On the Size of Chromatic Delaunay Mosaics

Ranita Biswas, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, Morteza Saghafian

Abstract: Given a locally finite set \(A \subseteq {{\mathbb R}}^d\) and a coloring \(\chi :A \rightarrow \{0,1,\ldots ,s\}\), we introduce the chromatic Delaunay mosaic of \(\chi \), which is a Delaunay mosaic in \({{\mathbb R}}^{d+s}\) that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with \(n = {{\#}{A}}\), and the coloring is random, then the chromatic Delaunay mosaic has \(O(n^{{\lceil d/2 \rceil }})\) cells in expectation. In contrast, for Delone sets and Poisson point processes in \({{\mathbb R}}^d\), the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in \({{\mathbb R}}^2\) all colorings of a well spread set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.

External IDs:dblp:journals/dcg/BiswasMDES26