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We propose three indexing schemes for storing a set S of N points in the plane, each moving along a linear trajectory, so that any query of the following form can be answered quickly: Given a rectangle R and a real value t; report all K points of S that lie inside R at time t: We first present an indexing structure that, for any given constant e > 0; uses(More)
We consider the problem of cutting a set of edges on a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running(More)
The straight skeleton of a polygon is a variant of the medial axis, introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an n-gon with T reflex vertices in time O(n'+& +ns~11+cr9~11+~), for any fixed E > 0, improving the previous best upper bound(More)
We prove an f2(n Ir " l) lower bound for the following problem: For some fixed linear equation in T variables, given a set of n real numbers, do any T of them satisfy th; ecpm-tion? Our lower bound holds in a restricted linear decision tree model, in which each decision is based on the sign of an arbitrary afline combination of T or fewer inputs. In this(More)
We describe the first algorithms to compute minimum cuts in surface-embedded graphs in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s,t)-cut in g<sup>O(g)</sup> n log n time. Except for the special case of planar graphs, for which O(n log(More)
Let <i>G</i> be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest non-separating cycle in G in O(g<sup>2</sup> n log n) time, exactly matching the fastest algorithm known for undirected graphs. We also describe an algorithm to compute the shortest non-contractible cycle in G in g<sup>O(g)</sup>n log n time,(More)
We present an algorithm to construct meshes suitable for space-time discontinuous Galerkin finite-element methods. Our method generalizes and improves the 'Tent Pitcher' algorithm of¨Ungör and Sheffer. Given an arbitrary simplicially meshed domain X of any dimension and a time interval [0, T ], our algorithm builds a simplicial mesh of the space-time domain(More)
We i n troduce a new method for nding several types of optimal k-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, which w ere based on high order Voronoi diagrams. Our technique allows us for(More)
We consider a class of geometric facility location problems in which the goal is to determine a set <i>X</i> of disks given by their centers <i>(t<sub>j</sub>)</i> and radii <i>(r<sub>j</sub>)</i> that cover a given set of demand points <i>Y&#8712;R</i><sup>2</sup> at the smallest possible cost. We consider cost functions of the form(More)