted
graphs in the Congested Clique model
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
- …