Approximate Greedy Clustering and Distance Selection for Graph Metrics

TL;DR

This paper introduces efficient algorithms for approximate greedy permutations and distance selection in finite metric spaces, focusing on graph and Euclidean metrics, with improved time complexities and approximation guarantees.

Contribution

It provides novel randomized and deterministic algorithms for approximate greedy permutations and distance selection, with specific improvements for graph and Euclidean metrics.

Findings

Expected time $O(rac{1}{ ext{eps}})(m+n) ext{log} n ext{log}(n/ ext{eps})$ for approximate greedy permutations in graphs.

Expected time $O( ext{eps}^{-2} n^{1+1/(1+ ext{eps})^2 + o(1)})$ for Euclidean spaces.

Near linear time approximation for distance selection in planar graph metrics.

Abstract

$\newcommand{\eps}{\varepsilon}$ In this paper, we consider two important problems defined on finite metric spaces, and provide efficient new algorithms and approximation schemes for these problems on inputs given as graph shortest path metrics or high-dimensional Euclidean metrics. The first of these problems is the greedy permutation (or farthest-first traversal) of a finite metric space: a permutation of the points of the space in which each point is as far as possible from all previous points. We describe randomized algorithms to find $(1+\eps)$-approximate greedy permutations of any graph with $n$ vertices and $m$ edges in expected time $O(\eps^{-1}(m+n)\log n\log(n/\eps))$, and to find $(1+\eps)$-approximate greedy permutations of points in high-dimensional Euclidean spaces in expected time $O(\eps^{-2} n^{1+1/(1+\eps)^2 + o(1)})$. Additionally we describe a deterministic algorithm to find exact greedy permutations of any graph with $n$ vertices and treewidth $O(1)$ in worst-case time $O(n^{3/2}\log^{O(1)} n)$. The second of the two problems we consider is distance selection: given $k \in [ \binom{n}{2} ]$, we are interested in computing the $k$th smallest distance in the given metric space. We show that for planar graph metrics one can approximate this distance, up to a constant factor, in near linear time.