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We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2-manifold. In particular, we show that the shortest set of loops that generate the fundamental group of any oriented combinatorial… (More)

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)

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)

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)

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)

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)

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 design a kinetic data structure for detecting collisions between two simple polygons in motion. In order to do so, we create a planar subdivision of the free space between the two polygons, called the external relative geodesic triangu-lation, which certifies their disjointness. We show how this subdivision can be maintained as a kinetic data structure… (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)