Preprocessing power weighted shortest path data using a s-Well Separated Pair Decomposition

TL;DR

This paper introduces algorithms for computing s-well separated pairs and K-nearest neighbors using power weighted shortest paths, combining them into a fused approach and discussing their data dependencies and open problems.

Contribution

It presents new algorithms for s-well separated pairs and K-nearest neighbors with power weighted shortest paths, and proposes a method to combine them into a unified algorithm.

Findings

Algorithms for s-well separated pairs and K-nearest neighbors are described.

The dependencies of these algorithms on input data are analyzed.

A fused algorithm combining both methods is introduced.

Abstract

For $s$ $>$ 0, we consider an algorithm that computes all $s$-well separated pairs in certain point sets in $\mathbb{R}^{n}$, $n$ $>1$. For an integer $K$ $>1$, we also consider an algorithm that is a permutation of Dijkstra's algorithm, that computes $K$-nearest neighbors using a certain power weighted shortest path metric in $\mathbb{R}^{n}$, $n$ $>$ $1$. We describe each algorithm and their respective dependencies on the input data. We introduce a way to combine both algorithms into a fused algorithm. Several open problems are given for future research.