Splitting a configuration in a simplex

This paper presents a new method of partition, named π-splitting, of a point set in d-dimensional space. Given a point G in a d-dimensional simplex T, T(G;i) is the subsimplex spanned by G and the ith facet of T. Let S be a set of n points in T, and let π be a sequence of nonnegative integers π1, ..., nd+1 satisfying σi=1d+1π1=n The π-splitter of (T, S) is a point G in T such that T(G;i) contains at least πi points of S in its closure for every i=1, 2, ..., d + 1. The associated dissection is the re-splitting. The existence of a π-splitting is shown for any (T, S) and π, and two efficient algorithms for finding such a splitting are given. One runs in O(d2n log n + d3n) time, and the other runs in O(n) time if the dimension d can be considered as a constant. Applications of re-splitting to mesh generation, polygonal-tour generation, and a combinatorial assignment problem are given. © 1993 Springer-Verlag New York Inc.