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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 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 give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓ p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different(More)
We present an algorithm to construct meshes suitable for spacetime 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 spacetime domain X(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)
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)
The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R 3 with spread ∆ has complexity O(∆ 3). This bound is tight in the worst case for all ∆ = O(√ n). In particular, the Delaunay triangulation of any dense point set has linear complexity. We(More)