Adaptive Massively Parallel Connectivity in Optimal Space

TL;DR

This paper introduces new algorithms for finding connected components in the Adaptive Massively Parallel Computation model, achieving faster round complexity with linear space in forests and general graphs, improving upon previous methods.

Contribution

The paper presents the first sub-logarithmic round algorithms for connected components in AMPC with linear space, and provides space-efficient solutions for constant-round scenarios.

Findings

Connected components can be found in $O(\log^* n)$ rounds in forests.

Expected $2^{O(\log^* n)}$ rounds in general graphs.

Improved space complexity for constant-round algorithms.

Abstract

We study the problem of finding connected components in the Adaptive Massively Parallel Computation (AMPC) model. We show that when we require the total space to be linear in the size of the input graph the problem can be solved in $O(\log^* n)$ rounds in forests (with high probability) and $2^{O(\log^* n)}$ expected rounds in general graphs. This improves upon an existing $O(\log \log_{m/n} n)$ round algorithm. For the case when the desired number of rounds is constant we show that both problems can be solved using $\Theta(m + n \log^{(k)} n)$ total space in expectation (in each round), where $k$ is an arbitrarily large constant and $\log^{(k)}$ is the $k$-th iterate of the $\log_2$ function. This improves upon existing algorithms requiring $\Omega(m + n \log n)$ total space.