Query processing of spatial objects: Complexity versus Redundancy

The management of complex spatial objects in applications, such as geography and cartography, imposes stringent new requirements on spatial database systems, in particular on efficient query processing. As shown before, the performance of spatial query processing can be improved by decomposing complex spatial objects into simple components. Up to now, only decomposition techniques generating a linear number of very simple components, e.g. triangles or trapezoids, have been considered. In this paper, we will investigate the natural trade-off between the complexity of the components and the redundancy, i.e. the number of components, with respect to its effect on efficient query processing. In particular, we present two new decomposition methods generating a better balance between the complexity and the number of components than previously known techniques. We compare these new decomposition methods to the traditional undecomposed representation as well as to the well-known decomposition into convex polygons with respect to their performance in spatial query processing. This comparison points out that for a wide range of query selectivity the new decomposition techniques clearly outperform both the undecomposed representation and the convex decomposition method. More important than the absolute gain in performance by a factor of up to an order of magnitude is the robust performance of our new decomposition techniques over the whole range of query selectivity

Bregman Voronoi Diagrams: Properties, Algorithms and Applications

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g. k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connexion with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation. Finally, we give some applications of Bregman Voronoi diagrams which are of interest in the context of computational geometry and machine learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures

The Littlewood-Gowers problem

We show that if A is a subset of Z/pZ (p a prime) of density bounded away from 0 and 1 then the A(Z/pZ)-norm (that is the l^1-norm of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times (log p)^{1/2 - \epsilon} as p tends to infinity. This improves on the exponent 1/3 in recent work of Green and Konyagin.Comment: 31 pp. Corrected typos. Updated references

On colouring point visibility graphs

In this paper we show that it can be decided in polynomial time whether or not the visibility graph of a given point set is 4-colourable, and such a 4-colouring, if it exists, can also be constructed in polynomial time. We show that the problem of deciding whether the visibility graph of a point set is 5-colourable, is NP-complete. We give an example of a point visibility graph that has chromatic number 6 while its clique number is only 4

On Generalizations of Network Design Problems with Degree Bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning tree), and (2) by incorporating `degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure

Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-complete. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.Comment: Accepted versio

Most vital segment barriers

We study continuous analogues of "vitality" for discrete network flows/paths, and consider problems related to placing segment barriers that have highest impact on a flow/path in a polygonal domain. This extends the graph-theoretic notion of "most vital arcs" for flows/paths to geometric environments. We give hardness results and efficient algorithms for various versions of the problem, (almost) completely separating hard and polynomially-solvable cases

A linear-time algorithm for constructing a circular visibility diagram

To computer circular visibility inside a simple polygon, circular arcs that emanate from a given interior point are classified with respect to the edges of the polygon they first intersect. Representing these sets of circular arcs by their centers results in a planar partition called the circular visibility diagram. An O(n) algorithm is given for constructing the circular visibility diagram for a simple polygon with n vertices.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41346/1/453_2005_Article_BF01206329.pd

Search for supersymmetry with a dominant R-parity violating LQDbar couplings in e+e- collisions at centre-of-mass energies of 130GeV to 172 GeV

A search for pair-production of supersymmetric particles under the assumption that R-parity is violated via a dominant LQDbar coupling has been performed using the data collected by ALEPH at centre-of-mass energies of 130-172 GeV. The observed candidate events in the data are in agreement with the Standard Model expectation. This result is translated into lower limits on the masses of charginos, neutralinos, sleptons, sneutrinos and squarks. For instance, for m_0=500 GeV/c^2 and tan(beta)=sqrt(2) charginos with masses smaller than 81 GeV/c^2 and neutralinos with masses smaller than 29 GeV/c^2 are excluded at the 95% confidence level for any generation structure of the LQDbar coupling.Comment: 32 pages, 30 figure

Search for the glueball candidates f0(1500) and fJ(1710) in gamma gamma collisions

Data taken with the ALEPH detector at LEP1 have been used to search for gamma gamma production of the glueball candidates f0(1500) and fJ(1710) via their decay to pi+pi-. No signal is observed and upper limits to the product of gamma gamma width and pi+pi- branching ratio of the f0(1500) and the fJ(1710) have been measured to be Gamma_(gamma gamma -> f0(1500)). BR(f0(1500)->pi+pi-) < 0.31 keV and Gamma_(gamma gamma -> fJ(1710)). BR(fJ(1710)->pi+pi-) < 0.55 keV at 95% confidence level.Comment: 10 pages, 3 figure