Random Sampling Applied to the MST Problem in the Node Congested Clique Model

TL;DR

This paper improves distributed algorithms for minimum spanning tree and forest problems in the Node Congested Clique model using sampling techniques and efficient protocols, reducing round complexity.

Contribution

It introduces faster algorithms for MST and SF in the NCC model, utilizing sampling and optimized communication protocols, with tighter analysis and new data structures.

Findings

MST algorithm improved to O(log^3 n) rounds

Spanning Forest algorithm achieved in O(log^2 n) rounds

Minimum Spanning Forest algorithm runs in O(log^2 n log Δ / log log n) rounds

Abstract

The Congested Clique model proposed by Lotker et al.[SICOMP'05] was introduced in order to provide a simple abstraction for overlay networks. Congested Clique is a model of distributed (or parallel) computing, in which there are $n$ players with unique identifiers from set [n], which perform computations in synchronous rounds. Each round consists of the phase of unlimited local computation and the communication phase. While communicating, each pair of players is allowed to exchange a single message of size $O(\log n)$ bits. Since, in a single round, each player can communicate with even $\Theta(n)$ other players, the model seems to be to powerful to imitate bandwidth restriction emerging from the underlying network. In this paper we study a restricted version of the Congested Clique model, the Node Congested Clique (NCC) model, proposed by Augustine et al.[arxiv1805], in which a player is allowed to send/receive only $O(\log n)$ messages per communication phase. More precisely, we provide communication primitives that improve the round complexity of the MST algorithm by Augustine et al. [arxiv1805] to $O(\log^3 n)$ rounds, and give an $O(\log^2 n)$ round algorithm solving the Spanning Forest (SF) problem. Furthermore, we present an approach based on the random sampling technique by Karger et al.[JACM'95] that gives an $O(\log^2 n \log \Delta / \log \log n)$ round algorithm for the Minimum Spanning Forest (MSF) problem. Besides the faster SF/ MSF algorithms we consider the key contributions to be - an efficient implementation of basic protocols in the NCC model - a tighter analysis of a special case of the sampling approach by Karger et al.[JACM'95] and related results by Pemmaraju and Sardeshmukh [FSTTCS'16] - efficient k-sparse recovery data structure that requires $O((k +\log n)\log n\log k)$ bits and provides recovery procedure that requires $O((k +\log n)\log k)$ steps