The dynamic trees problem is to maintain a tree under edge updates while supporting queries like connectivity queries or path queries. Despite the first data structure for this fundamental problem---the link-cut tree---being invented 40 years ago, our experiments reveal that they are still the fastest sequential data structure for the problem. However, link-cut trees cannot support parallel batch-dynamic updates and have limitations on the kinds of queries they support.

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.

Single-linkage clustering is a popular form of hierarchical agglomerative clustering (HAC) where the distance between two clusters is defined as the minimum distance between any pair of points across the two clusters. In single-linkage HAC, the output is typically the single-linkage dendrogram (SLD), which is the binary tree representing the hierarchy of clusters formed by iteratively contracting the two closest clusters. In the dynamic setting, prior work has only studied maintaining a minimum spanning forest over the data since single-linkage HAC reduces to computing the SLD on the minimum spanning forest of the data.

We study the problem of dynamically maintaining the connected components of an undirected graph subject to edge insertions and deletions. We give the first parallel algorithm for the problem that is work-efficient, supports batches of updates, runs in polylogarithmic depth, and uses only linear total space. The existing algorithms for the problem either use super-linear space, do not come with strong theoretical bounds, or are not parallel. On the empirical side, we provide the first implementation of the cluster forest algorithm , the first linear-space and polylogarithmic update time algorithm for dynamic connectivity. Experimentally, we find that our algorithm uses up to 19.7× less space and is up to 6.2× faster than the level-set algorithm of Holm, de Lichten-berg, and Thorup, arguably the most widely-implemented dynamic connectivity algorithm with strong theoretical guarantees.