AI 中文总结
本文提出MOJOS框架，利用Treaps构建批量动态序列，实现首个理论上高效的批量并行Link-Cut树，支持路径查询且深度为O(log n)，性能优于现有并行批量动态树。
AI 中文摘要
并行批量动态树是近期动态图算法理论和实践进展中的基本构建块。然而，所有现有的并行批量动态树数据结构，包括Euler Tour树、UFO树、拓扑树和耙压缩树，在顺序设置中均显著逊色于Link-Cut树，后者作为顺序最先进技术已有40多年。尽管Link-Cut树在顺序设置中表现出色，设计高效的批量并行Link-Cut树仍是一个重大开放问题。在本文中，我们通过引入MOJOS（一个统一框架，用于理论和实践高效的并行批量动态树）来弥合这一差距。我们利用Euler Tour树和Link-Cut树都依赖于支持拆分和合并的公共动态序列抽象这一事实。我们引入了一种基于Treaps构建的新批量动态序列，该序列实现了最优的工作量和深度，并在批量更新、查询和内存使用方面优于现有的并行跳表和Treap实现。借助MOJOS，我们开发了一种新的批量并行Euler Tour树算法，该算法优于先前支持子树查询的批量动态树实现。与先前依赖跳表表示循环序列的批量并行Euler Tour树不同，MOJOS允许任何批量动态序列数据结构作为即插即用的替代品。最后，我们开发了第一个理论上高效的批量并行Link-Cut树，这也是第一个支持路径查询的批量动态数据结构，在二进制分叉模型中实现批量更新的$O(\log n)$深度。我们的Link-Cut树实现优于所有已知的支持路径查询的并行批量动态树数据结构。
英文摘要
Parallel batch-dynamic trees are a fundamental building block in recent theoretical and practical advances in dynamic graph algorithms. However, all existing parallel batch-dynamic tree data structures, including Euler tour trees, UFO trees, topology trees, and rake-compress trees, are all significantly outperformed in the sequential setting by link-cut trees, which have been the sequential state-of-the-art for over 40 years. Despite their excellent performance in the sequential setting, designing efficient batch-parallel link-cut trees has remained a major open problem. In this paper, we close this gap by introducing MOJOS, a unified framework for theoretically- and practically-efficient parallel batch-dynamic trees. We exploit the fact that both Euler tour trees and link-cut trees rely on a common dynamic sequence abstraction that supports splitting and joining. We introduce a new batch-dynamic sequence built using treaps that achieves optimal work and depth, and outperforms existing parallel skip list and treap implementations for batch updates, queries, and memory usage. With MOJOS, we develop a new batch-parallel Euler tour tree algorithm that outperforms prior batch-dynamic tree implementations supporting subtree queries. Unlike prior batch-parallel Euler tour trees which rely on skip list's ability to represent cyclic sequences, MOJOS allows any batch-dynamic sequence data structure to be used as a drop-in replacement. Finally, we develop the first theoretically-efficient batch-parallel link-cut tree, which is also the first batch-dynamic data structure supporting path queries to achieve $O(\log n)$ depth for batch updates in the binary-forking model. Our link-cut tree implementation outperforms all known parallel batch-dynamic tree data structures supporting path queries.
