Parallel Batch-Dynamic Graph Connectivity
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With the rapid growth of graph datasets over the past decade, a new kind of dynamic algorithm, supporting the ability to ingest batches of updates and exploit parallelism is needed in order to efficiently process large streams of updates. In this paper, we study batch and parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in sequential setting. Perhaps the best known sequential algorithm is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves O(^2 n) amortized time per edge insertion or deletion, and O( n) time per query. In this paper, 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( n (1 + n / Δ)) expected amortized work per edge insertion and deletion and O(^3 n) depth w.h.p. Our algorithm answers a batch of k connectivity queries in O(k (1 + n/k)) expected work and O( n) depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.
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