A Simple and Efficient Method to Compute a Single Linkage Dendrogram

TL;DR

This paper presents a simple, efficient recursive method to compute a single linkage dendrogram using Prim's algorithm for MST, avoiding the need to store the full distance matrix.

Contribution

It introduces a recursive partitioning algorithm that efficiently computes the dendrogram directly from Prim's MST without extra computational cost.

Findings

Uses Prim's algorithm to compute MST efficiently

Avoids storing the full distance matrix

Provides a straightforward recursive splitting method

Abstract

We address the problem of computing a single linkage dendrogram. A possible approach is to: (i) Form an edge weighted graph $G$ over the data, with edge weights reflecting dissimilarities. (ii) Calculate the MST $T$ of $G$. (iii) Break the longest edge of $T$ thereby splitting it into subtrees $T_L$, $T_R$. (iv) Apply the splitting process recursively to the subtrees. This approach has the attractive feature that Prim's algorithm for MST construction calculates distances as needed, and hence there is no need to ever store the inter-point distance matrix. The recursive partitioning algorithm requires us to determine the vertices (and edges) of $T_L$ and $T_R$. We show how this can be done easily and efficiently using information generated by Prim's algorithm without any additional computational cost.