A Surprisingly Simple Method for Distributed Euclidean-Minimum Spanning Tree / Single Linkage Dendrogram Construction from High Dimensional Embeddings via Distance Decomposition

We introduce a decomposition method for the distributed calculation of exact Euclidean Minimum Spanning Trees in high dimensions (where sub-quadratic algorithms are not effective), or more generalized geometric-minimum spanning trees of complete graphs, where for each vertex v∈ V in the graph $G=(V,E)$ is represented by a vector in v⃗∈ Rⁿ, and each for any edge, the the weight of the edge in the graph is given by a symmetric binary `distance' function between the representative vectors w(,y\) = d(x⃗,y⃗). This is motivated by the task of clustering high dimensional embeddings produced by neural networks, where low-dimensional algorithms are ineffective; such geometric-minimum spanning trees find applications as a subroutine in the construction of single linkage dendrograms, as the two structures can be converted between each other efficiently.