Simple Construction of Greedy Trees and Greedy Permutations

TL;DR

The paper tackles the construction of greedy permutations and their associated greedy trees in doubling metrics. It presents a simple variation of Clarkson's algorithm built around finite Voronoi diagrams and a new proximity structure, the greedy tree, to achieve deterministic $2^{O(d)}n\log n$ time and linear space. A key insight is that greedy trees can be constructed and manipulated more easily than the full greedy permutation, enabling linear-time merging of trees and a refining procedure that yields a $(1+1/n)$-approximate greedy permutation. The results offer a practical, deterministic alternative to prior randomized or highly complex approaches, with implications for proximity search, sampling, and $k$-center clustering in low-dimensional metric spaces.

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 $2^{O(d)}n\log Δ$ 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}

Figures (7)

• Figure 1: Each panel depicts a prefix of the greedy permutation and the corresponding net.
• Figure 2: A greedy permutation is computed via incremental Voronoi construction. The shaded regions cover points that are closer to the new site than its previous nearest site.
• Figure 3: On the left is depicted a geometric representation of the greedy tree, with the corresponding data structure on the right.
• Figure 4: A node $x\in V(a)$ is touched when site $b'$ is inserted. The predecessor of $b'$ is $b$. We bound the packing of the set $X_r$ by finding a lower bound for $\mathbf{d}(b',b)$.
• Figure 5: Point location of greedy tree nodes.

Theorems & Definitions (53)

• Lemma 2.1
• Lemma 4.0
• Theorem 4.0
• proof
• Lemma 5.0
• proof
• Lemma 5.0
• Lemma 5.0
• Lemma 5.0
• Lemma 5.0