Fast Algorithm for Finding Maximum Distance with Space Subdivision in E2

Finding an exact maximum distance of two points in the given set is a fundamental computational problem which is solved in many applications. This paper presents a fast, simple to implement and robust algorithm for finding this maximum distance of two points in E2. This algorithm is based on a polar subdivision followed by division of remaining points into uniform grid. The main idea of the algorithm is to eliminate as many input points as possible before finding the maximum distance. The proposed algorithm gives the significant speed up compared to the standard algorithm

Polyhedral computational geometry for averaging metric phylogenetic trees

This paper investigates the computational geometry relevant to calculations of the Frechet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability measures on spaces of nonpositive curvature developed by Sturm. We show that the combinatorics of geodesics with a specified fixed endpoint in tree space are determined by the location of the varying endpoint in a certain polyhedral subdivision of tree space. The variance function associated to a finite subset of tree space has a fixed $C^\infty$ algebraic formula within each cell of the corresponding subdivision, and is continuously differentiable in the interior of each orthant of tree space. We use this subdivision to establish two iterative methods for producing sequences that converge to the Frechet mean: one based on Sturm's Law of Large Numbers, and another based on descent algorithms for finding optima of smooth functions on convex polyhedra. We present properties and biological applications of Frechet means and extend our main results to more general globally nonpositively curved spaces composed of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5, added counter example for polyhedrality of vistal subdivision in general CAT(0) cubical complexes; v1: 43 pages, 5 figure

Distance-Sensitive Planar Point Location

Let $\mathcal{S}$ be a connected planar polygonal subdivision with $n$ edges that we want to preprocess for point-location queries, and where we are given the probability $\gamma_i$ that the query point lies in a polygon $P_i$ of $\mathcal{S}$. We show how to preprocess $\mathcal{S}$ such that the query time for a point~$p\in P_i$ depends on~$\gamma_i$ and, in addition, on the distance from $p$ to the boundary of~$P_i$---the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time $O\left(\min \left(\log n, 1 + \log \frac{\mathrm{area}(P_i)}{\gamma_i \Delta_{p}^2}\right)\right)$, where $\Delta_{p}$ is the shortest Euclidean distance of the query point~$p$ to the boundary of $P_i$. Our structure uses $O(n)$ space and $O(n \log n)$ preprocessing time. It is based on a decomposition of the regions of $\mathcal{S}$ into convex quadrilaterals and triangles with the following property: for any point $p\in P_i$, the quadrilateral or triangle containing~$p$ has area $\Omega(\Delta_{p}^2)$. For the special case where $\mathcal{S}$ is a subdivision of the unit square and $\gamma_i=\mathrm{area}(P_i)$, we present a simpler solution that achieves a query time of $O\left(\min \left(\log n, \log \frac{1}{\Delta_{p}^2}\right)\right)$. The latter solution can be extended to convex subdivisions in three dimensions

Ray casting implicit fractal surfaces with reduced affine arithmetic

A method is presented for ray casting implicit surfaces defined by fractal combinations of procedural noise functions. The method is robust and uses affine arithmetic to bound the variation of the implicit function along a ray. The method is also efficient due to a modification in the affine arithmetic representation that introduces a condensation step at the end of every non-affine operation. We show that our method is able to retain the tight estimation capabilities of affine arithmetic for ray casting implicit surfaces made from procedural noise functions while being faster to compute and more efficient to store

A Multi-variate Discrimination Technique Based on Range-Searching

We present a fast and transparent multi-variate event classification technique, called PDE-RS, which is based on sampling the signal and background densities in a multi-dimensional phase space using range-searching. The employed algorithm is presented in detail and its behaviour is studied with simple toy examples representing basic patterns of problems often encountered in High Energy Physics data analyses. In addition an example relevant for the search for instanton-induced processes in deep-inelastic scattering at HERA is discussed. For all studied examples, the new presented method performs as good as artificial Neural Networks and has furthermore the advantage to need less computation time. This allows to carefully select the best combination of observables which optimally separate the signal and background and for which the simulations describe the data best. Moreover, the systematic and statistical uncertainties can be easily evaluated. The method is therefore a powerful tool to find a small number of signal events in the large data samples expected at future particle colliders.Comment: Submitted to NIM, 18 pages, 8 figure

Computational Approaches to Lattice Packing and Covering Problems

We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the sense that they approximate optimal covering lattices and optimal packing-covering lattices within any desired accuracy. Both algorithms involve semidefinite programming and are based on Voronoi's reduction theory for positive definite quadratic forms, which describes all possible Delone triangulations of Z^d. In practice, our implementations reproduce known results in dimensions d <= 5 and in particular solve the two problems in these dimensions. For d = 6 our computations produce new best known covering as well as packing-covering lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our approach leads to new best known covering lattices. Although we use numerical methods, we made some effort to transform numerical evidences into rigorous proofs. We provide rigorous error bounds and prove that some of the new lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in Discrete and Computational Geometry, see also http://fma2.math.uni-magdeburg.de/~latgeo