Sublinear Algorithms for Estimating Single-Linkage Clustering Costs

Abstract

Single-linkage clustering (SLC) is a fundamental method for hierarchical data analysis. In the distance setting, a k-clustering produced by SLC can be obtained by computing a minimum spanning tree (MST) and deleting its k-1 heaviest edges. This naturally induces a cost profile for the SLC hierarchy: for each k ∈ [n], we define cost_k to be the weight of the resulting k-component spanning forest, equivalently, the minimum total weight of any spanning forest with exactly k connected components. The corresponding SLC cost profile is (cost_1,…,cost_n), and the scalar quantity cost(G) = ∑_{k=1}^n cost_k is the area under this profile.
We study the problem of approximating these quantities in sublinear time. We assume that the input is a weighted graph G of average degree d with edge weights in {1,… ,W}, accessed through adjacency-list queries; missing edges are treated as having infinite distance. Our main result is a sampling-based algorithm that outputs a succinct sketch of the entire SLC cost profile in the distance setting. The algorithm runs in Õ(d√W/ε³) time and returns a sketch from which one can derive estimates (cost̂_1,…,cost̂_n) satisfying ∑_{k=1}^n |cost̂_k-cost_k| ≤ ε cost(G).
Thus, we obtain an 𝓁₁ approximation to the full profile whose error is at most an ε-fraction of the area under the true profile. In particular, this yields a (1±ε)-approximation to cost(G) within the same running time. We also prove a nearly matching lower bound of Ω(d√W/ε²) queries for estimating cost(G).
We further extend our results to the similarity setting, where SLC is defined via a maximum spanning tree. In this case, we obtain algorithms with running time Õ(dW/ε³) for both the profile and the total cost, together with a nearly matching lower bound of Ω(dW/ε²) queries. These bounds reveal a genuine separation between the distance and similarity settings. Finally, we extend our algorithms to metric spaces, where we obtain Õ(n/ε⁷)-query algorithms for both distance and similarity metrics, and we complement our theory with experimental validation.