Self-Improving Voronoi Construction for a Hidden Mixture of Product Distributions

TL;DR

This paper introduces a self-improving algorithm for Voronoi diagram construction under convex distance functions, efficiently adapting to hidden mixture distributions with limited prior knowledge, and achieving near-optimal expected running times.

Contribution

It presents a novel self-improving approach for Voronoi diagrams that learns from input distributions and improves efficiency over time, especially for mixtures of product distributions.

Findings

Achieves expected running time close to entropy of the output distribution.

Improves to linear expected time for Euclidean metrics.

Handles hidden mixture distributions with limited prior information.

Abstract

We propose a self-improving algorithm for computing Voronoi diagrams under a given convex distance function with constant description complexity. The $n$ input points are drawn from a hidden mixture of product distributions; we are only given an upper bound $m = o(\sqrt{n})$ on the number of distributions in the mixture, and the property that for each distribution, an input instance is drawn from it with a probability of $\Omega(1/n)$. For any $\varepsilon \in (0,1)$, after spending $O\bigl(mn\log^{O(1)} (mn) + m^{\varepsilon} n^{1+\varepsilon}\log(mn)\bigr)$ time in a training phase, our algorithm achieves an $O\bigl(\frac{1}{\varepsilon}n\log m + \frac{1}{\varepsilon}n2^{O(\log^* n)} + \frac{1}{\varepsilon}H\bigr)$ expected running time with probability at least $1 - O(1/n)$, where $H$ is the entropy of the distribution of the Voronoi diagram output. The expectation is taken over the input distribution and the randomized decisions of the algorithm. For the Euclidean metric, the expected running time improves to $O\bigl(\frac{1}{\varepsilon}n\log m + \frac{1}{\varepsilon}H\bigr)$.