Searching for the quark-gluon plasma

The claims for production of high energy densities and possible new states of matter in collisions of nuclei by George F. Bertsch (Science, {\bf 265} (1994) 480-481) are examined and compared with simple explanations of the data which have appeared in the literature

Translating a Regular Grid over a Point Set

We consider the problem of translating a (finite or infinite) square grid G over a set S of n points in the plane in order to maximize some objective function. We say that a grid cell is k-occupied if it contains k or more points of 5. The main set of problems we study have to do with translating an infinite grid so that the number of fe-occupied cells is maximized or minimized. For these problems we obtain running times of the form O(kn polylog n). We also consider the problem of translating a finite size grid, with m cells, in order to maximize the number of fe-occupied cells. Here we obtain a running time of the form O(knm polylog nm)

A generalized Winternitz Theorem

We prove that, for every simple polygon P having k ≥ 1 reflex vertices, there exists a point q ε P such that every half-polygon that contains q contains nearly 1/2(k + 1) times the area of P. We also give a family of examples showing that this result is the best possible

Feature Based Cut Detection with Automatic Threshold Selection

There has been much work concentrated on creating accurate shot boundary detection algorithms in recent years. However a truly accurate method of cut detection still eludes researchers in general. In this work we present a scheme based on stable feature tracking for inter frame differencing. Furthermore, we present a method to stabilize the differences and automatically detect a global threshold to achieve a high detection rate. We compare our scheme against other cut detection techniques on a variety of data sources that have been specifically selected because of the difficulties they present due to quick motion, highly edited sequences and computer-generated effects

Constrained generalized Delaunay graphs are plane spanners

We look at generalized Delaunay graphs in the constrained setting by introducing line segments which the edges of the graph are not allowed to cross. Given an arbitrary convex shape C, a constrained Delaunay graph is constructed by adding an edge between two vertices p and q if and only if there exists a homothet of C with p and q on its boundary that does not contain any other vertices visible to p and q. We show that, regardless of the convex shape C used to construct the constrained Delaunay graph, there exists a constant t (that depends on C) such that it is a plane t-spanner of the visibility graph

Computing the greedy spanner in near-quadratic time

It is well-known that the greedy algorithm produces high quality spanners and therefore is used in several applications. However, for points in d-dimensional Euclidean space, the greedy algorithm has cubic running time. In this paper we present an algorithm that computes the greedy spanner (spanner computed by the greedy algorithm) for a set of n points from a metric space with bounded doubling dimension in time using space. Since the lower bound for computing such spanners is Ω(n 2), the time complexity of our algorithm is optimal to within a logarithmic factor

A polynomial bound for untangling geometric planar graphs

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n^\epsilon vertices fixed. We answer this question in the affirmative with \epsilon=1/4. The previous best known bound was \Omega((\log n / \log\log n)^{1/2}). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170 2007] by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than 3(n^{1/2}-1) vertices fixed. Moreover, we improve the lower bound to (n/2)^{1/2}.Comment: 14 pages, 7 figure

Station layouts in the presence of location constraints

In wireless communication, the signal of a typical broadcast station is transmited from a broadcast center p and reaches objects at a distance, say, R from it. In addition there is a radius r, r < R, such that the signal originating from the center of the station is so strong that human habitation within distance r from the center p should be avoided. Thus every station determines a region which is an “annulus of permissible habitation". We consider the following station layout (SL) problem: Cover a given (say, rectangular) planar region which includes a collection of orthogonal buildings with a minimum number of stations so that every point in the region is within the reach of a station, while at the same time no building is within the dangerous range of a station. We give algorithms for computing such station layouts in both the one-and two-dimensional cases

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]

On spanning properties of various delaunay graphs

A geometric graph G is a graph whose vertices are points in the plane and whose edges are line segments weighted by the Euclidean distance between their endpoints. In this setting, a t-spanner of G is a connected spanning subgraph G′ with the property that for every pair of vertices x; y, the shortest path from x to y in G′ has weight at most t ≥ 1 times the shortest path from x to y in G. The parameter t is commonly referred to as the spanning ratio or the stretch factor. Among the many beautiful properties that Delaunay graphs possess, a constant spanning ratio is one of them. We provide a comprehensive overview of various results concerning the spanning ratio among other properties of different types of Delaunay graphs and their subgraphs