平面上彩色单纯形的深度和中线算法
Algorithms for Colourful Simplicial Depth and Medians in the Plane
摘要：
平面上一个点x相对于k个色类中n个点的配置的色单形深度，正好是顶点来自3个不同色类的闭单形（三角形）的数量，这些顶点的凸包中包含x。我们考虑的问题，有效地计算彩色单纯深度的一个点x，并找到一个点，称为中位数，最大限度地提高彩色单纯深度。对于计算x的颜色单纯深度，我们的算法一般运行时间为O（n log（n）+ k n），如果点在x周围排序，则为O（kn）。时间复杂度为O（n^4）。为了比较，这些问题的单色版本的最佳已知算法的运行时间通常是O（n log（n）），如果点在单色深度x周围排序，则改进为O（n），并且查找单色中值为O（n^4）。
英文摘要：
The colourful simplicial depth of a point x in the plane relative to a configuration of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point, called a median, that maximizes colourful simplicial depth. For computing the colourful simplicial depth of x, our algorithm runs in time O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n^4). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O(n log(n)) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n^4) for finding a monochrome median.
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