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.

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.

Average linkage Hierarchical Agglomerative Clustering (HAC) is an extensively studied and applied method for hierarchical clustering. Recent applications to massive datasets have driven significant interest in near-linear-time and efficient parallel algorithms for average linkage HAC. We provide hardness results that rule out such algorithms. On the sequential side, we establish a runtime lower bound of $n^{3/2-ε}$ on $n$ node graphs for sequential combinatorial algorithms under standard fine-grained complexity assumptions. This essentially matches the best-known running time for average linkage HAC. On the parallel side, we prove that average linkage HAC likely cannot be parallelized even on simple graphs by showing that it is CC-hard on trees of diameter $4$. On the possibility side, we demonstrate that average linkage HAC can be efficiently parallelized (i.e., it is in NC) on paths and can be solved in near-linear time when the height of the output cluster hierarchy is small.

We give an algorithm for Centroid-Linkage Hierarchical Agglomerative Clustering (HAC), which computes a $c$-approximate clustering in roughly $n^{1+O(1/c^2)}$ time. We obtain our result by combining a new centroid-linkage HAC algorithm with a novel fully dynamic data structure for nearest neighbor search which works under adaptive updates. We also evaluate our algorithm empirically. By leveraging a state-of-the-art nearest-neighbor search library, we obtain a fast and accurate centroid-linkage HAC algorithm. Compared to an existing state-of-the-art exact baseline, our implementation maintains the clustering quality while delivering up to a $36\times$ speedup due to performing fewer distance comparisons.

Computing a Single-Linkage Dendrogram (SLD) is a key step in the classic single-linkage hierarchical clustering algorithm. Given an input edge-weighted tree T, the SLD of T is a binary dendrogram that summarizes the n-1 clusterings obtained by contracting the edges of T in order of weight. Existing algorithms for computing the SLD all require Ømega(nłog n) work where n = |T|. Furthermore, to the best of our knowledge no prior work provides a parallel algorithm obtaining non-trivial speedup for this problem. In this paper, we design faster parallel algorithms for computing SLDs both in theory and in practice based on new structural results about SLDs. In particular, we obtain a deterministic output-sensitive parallel algorithm based on parallel tree contraction that requires O(n łog h) work and O(łog^2 n łog^2 h) depth, where h is the height of the output SLD. We also give a deterministic bottom-up algorithm for the problem inspired by the nearest-neighbor chain algorithm for hierarchical agglomerative clustering, and show that it achieves O(nłog h) work and O(h łog n) depth. Our results are based on a novel divide-and-conquer framework for building SLDs, inspired by divide-and-conquer algorithms for Cartesian trees. Our new algorithms can quickly compute the SLD on billion-scale trees, and obtain up to 150x speedup over the highly-efficient Union-Find algorithm typically used to compute SLDs in practice.

The current best approximation algorithms for k-median rely on first obtaining a structured fractional solution known as a bi-point solution, and then rounding it to an integer solution. We improve this second step by unifying and refining previous approaches. We describe a hierarchy of increasingly-complex partitioning schemes for the facilities, along with corresponding sets of algorithms and factor-revealing non-linear programs. We prove that the third layer of this hierarchy is a 2.613-approximation, improving upon the current best ratio of 2.675, while no layer can be proved better than 2.588 under the proposed analysis. On the negative side, we give a family of bi-point solutions which cannot be approximated better than the square root of the golden ratio, even if allowed to open k + o(k) facilities. This gives a barrier to current approaches for obtaining an approximation better than . Altogether we reduce the approximation gap of bi-point solutions by two thirds.

The Feedback Vertex Set problem is undoubtedly one of the most well-studied problems in Parameterized Complexity. In this problem, given an undirected graph $G$ and a non-negative integer $k$, the objective is to test whether there exists a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is a forest. After a long line of improvement, recently, Li and Nederlof [SODA, 2020] designed a randomized algorithm for the problem running in time $\mathcal{O}^{\star}(2.7^k)$. In the Parameterized Complexity literature, several problems around Feedback Vertex Set have been studied. Some of these include Independent Feedback Vertex Set (where the set $S$ should be an independent set in $G$), Almost Forest Deletion and Pseudoforest Deletion. In Pseudoforest Deletion, each connected component in $G-S$ has at most one cycle in it. However, in Almost Forest Deletion, the input is a graph $G$ and non-negative integers $k,\ell \in \mathbb{N}$, and the objective is to test whether there exists a vertex subset $S$ of size at most $k$, such that $G-S$ is $\ell$ edges away from a forest. In this paper, using the methodology of Li and Nederlof [SODA, 2020], we obtain the current fastest algorithms for all these problems. In particular we obtain following randomized algorithms. 1) Independent Feedback Vertex Set can be solved in time $\mathcal{O}^{\star}(2.7^k)$. 2) Pseudo Forest Deletion can be solved in time $\mathcal{O}^{\star}(2.85^k)$. 3) Almost Forest Deletion can be solved in $\mathcal{O}^{\star}(\min\{2.85^k \cdot 8.54^\ell,2.7^k \cdot 36.61^\ell,3^k \cdot 1.78^\ell\})$.