Simple Construction of Greedy Trees and Greedy Permutations

Abstract

\begin{abstract} Greedy permutations, also known as Gonzalez Orderings or Farthest Point Traversals are a standard way to approximate \(k\)-center clustering and have many applications in sampling and approximating metric spaces. A greedy tree is an added structure on a greedy permutation that tracks the (approximate) nearest predecessor. Greedy trees have applications in proximity search as well as in topological data analysis. For metrics of doubling dimension \(d\), a \(2^{O(d)}n\log n\) time algorithm is known, but it is randomized and also, quite complicated. Its construction involves a series of intermediate structures and \(O(n \log n)\) space. In this paper, we show how to construct greedy permutations and greedy trees using a simple variation of an algorithm of Clarkson that was shown to run in \(2^{O(d)}n\log \Delta\) time, where the spread \(\spread\) is the ratio of largest to smallest pairwise distances. The improvement comes from the observation that the greedy tree can be constructed more easily than the greedy permutation. This leads to a linear time algorithm for merging two approximate greedy trees and thus, an \(2^{O(d)}n \log n\) time algorithm for computing the tree. Then, we show how to extract a \((1+\frac{1}{n})\)-approximate greedy permutation from the approximate greedy tree in the same asymptotic running time. \end{abstract}