Batch-Parallel Euler Tour Trees

Abstract

The dynamic trees problem is to maintain a forest undergoing edge insertions and deletions while supporting queries for information such as connectivity. There are many existing data structures for this problem, but few of them are capable of exploiting parallelism in the batch-setting, in which large batches of edges are inserted or deleted from the forest at once. In this paper, we demonstrate that the Euler tour tree, an existing sequential dynamic trees data structure, can be parallelized in the batch setting. For a batch of $k$ updates over a forest of $n$ vertices, our parallel Euler tour trees perform $O(k \log (1 + n/k))$ expected work with $O(\log n)$ depth with high probability. Our work bound is asymptotically optimal, and we improve on the depth bound achieved by Acar et al. for the batch-parallel dynamic trees problem. The main building block for parallelizing Euler tour trees is a batch-parallel skip list data structure, which we believe may be of independent interest. Euler tour trees require a sequence data structure capable of joins and splits. Sequentially, balanced binary trees are used, but they are difficult to join or split in parallel. We show that skip lists, on the other hand, support batches of joins or splits of size $k$ over $n$ elements with $O(k \log (1 + n/k))$ work in expectation and $O(\log n)$ depth with high probability. We also achieve the same efficiency bounds for augmented skip lists, which allows us to augment our Euler tour trees to support subtree queries. Our data structures achieve between 67--96x self-relative speedup on 72 cores with hyper-threading on large batch sizes. Our data structures also outperform the fastest existing sequential dynamic trees data structures empirically.