Parallel Batch-Dynamic Maximal Independent Set

TL;DR

This paper presents the first efficient parallel batch-dynamic algorithm for maintaining a maximal independent set in graphs, outperforming previous sequential algorithms in work and parallel time.

Contribution

It introduces a novel parallel batch-dynamic MIS algorithm with polylogarithmic depth and work efficiency, along with a new analysis of batch influence sets.

Findings

Expected work is O(b log^3 n) for batch size b

Algorithm maintains a lexicographically first MIS

Depth is polylogarithmic, matching static MIS bounds

Abstract

We develop the first theoretically-efficient algorithm for maintaining the maximal independent set (MIS) of a graph in the parallel batch-dynamic setting. In this setting, a graph is updated with batches of edge insertions/deletions, and for each batch a parallel algorithm updates the maximal independent set to agree with the new graph. A batch-dynamic algorithm is considered efficient if it is work efficient (i.e., does no more asymptotic work than applying the updates sequentially) and has polylogarithmic depth (parallel time). In the sequential setting, the best known dynamic algorithms for MIS, by Chechik and Zhang (CZ) [FOCS19] and Behnezhad et al. (BDHSS) [FOCS19], take $O(\log^4 n)$ time per update in expectation. For a batch of $b$ updates, our algorithm has $O(b \log^3 n)$ expected work and polylogarithmic depth with high probability (whp). It therefore outperforms the best algorithm even in the sequential dynamic case ($b = 1)$. As with the sequential dynamic MIS algorithms of CZ and BDHSS, our solution maintains a lexicographically first MIS based on a random ordering of the vertices. Their analysis relied on a result of Censor-Hillel, Haramaty and Karnin [PODC16] that bounded the ``influence set" for a single update, but surprisingly, the influence of a batch is not simply the union of the influence of each update therein. We therefore develop a new approach to analyze the influence set for a batch of updates. Our construction of the batch influence set is natural and leads to an arguably simpler analysis than prior work. We then instrument this construction to bound the work of our algorithm. To argue our depth is polylogarithmic, we prove that the number of subrounds our algorithm takes is the same as depth bounds on parallel static MIS.