Parallel Batch-Dynamic Graph Connectivity

Umut A. Acar, Daniel Anderson, Guy E. Blelloch, Laxman Dhulipala

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 best sequential algorithm for dynamic connectivity is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves O(łog2 n) amortized time per edge insertion or deletion, and O(łog 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 Δ is the average batch size of all deletions, our algorithm achieves O(łog n łog(1+n / Δ)) expected amortized work per edge insertion and deletion and O(łog3 n) depth w.h.p. Our algorithm answers a batch of k connectivity queries in O(k łog(1 + n/k)) expected work and O(łog n) depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.