The Flip Diameter of Rectangulations and Convex Subdivisions

We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of $n$ points. This bound is the best possible for some point sets, while $\Theta(n)$ operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of $n$ points in the plane.Comment: 17 pages, 12 figures, an extended abstract has been presented at LATIN 201

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

Coverage with k-transmitters in the presence of obstacles

For a fixed integer k ≥ 0, a k-transmitter is an omnidirectional wireless transmitter with an infinite broadcast range that is able to penetrate up to k “walls”, represented as line segments in the plane. We develop lower and upper bounds for the number of k-transmitters that are necessary and sufficient to cover a given collection of line segments, polygonal chains and polygons.Postprint (published version