The floodlight problem

Given three angles summing to 2, given n points in the plane and a tripartition k1 + k2 + k3 = n, we can tripartition the plane into three wedges of the given angles so that the i-th wedge contains ki of the points. This new result on dissecting point sets is used to prove that lights of specied angles not exceeding can be placed at n xed points in the plane to illuminate the entire plane if and only if the angles sum to at least 2. We give O(n log n) algorithms for both these problems

Bounded-Degree Polyhedronization of Point Sets

Abstract In 1994 Grünbaum showed that, given a point set S in R 3 , it is always possible to construct a polyhedron whose vertices are exactly S. Such a polyhedron is called a polyhedronization of S. Agarwal et al. extended this work in 2008 by showing that there always exists a polyhedronization that can be decomposed into a union of tetrahedra (tetrahedralizable). In the same work they introduced the notion of a serpentine polyhedronization for which the dual of its tetrahedralization is a chain. In this work we present a randomized algorithm running in O(n log 6 n) expected time that constructs a serpentine polyhedronization that has vertices with degree at most 7, answering an open question by Agarwal et al

Abstract Disjoint Segments have Convex Partitions with 2-Edge Connected Dual Graphs

The empty space around n disjoint line segments in the plane can be partitioned into n + 1 convex faces by extending the segments in some order. The dual graph of such a partition is the plane graph whose vertices correspond to the n+1 convex faces, and every segment endpoint corresponds to an edge between the two incident faces on opposite sides of the segment. We construct, for every set of n disjoint line segments in the plane, a convex partition whose dual graph is 2-edge connected.

Depth Explorer - A Software Tool for Analysis of Depth Measures

Data depth is an analysis method measuring how central a point is relative to a cloud of data points. Given a cloud of data points in R 2, the most central point, the median