# Simple Concurrent Labeling Algorithms for Connected Components

## TL;DR

This paper introduces simple, efficient concurrent algorithms for computing connected components in graphs, achieving optimal or near-optimal bounds in parallel models, and clarifies previous analysis errors.

## Contribution

It presents new simple algorithms for connected components with tight bounds, improving on existing methods and correcting prior analysis errors.

## Key findings

- Algorithms run in Θ(log n) steps with Θ(m log n) work
- Alternative algorithms run in O(log^2 n) steps with O(m log^2 n) work
- Previous algorithms had gaps and errors in their analysis

## Abstract

We study a class of simple algorithms for concurrently computing the connected components of an $n$-vertex, $m$-edge graph. Our algorithms are easy to implement in either the COMBINING CRCW PRAM or the MPC computing model. For two related algorithms in this class, we obtain $\Theta(\lg n)$ step and $\Theta(m \lg n)$ work bounds. For two others, we obtain $O(\lg^2 n)$ step and $O(m \lg^2 n)$ work bounds, which are tight for one of them. All our algorithms are simpler than related algorithms in the literature. We also point out some gaps and errors in the analysis of previous algorithms. Our results show that even a basic problem like connected components still has secrets to reveal.