[Paper Review] MST in O(1) Rounds of the Congested Clique

Tomasz Jurdziński, Krzysztof Nowicki|arXiv (Cornell University)|Jul 26, 2017

Complexity and Algorithms in Graphs14 references3 citations

TL;DR

This paper presents a randomized distributed algorithm that computes the Minimum Spanning Tree (MST) in O(1) rounds in the congested clique model, with high probability. It achieves this by introducing a novel sparsification technique that combines connected components of sparse subgraphs, reducing the problem to two restricted instances: one with low-degree graphs and another with a known partition into O(n/log log n) components, both solvable in O(1) rounds using existing and new routing and sampling techniques.

ABSTRACT

We present a distributed randomized algorithm finding Minimum Spanning Tree (MST) of a given graph in O(1) rounds, with high probability, in the Congested Clique model. The input graph in the Congested Clique model is a graph of n nodes, where each node initially knows only its incident edges. The communication graph is a clique with limited edge bandwidth: each two nodes (not necessarily neighbours in the input graph) can exchange $O(\log n)$ bits. As in previous works, the key part of the MST algorithm is an efficient Connected Components (CC) algorithm. However, unlike the former approaches, we do not aim at simulating the standard Boruvka algorithm, at least at initial stages of the CC algorithm. Instead, we develop a new technique which combines connected components of sample sparse subgraphs of the input graph in order to accelerate the process of uncovering connected components of the original input graph. More specifically, we develop a sparsification technique which reduces an initial CC problem in $O(1)$ rounds to its two restricted instances. The former instance has a graph with maximal degree $O(\log \log n)$ as the input -- here our sample-combining technique helps. In the latter instance, a partition of the input graph into $O(n/\log \log n)$ connected components is known. This gives an opportunity to apply previous algorithms to determine connected components in $O(1)$ rounds. Our result addresses the problem from and the $O(\log \log n)$ algorithm of Lotker et al. [SPAA 2003; SICOMP 2005], improves over previous $O(\log* n)$ algorithm of Ghaffari et al. [PODC 2016] and $O(\log \log \log n)$ algorithm of Hegeman et al. [PODC 2015] . It also determines $Θ(1)$ round complexity in the congested clique for MST, as well as other graph problems, including bipartiteness, cut verification, s-t connectivity and cycle containment.

Motivation & Objective

• To resolve the open problem of achieving O(1) round complexity for MST in the congested clique model.
• To improve upon prior randomized bounds of O(log* n) and O(log log log n) rounds for MST.
• To establish O(1) round complexity for related graph problems such as connectivity, bipartiteness, and cycle containment.
• To develop a new technique for accelerating connected components computation via sparse subgraph sampling and merging.
• To demonstrate feasibility of simulating multiple algorithm instances in parallel using efficient routing and proxy-based computation.

Proposed method

• Introduce a sparsification technique that reduces the original MST problem to two restricted subproblems: low-degree graphs and partitioned graphs.
• Use random sampling to construct sparse subgraphs with maximum degree O(log log n), enabling efficient component merging.
• Simultaneously execute a modified Boruvka-style algorithm on √n such sparse subgraphs using proxy nodes to manage distributed computation.
• Leverage Lenzen’s routing lemma to deliver O(n) messages per node in O(1) rounds, enabling efficient inter-node communication across multiple parallel instances.
• Use proxy nodes to maintain and propagate spanning forest information across multiple subgraph instances, then route parent pointers back to original nodes in O(1) rounds.
• Apply a reduction from MST to connected components (CC), and solve CC in O(1) rounds by combining results from the two restricted instances.

Experimental results

Research questions

• RQ1 Can the MST problem be solved in O(1) rounds in the congested clique model using randomized algorithms?
• RQ2 Is it possible to accelerate connected components computation by combining components from sparse subgraphs rather than simulating standard Boruvka steps?
• RQ3 What is the round complexity of MST and related graph problems in the congested clique model when limited to O(1) rounds?
• RQ4 How can multiple instances of a CC algorithm be executed in parallel without increasing round complexity?
• RQ5 Can proxy-based computation and efficient routing be used to simulate complex distributed algorithms in O(1) rounds?

Key findings

• The paper establishes O(1) round complexity for MST in the congested clique model using a randomized algorithm, with high probability.
• The algorithm reduces the MST problem to two restricted instances: one with graphs of maximum degree O(log log n), and another with a known partition into O(n / log log n) components.
• The connected components of low-degree graphs are computed in O(1) rounds using a novel sampling and merging technique that avoids full Boruvka simulation.
• The algorithm achieves O(1) round complexity for several verification problems, including bipartiteness, cut verification, s-t connectivity, and cycle containment, via reductions to connectivity.
• The solution leverages Lenzen’s routing lemma to deliver O(n) messages per node in O(1) rounds, enabling efficient parallel execution of √n algorithm instances.
• The approach enables simulation of multiple algorithm instances in parallel using proxy nodes, with parent information from spanning forests routed back to original nodes in O(1) rounds.