We show that finding the optimistic shortest path on an uncertain terrain is NP-hard using a reduction similar to Canny and Reif's reduction of 3SAT to 3D Euclidean shortest path. Shortest path problems are a well-studied class of problems in theoretical computer science. One particularly applicable type of shortest path problem is to find the geodesic… (More)
We present new results for three problems dealing with a set <i>P</i> of <i>n</i> convex constant-complexity fat polyhedra in 3-space. (i) We describe a data structure for vertical ray shooting in <i>P</i> that has <i>O</i>(log<sup>2</sup> <i>n</i>) query time and uses <i>O</i>(<i>n</i> log<sup>2</sup> <i>n</i>) storage. (ii) We give an algorithm to compute… (More)
In this paper we consider imprecise terrains, that is, triangulated terrains with a vertical error interval in the vertices. In particular, we study the problem of removing as many local extrema (minima and maxima) as possible from the terrain. We show that removing only minima or only maxima can be done optimally in O(n log n) time, for a terrain with n… (More)
We present a data structure for ray-shooting queries in a set of convex fat polyhedra of total complexity <i>n</i> in <i>R</i><sup>3</sup>. The data structure uses <i>O(n</i><sup>2+ε</sup>) storage and preprocessing time, and queries can be answered in <i>O</i>(log<sup>2</sup> <i>n</i>) time. A trade-off between storage and query time is also possible:… (More)
We study optimization problems in an imprecision model for polyhedral terrains. An imprecise terrain is given by a triangulated point set where the height component of the vertices is specified by an interval of possible values. We restrict ourselves to 1.5-dimensional terrains: an imprecise terrain is given by an x-monotone polyline, and the y-coordinate… (More)
We investigate the problem of creating fast evacuation plans for buildings that are modeled as grid polygons, possibly containing exponentially many cells. We study this problem in two contexts: the " confluent " context in which the routes to exits remain fixed over time, and the " non-confluent " context in which routes may change. Confluent evacuation… (More)
We propose a new model of realistic input: k-guardable objects. An object is k-guardable if its boundary can be seen by k guards in the interior of the object. In this abstract, we describe a simple algorithm for trian-gulating k-guardable polygons. Our algorithm, which is easily implementable, takes linear time assuming that k is constant.
We study the problem of cutting a set of rods (line segments in ℝ<sup>3</sup>) into fragments, using a minimum number of cuts, so that the resulting set of fragments admits a depth order. We prove that this problem is NP-complete, even when the rods have only three distinct orientations. We also give a polynomial-time approximation algorithm with no… (More)