Fast Distributed Algorithms for Connectivity and MST in Large Graphs
release_iyna6b7jdzaq7iz3ccwaiwgusa
by
Gopal Pandurangan, Peter Robinson, Michele Scquizzato
2015
Abstract
Motivated by the increasing need to understand the algorithmic foundations of
distributed large-scale graph computations, we study a number of fundamental
graph problems in a message-passing model for distributed computing where k
≥ 2 machines jointly perform computations on graphs with n nodes
(typically, n ≫ k). The input graph is assumed to be initially randomly
partitioned among the k machines, a common implementation in many real-world
systems. Communication is point-to-point, and the goal is to minimize the
number of communication rounds of the computation.
Our main result is an (almost) optimal distributed randomized algorithm for
graph connectivity. Our algorithm runs in Õ(n/k^2) rounds
(Õ notation hides a (n) factor and an additive
(n) term). This improves over the best previously known bound of
Õ(n/k) [Klauck et al., SODA 2015], and is optimal (up to a
polylogarithmic factor) in view of an existing lower bound of
Ω̃(n/k^2). Our improved algorithm uses a bunch of techniques,
including linear graph sketching, that prove useful in the design of efficient
distributed graph algorithms. Using the connectivity algorithm as a building
block, we then present fast randomized algorithms for computing minimum
spanning trees, (approximate) min-cuts, and for many graph verification
problems. All these algorithms take Õ(n/k^2) rounds, and are optimal
up to polylogarithmic factors. We also show an almost matching lower bound of
Ω̃(n/k^2) rounds for many graph verification problems by
leveraging lower bounds in random-partition communication complexity.
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