Jeff Erickson

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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)
We propose three indexing schemes for storing a set <italic>S</italic> of <italic>N</italic> points in the plane, each moving along a linear trajectory, so that a query of the following form can be answered quickly: Given a rectangle <italic>R</italic> and a real value <italic>t<subscrpt>q</subscrpt></italic>, report all <italic>K</italic> points of(More)
We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in IR, is any point contained in any hyperplane? We de ne a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time (n logm + nm(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; ecpmtion? 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 consider the complexity of Delaunay triangulations of sets of point s in $\Real^3$ under certain practical geometric constraints. The \emph{spread} of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of $n$ points in~$\Real^3$ with spread $\Delta$ has complexity(More)
We introduce 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 were based on high order Voronoi diagrams. Our technique allows us for the(More)
This paper addresses the problem of capturing an arbitrary convex object P in the plane with three congruent disc-shaped robots. Given two stationary robots in contact with P, we characterize the set of positions of a third robot, the so-called capture region, that prevent P from escaping to infinity via continuous rigid motion. We show that the computation(More)
1 Introduction WC show how to preprocess a set S of points in lRd into an external memory data structure that efficiently supports linear-constraint queries. Each query is in the form of a linear constraint aa x < b; the data structure must report all the points of S that satisfy the constraint. Our goal is to minimize the number of disk blocks required to(More)