# Parallel Batch-Dynamic Graph Connectivity

## TL;DR

This paper introduces the first parallel batch-dynamic graph connectivity algorithm that is work-efficient and faster for large batch sizes, improving upon sequential algorithms in terms of parallel efficiency and scalability.

## Contribution

It presents a novel parallel batch-dynamic connectivity algorithm with provable work efficiency and improved performance for large batch updates, filling a gap in parallel graph algorithms.

## Key findings

- Achieves $O( ext{log} n ext{log}(1 + n / ext{Δ}))$ expected work per update
- Answers batch connectivity queries in $O(k ext{log}(1 + n/k))$ expected work
- Provides $O( ext{log} n)$ depth with high probability

## Abstract

In this paper, we study batch parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in the sequential setting. The most well known sequential algorithm for dynamic connectivity is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves $O(\log^2 n)$ amortized time per edge insertion or deletion, and $O(\log n / \log\log n)$ time per query. We design a parallel batch-dynamic connectivity algorithm that is work-efficient with respect to the HDT algorithm for small batch sizes, and is asymptotically faster when the average batch size is sufficiently large. Given a sequence of batched updates, where $\Delta$ is the average batch size of all deletions, our algorithm achieves $O(\log n \log(1 + n / \Delta))$ expected amortized work per edge insertion and deletion and $O(\log^3 n)$ depth w.h.p. Our algorithm answers a batch of $k$ connectivity queries in $O(k \log(1 + n/k))$ expected work and $O(\log n)$ depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.