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We study several coloring problems for geometric range-spaces. In addition to their theoretical interest, some of these problems arise in sensor networks. Given a set of points in R 2 or R 3 , we want to color them such that every region of a certain family (e.g., every disk containing at least a certain number of points) contains points of many (say, k)(More)
Consider two orthogonal closed chains on a cylinder. These chains are monotone with respect to the tangential Θ direction. We wish to rigidly move one chain so that the total area between the two is minimized. This minimization is a geometric measure of similarity between two melodies proposed by´O Maidín. The Θ direction represents time and the axial(More)
Several measures of data depth have been proposed, each attempting to maintain certain robustness properties. This paper lists the main approaches known to the computer science community. Properties and algorithms are mentioned, for computing the depth of a point and the location of the deepest point.
— Modular robots consist of many small units that attach together and can perform local motions. By combining these motions, we can achieve a reconfiguration of the global shape. The term modular comes from the idea of grouping together a fixed number of units into a module, which behaves as a larger individual component. Recently, a fair amount of research(More)
A highway H is a line in the plane on which one can travel at a greater speed than in the remaining plane. One can choose to enter and exit H at any point. The highway time distance between a pair of points is the minimum time required to move from one point to the other, with optional use of H. The highway hull H(S, H) of a point set S is the minimal set(More)
Consider two orthogonal closed chains on a cylinder. The chains are monotone with respect to the angle Θ. We wish to rigidly move one chain so that the total area between the two chains is minimized. This is a geometric measure of similarity between two repeating melodies proposed by´O Maidín. We present an O(n) time algorithm to compute this measure if Θ(More)
Given a set S of n points in R 2 , the Oja depth of a point is the sum of the areas of all triangles formed by and two elements of S. A point in R 2 with minimum depth is an Oja median. We show how an Oja median may be computed in O(n log 3 n) time. In addition, we present an algorithm for computing the Fermat-Torricelli points of n lines in O(n) time.(More)
We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any locally finite αk-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth. The question is motivated by a sensor network problem, in which a region has to be(More)
In this paper we propose a novel algorithm that, given a source robot S and a target robot T , reconfigures S into T. Both S and T are robots composed of n atoms arranged in 2 × 2 × 2 meta-modules. The reconfiguration involves a total of O(n) atom operations (expand, contract, attach, detach) and is performed in O(n) parallel steps. This improves on(More)
We consider a model of reconfigurable robot, introduced and prototyped by the robotics community. The robot consists of independently manipulable unit-square atoms that can extend/contract arms on each side and attach/detach from neighbors. The optimal worst-case number of sequential moves required to transform one connected configuration to another was(More)