论文标题
关于彩色Delaunay Mosaics的大小
On the Size of Chromatic Delaunay Mosaics
论文作者
论文摘要
给定一个局部有限的集合$ a \ subseteq \ mathbb {r}^d $和一个着色$χ\ colon a \ to \ {0,1，\ ldots，s \} $，我们介绍了$ c $ $ n $ n $ ns $ n Math Bbb的色度Delaunay mosaic of $ can不同颜色的点混合。我们的主要结果是关于色度Delaunay Mosaic的大小的界限，我们假设$ d $和$ s $是常数。例如，如果$ a $在$ n = \＃{a} $中是有限的，并且着色是随机的，则色度Delaunay Mosaic具有$ O（n^{\ lceil {d/2} \ rceil}）$单元格。相比之下，对于$ \ mathbb {r}^d $中的DeLone Sets和Poisson Point过程，封闭的球内的预期单元格仅是该球中的点数的恒定时间。此外，在$ \ mathbb {r}^2 $中，$ n $点的所有着色都具有尺寸$ o（n）$的色度delaunay马赛克。这鼓励在应用中使用色度的Delaunay Mosaics。
Given a locally finite set $A \subseteq \mathbb{R}^d$ and a coloring $χ\colon A \to \{0,1,\ldots,s\}$, we introduce the chromatic Delaunay mosaic of $χ$, which is a Delaunay mosaic in $\mathbb{R}^{s+d}$ 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 dense set of $n$ points have chromatic Delaunay mosaics of size $O(n)$. This encourages the use of chromatic Delaunay mosaics in applications.
