For 2D or 3D meshes that represent a continuous function to the reals, the contours-or isosurfaces-of a specified value are an important way to visualize it. To find such contours, a seed set can be used for the starting points from which the traversal of the contours can start. This paper gives the first methods to obtain seed sets that are provably small in size. They are based on a variant of the contour tree (or topographic change tree). We give a new, simple algorithm to compute such a tree in regular and irregular meshes that requires O(rt log n) time in 2D for meshes with n elements, and in 0( n2) time in higher dimensions. The additional storage overhead is proportiid to the maximum size of any contour (linear in the worst case, but typically less). Given the contour tree, a minimum size seed set can be computed in polynomial time and storage. Since in practice at most linear storage is allowed, we develop a simple approximation aIgorithm giving a seed set of size at most twice the size of the minimum. It requires O(n log2 n) time in 2D and 0(n2) time otherwise, and requires linear storage. We also give experimental results, showing the size of the seed sets and supporting the claim that sublinear storage is used.

We consider a competitive facility location problem with two players. Players 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. We present a winning strategy for the second player, where the arena is a circle or a line segment. We permit variations where players can play more than one point at a time, and show that the ÿrst player can ensure that the second player wins by an arbitrarily small margin.

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In this paper we show that in sorting-based applications of parametric search, Quicksort can replace the parallel sorting algorithms that are usually advocated. Because of the simplicity of Quicksort, this may lead to applications of parametric search that are not only efficient in theory, but in practice as well. Also, we argue that Cole's optimization of certain parametric-search algorithms may be unnecessary under realistic assumptions about the input. Furthermore, we present a generic, flexible, and easy-to-use framework that greatly simplifies the implementation of algorithms based on parametric search. We use our framework to implement an algorithm that solves the Fréchet-distance problem. The implementation based on parametric search is faster than the binary-search approach that is often suggested as a practical replacement for the parametric-search technique.

Let P be a simple polygon. We define a witness set W to be a set of points such that if any (prospective) guard set G guards W , then it is guaranteed that G guards P. Not all polygons admit a finite witness set. If a finite minimal witness set exists, then it cannot contain any witness in the interior of P ; all witnesses must lie on the boundary of P , and there can be at most one witness in the interior of every edge. We give an algorithm to compute a minimum witness set for P in O(n 2 log n) time, if such a set exists, or to report the non-existence within the same time bounds.