Deterministic and Work-Efficient Parallel Batch-Dynamic Trees in Low Span

TL;DR

This paper introduces the first deterministic, work-efficient parallel algorithm for maintaining dynamic trees with low span, improving reliability and worst-case guarantees over previous randomized methods.

Contribution

It provides the first deterministic, work-efficient parallel algorithm for batch-dynamic trees with low span, applicable to dynamic graph algorithms.

Findings

Deterministic worst-case $O(k \, \log(1 + n/k))$ work for batch updates.

Achieves $O(\\log n \\log^{(c)} k)$ span for any constant $c$.

Improves randomized span bounds from $O(\log n \log^* n)$ to $O(\log n)$.

Abstract

Dynamic trees are a well-studied and fundamental building block of dynamic graph algorithms dating back to the seminal work of Sleator and Tarjan [STOC'81, (1981), pp. 114-122]. The problem is to maintain a tree subject to online edge insertions and deletions while answering queries about the tree, such as the heaviest weight on a path, etc. In the parallel batch-dynamic setting, the goal is to process batches of edge updates work efficiently in low ($\text{polylog}\ n$) span. Two work-efficient algorithms are known, batch-parallel Euler Tour Trees by Tseng et al. [ALENEX'19, (2019), pp. 92-106] and parallel Rake-Compress (RC) Trees by Acar et al. [ESA'20, (2020), pp. 2:1-2:23]. Both however are randomized and work efficient in expectation. Several downstream results that use these data structures (and indeed to the best of our knowledge, all known work-efficient parallel batch-dynamic graph algorithms) are therefore also randomized. In this work, we give the first deterministic work-efficient solution to the problem. Our algorithm maintains a dynamic parallel tree contraction subject to batches of $k$ edge updates deterministically in worst-case $O(k \log(1 + n/k))$ work and $O(\log n \log^{(c)} k)$ span for any constant $c$. This allows us to implement parallel batch-dynamic RC-Trees with worst-case $O(k \log(1 + n/k))$ work updates and queries deterministically. Our techniques that we use to obtain the given span bound can also be applied to the state-of-the-art randomized variant of the algorithm to improve its span from $O(\log n \log^* n)$ to $O(\log n)$.