A Practical Approach for Computing the Diameter of a Point Set

TL;DR

This work addresses the practical computation of the diameter of a point set in $\mathbb{R}^d$ by introducing a simple, robust approximation algorithm that often yields the exact diameter quickly. The approach maintains a hierarchical fair-split decomposition and a dynamic candidate set, yielding a $(1-\\varepsilon)$-approximation in $O((n + 1/\\varepsilon^{3})\\log{1/\\varepsilon})$ time in 3D and extending to higher dimensions, with refined analyses giving improved bounds for related variants. Empirically, the method dominates many prior schemes on real-world inputs and offers flexible trade-offs between accuracy and speed, while remaining implementable within hours. The results highlight the practical ease of use and the algorithm’s adaptability to input hardness, suggesting it as a practical default for diameter computations in applied settings.

Abstract

We present an approximation algorithm for computing the diameter of a point-set in $\Re^d$. The new algorithm is sensitive to the ``hardness'' of computing the diameter of the given input, and for most inputs it is able to compute the exact diameter extremely fast. The new algorithm is simple, robust, has good empirical performance, and can be implemented quickly. As such, it seems to be the algorithm of choice in practice for computing/approximating the diameter.

Figures (3)

• Figure 3.1: Illustration of how AprxDiameter works on a densely sampled ellipse. As the algorithm progresses the pairs-decomposition become finer, and pairs that are too short are being thrown away. The dark regions are nodes that all the pairs associated with them were thrown away. As time progresses, the pairs maintained by the algorithm converge to the real diameter. It seems that for all "real" inputs, after several iterations, only a small fraction of the input points would be contained in the active nodes maintained by the algorithm.
• Figure 4.1: Illustration of the proof of \ref{['lemma:packing']}
• Figure 4.2: Illustration of the proof of \ref{['lemma:pairs:in:cluster']}. The cells that participate in a cluster $U_j$ are centered around $p_j$ and $q_j$. By \ref{['lemma:close:to:boundary']} all such nodes must lie close to the surface of the convex-hull ${\mathcal{CH}}(P)$.

Theorems & Definitions (14)

• Lemma 3.1
• Corollary 4.1
• Lemma 4.3
• Corollary 4.4
• Lemma 4.6: ck-dmpsa-95, Lemma 4.1
• Lemma 4.7
• Lemma 4.8
• Lemma 4.9
• Lemma 4.10
• Lemma 4.11