[Paper Review] Almost Optimal Distributed Algorithms for Large-Scale Graph Problems.

Gopal Pandurangan, Peter Robinson+1 more|arXiv (Cornell University)|Mar 9, 2015

Complexity and Algorithms in Graphs29 references3 citations

TL;DR

This paper presents an almost optimal randomized distributed algorithm for graph connectivity in a k-machine model, achieving a runtime of $\tilde{O}(n/k^2)$ rounds using linear graph sketching and other techniques. The approach improves upon prior bounds and is optimal up to polylogarithmic factors, with extensions to minimum spanning trees, min-cuts, and graph verification problems all running in $\tilde{O}(n/k^2)$ rounds.

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 $k \geq 2$ machines jointly perform computations on graphs with $n$ nodes (typically, $n \gg k$). The graph is assumed to be randomly partitioned among the $k$ machines, a common implementation in many real-world systems. The communication is point-to-point, and the goal is to minimize the time complexity, i.e., 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 $ ilde{O}(n/k^2)$ rounds ($ ilde{O}$ notation hides a $ ext{polylog}(n)$ factor and an additive $ ext{polylog}(n)$ term). This improves over the best previously known bound of $ ilde{O}(n/k)$ [Klauck et al., SODA 2015], and is optimal (up to a polylogarithmic factor) in view of the existing lower bound of $ ilde{\Omega}(n/k^2)$. Our improved algorithm uses a bunch of techniques, including linear graph sketching, which for the first time is applied in a non-trivial way in distributed computing. We then present fast randomized algorithms for computing minimum spanning trees, (approximate) min-cuts, and for many graph verification problems. All these algorithms take $ ilde{O}(n/k^2)$ rounds, and are optimal up to polylogarithmic factors. We also show an almost matching lower bound of $ ilde{\Omega}(n/k^2)$ for many graph verification problems.

Motivation & Objective

• To address the growing need for efficient distributed algorithms in large-scale graph computations across multiple machines.
• To minimize communication rounds in a message-passing model where graphs are randomly partitioned among k machines.
• To achieve near-optimal time complexity for fundamental graph problems such as connectivity, MST, min-cut, and verification.
• To establish tight upper and lower bounds for distributed graph problems in the k-machine model.

Proposed method

• Employing linear graph sketching for the first time in a non-trivial way in distributed computing to enable efficient communication.
• Designing a randomized algorithm that achieves $\tilde{O}(n/k^2)$ rounds for graph connectivity by leveraging sketching and distributed aggregation.
• Extending the sketching-based approach to compute minimum spanning trees and approximate min-cuts within the same round complexity.
• Using randomized verification techniques to check graph properties efficiently in $\tilde{O}(n/k^2)$ rounds.
• Applying reductions and lower bound techniques to show that $\tilde{\Omega}(n/k^2)$ is a nearly matching lower bound for many verification problems.

Experimental results

Research questions

• RQ1 Can graph connectivity be solved in $\tilde{O}(n/k^2)$ rounds in a k-machine distributed model with random graph partitioning?
• RQ2 What techniques enable a significant improvement over the previous $\tilde{O}(n/k)$ bound for connectivity?
• RQ3 To what extent can linear graph sketching be used to design efficient distributed algorithms for graph problems?
• RQ4 Are the round complexities of $\tilde{O}(n/k^2)$ optimal for MST, min-cut, and graph verification up to polylogarithmic factors?

Key findings

• The proposed algorithm for graph connectivity runs in $\tilde{O}(n/k^2)$ rounds, improving upon the prior $\tilde{O}(n/k)$ bound.
• The algorithm is optimal up to polylogarithmic factors, matching the existing $\tilde{\Omega}(n/k^2)$ lower bound.
• Minimum spanning tree and approximate min-cut problems are solved in $\tilde{O}(n/k^2)$ rounds using the same sketching-based approach.
• Graph verification problems, including connectivity and cut verification, are shown to require at least $\tilde{\Omega}(n/k^2)$ rounds, establishing near-tight bounds.