[Paper Review] Super-fast MST Algorithms in the Congested Clique using $o(m)$ Messages

Sriram V. Pemmaraju, Vivek B. Sardeshmukh|arXiv (Cornell University)|Oct 12, 2016

Complexity and Algorithms in Graphs7 references4 citations

TL;DR

This paper presents the first super-fast minimum spanning tree (MST) algorithm in the Congested Clique model with sub-quadratic message complexity, achieving $O( ext{poly}( ext{log}^* n))$ rounds and $ ilde{O}( ext{poly}(n))$ messages. It leverages component-wise edge sampling and $ ilde{O}(n)$-message linear graph sketches to reduce communication while maintaining near-optimal round complexity.

ABSTRACT

In a sequence of recent results (PODC 2015 and PODC 2016), the running time of the fastest algorithm for the \emph{minimum spanning tree (MST)} problem in the \emph{Congested Clique} model was first improved to $O(\log \log \log n)$ from $O(\log \log n)$ (Hegeman et al., PODC 2015) and then to $O(\log^* n)$ (Ghaffari and Parter, PODC 2016). All of these algorithms use $Θ(n^2)$ messages independent of the number of edges in the input graph. This paper positively answers a question raised in Hegeman et al., and presents the first "super-fast" MST algorithm with $o(m)$ message complexity for input graphs with $m$ edges. Specifically, we present an algorithm running in $O(\log^* n)$ rounds, with message complexity $ ilde{O}(\sqrt{m \cdot n})$ and then build on this algorithm to derive a family of algorithms, containing for any $\varepsilon$, $0 < \varepsilon \le 1$, an algorithm running in $O(\log^* n/\varepsilon)$ rounds, using $ ilde{O}(n^{1 + \varepsilon}/\varepsilon)$ messages. Setting $\varepsilon = \log\log n/\log n$ leads to the first sub-logarithmic round Congested Clique MST algorithm that uses only $ ilde{O}(n)$ messages. Our primary tools in achieving these results are (i) a component-wise bound on the number of candidates for MST edges, extending the sampling lemma of Karger, Klein, and Tarjan (Karger, Klein, and Tarjan, JACM 1995) and (ii) $Θ(\log n)$-wise-independent linear graph sketches (Cormode and Firmani, Dist.~Par.~Databases, 2014) for generating MST candidate edges.

Motivation & Objective

• To address the open question of whether super-fast MST algorithms in the Congested Clique model can achieve sub-quadratic message complexity.
• To design an MST algorithm that maintains $O(\log^* n)$ round complexity while reducing message count below $\Theta(n^2)$.
• To demonstrate that low message complexity is compatible with super-fast distributed computation in the Congested Clique model.
• To provide a family of algorithms parameterized by $\varepsilon$ that trade off round complexity and message count.

Proposed method

• Extends the sampling lemma of Karger, Klein, and Tarjan to bound the number of candidate MST edges per connected component.
• Uses $\Theta(\log n)$-wise independent linear graph sketches to generate MST candidate edges efficiently.
• Employs a low-message routing scheme (RSG) to redistribute keys and messages among $\tilde{O}(\sqrt{m})$ nodes.
• Reduces the key sorting problem to a smaller clique of size $\tilde{O}(\sqrt{k})$ using redistribution and Lenzen’s $O(1)$-round sorting on the subset.
• Applies a distributed sorting subroutine that uses $O(k)$ messages for $k = O(n^{2-\varepsilon})$ keys, enabling efficient global ranking of edge weights.
• Combines sketching and routing primitives to build a family of algorithms with tunable trade-offs between rounds and messages.

Experimental results

Research questions

• RQ1 Can a super-fast MST algorithm in the Congested Clique model achieve $o(m)$ message complexity?
• RQ2 Is it possible to maintain $O(\log^* n)$ round complexity while reducing message count below $\Theta(n^2)$?
• RQ3 Can component-wise sampling and linear sketches be used to generate MST candidates with low communication cost?
• RQ4 What is the trade-off between round complexity and message complexity in low-message MST algorithms?
• RQ5 Can a sub-logarithmic round MST algorithm be built with only $\tilde{O}(n)$ messages?

Key findings

• The paper presents an $O(\log^* n)$-round MST algorithm with $\tilde{O}(\sqrt{m \cdot n})$ message complexity, which is $o(m)$ for super-linear graphs ($m = \omega(n \cdot \text{poly}(\log n))$).
• A family of algorithms is constructed with $O(\log^* n / \varepsilon)$ rounds and $\tilde{O}(n^{1+\varepsilon}/\varepsilon)$ messages for any $\varepsilon \in (0,1]$.
• Setting $\varepsilon = \log \log n / \log n$ yields a sub-logarithmic round algorithm using only $\tilde{O}(n)$ messages.
• The algorithm achieves $O(\log^* n)$ rounds with $\tilde{O}(m)$ messages, improving on prior $\Theta(n^2)$ message complexity.
• The distributed sorting subroutine reduces message complexity to $O(k)$ for $k = O(n^{2-\varepsilon})$ keys, enabling efficient global edge ranking.
• The results resolve an open question from Hegeman et al. by showing that super-fast MST algorithms can exist with $\tilde{O}(n)$ messages.