Otfried Cheong

Competitive facility location: the Voronoi game

Abstract. We consider a competitive facility location problem with two players.Pla yers alternate placing points, one at a time, into the playing arena, until each of them has placed n points.The arena is then subdivided according to the nearest-neighbor rule, and the player whose points control the larger area wins.W e present a winning strategy for the second player, where the arena is a circle or a line segment.

On finding a guard that sees most and a shop that sells most

We present a near-quadratic time algorithm that computes a point inside a simple polygon P in the plane having approximately the largest visibility polygon inside P, and a near-linear time algorithm for finding the point that will have approximately the largest Voronoi region when added to an n-point set in the plane. We apply the same technique to find the translation that approximately maximizes the area of intersection of two polygonal regions in near-quadratic time, and the rigid motion doing so in near-cubic time.

The One-Round Voronoi Game

Abstract In the one-round Voronoi game, the first player chooses an n-point set W in a square Q, and then the second player places another n-point set B into Q. The payoff for the second player is the fraction of the area of Q occupied by the regions of the points of B in the Voronoi diagram of W \cup B. We give a (randomized) strategy for the second player that always guarantees him a payoff of at least ½ + α, for a constant α > 0 and every large enough n. This contrasts with the one-dimensional situation, with Q=[0,1], where the first player can always win more than ½.

Voronoi diagrams on the sphere

A warning panel trailer in which the warning panel, with the plurality of discrete electrical image display means, is raised to various operative positions along a column support and lowered to an inoperative position in an elongated seat on top of a platform of the trailer. The seat extends between the front and back of the trailer so that the travelling trailer stores the warning panel with an edge facing the direction of travel and the plane of the panel facing away from such direction of travel. The warning panel is lowered and raised by a winched cable which turns around a lower pulley assembly adjacent the platform of the trailer and an upper, pivotally mounted pulley assembly which rotates with the warning panel when selectively positioning the warning panel.

The Voronoi Diagram of Curved Objects

Abstract Voronoi diagrams of curved objects can show certain phenomena that are often considered artifacts: The Voronoi diagram is not connected; there are pairs of objects whose bisector is a closed curve or even a two-dimensional object; there are Voronoi edges between different parts of the same site (so-called self-Voronoi-edges); these self-Voronoi-edges may end at seemingly arbitrary points not on a site, and, in the case of a circular site, even degenerate to a single isolated point. We give a systematic study of these phenomena, characterizing their differential-geometric and topological properties. We show how a given set of curves can be refined such that the resulting curves define a “well-behaved” Voronoi diagram. We also give a randomized incremental algorithm to compute this diagram. The expected running time of this algorithm is O(n log n).

On simplifying dot maps

Dot maps--drawings of point sets--are a well known cartographic method to visualize density functions over an area. We study the problem of simplifying a given dot map: given a set P of points in the plane, we want to compute a smaller set Q of points whose distribution approximates the distribution of the original set P.We formalize this using the concept of e-approximations, and we give efficient algorithms for computing the approximation error of a set Q of m points with respect to a set P of n points (with m ≤ n) for certain families of ranges, namely unit squares, arbitrary squares, and arbitrary rectangles.If the family R of ranges is the family of all possible unit squares, then we compute the approximation error of Q with respect to P in O(n log n) time. If R is the family of all possible rectangles, we present an O(mn log n) time algorithm. If R is the family of all possible squares, then we present a simple O(m2n + n log n) algorithm and an O(n2 √n log n) time algorithm which is more efficient in the worst case.Finally, we develop heuristics to compute good approximations, and we evaluate our heuristics experimentally.

Competitive Facility Location along a Highway

We consider a competitive facility location problem with two players.Pla yers alternate placing points, one at a time, into the playing arena, until each of them has placed n points.The arena is then subdivided according to the nearest-neighbor rule, and the player whose points control the larger area wins.W e present a winning strategy for the second player, where the arena is a circle or a line segment.

Geometric permutations of disjoint unit spheres

We show that a set of n disjoint unit spheres in Rd admits at most two distinct geometric permutations if n ≥ 9, and at most three if 3 ≤ n ≤ 8. This result improves a Helly-type theorem on line transversals for disjoint unit spheres in R3: if any subset of size at most 18 of a family of such spheres admits a line transversal, then there is a line transversal for the entire family.

Constructing optimal highways

For two points p and q in the plane, a (unbounded) line h, called a highway, and a real v > 1, we define the travel time (also known as the city distance) from p and q to be the time needed to traverse a quickest path from p to q, where the distance is measured with speed v on h and with speed 1 in the underlying metric elsewhere. Given a set S of n points in the plane and a high-way speed v, we consider the problem of finding an axis-parallel line, the highway, that minimizes the maximum travel time over all pairs of points in S. We achieve a linear-time algorithm both for the L1- and the Euclidean metric as the underlying metric. We also consider the problem of computing an optimal pair of highways, one being horizontal, one vertical.

The one-round Voronoi game

(MATH) In the one-round Voronoi game, the first player chooses an n-point set