Data Structures In Computational Geometry Research Articles

Wavelet Tree ensembles with Machine Learning and its classification

Wavelet trees are compact data structures in computational geometry. In the past, it was used as an essential tool for handling the size of data, data compression, indexing and for many more applications. Machine learning algorithms are used for classification of data and its analysis. In this article, we discuss the scope of machine learning with wavelet trees, wavelet entropy, wavelet matrix and wavelet packets. The study concludes that machine learning applications with wavelet tree is a better choice in terms storage and classification of data. The proposed methodology consists of three techniques for making the data more efficient. It consists of LZW Compression techniques, Wavelet tree, and machine learning algorithm SVM. In this methodology compression with classification process is done for datasets. This proposed methodology performs with machine learningalgorithms in terms of classification of data. In future this method can be used for efficient searching and indexing of large data sets. The classified and compressed dataset perform the indexing with wavelet tree takes less searching time.

Parallel Voronoi Computation for Physics-Based Simulations

Voronoi diagrams are fundamental data structures in computational geometry, with applications in areas such as physics-based simulations. For non-Euclidean distances, the Voronoi diagram must be performed over a grid-graph, where the edges encode the required distance information. The major bottleneck in this case is a shortest path algorithm that must be computed multiple times during the simulation. The authors present a GPU algorithm for solving the shortest path problem from multiple sources using a generalized distance function. Their algorithm was designed to leverage the grid-based nature of the underlying graph that represents the deformable objects. Experimental results report speed-ups up to 65x over a current reference sequential method.

GEODESIC-PRESERVING POLYGON SIMPLIFICATION

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon [Formula: see text] by a polygon [Formula: see text] such that (1) [Formula: see text] contains [Formula: see text], (2) [Formula: see text] has its reflex vertices at the same positions as [Formula: see text], and (3) the number of vertices of [Formula: see text] is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both [Formula: see text] and [Formula: see text], our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of [Formula: see text]. We describe several of these applications (including linear-time post-processing steps that might be necessary).

Computing the conjugate of convex piecewise linear-quadratic bivariate functions

We present a new algorithm to compute the Legendre–Fenchel conjugate of convex piecewise linear-quadratic (PLQ) bivariate functions. The algorithm stores a function using a (primal) planar arrangement. It then computes the (dual) arrangement associated with the conjugate by looping through vertices, edges, and faces in the primal arrangement and building associated dual vertices, edges, and faces. Using optimal computational geometry data structures, the algorithm has a linear time worst-case complexity. We present the algorithm, and illustrate it with numerical examples. We proceed to build a toolbox for convex bivariate PLQ functions by implementing the addition, and scalar multiplication operations. Finally, we compose these operators to compute classical convex analysis operators such as the Moreau envelope, and the proximal average.

Cuttings for Disks and Axis-Aligned Rectangles in Three-Space

We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r∈ℕ, a \(\frac{1}{r}\) -cutting is a covering of the space with simplices such that the interior of each simplex intersects at most n/r objects. For n pairwise disjoint disks in ℝ3 and a parameter r∈ℕ, we construct a \(\frac{1}{r}\) -cutting of size O(r 2). For n axis-aligned rectangles in ℝ3, we construct a \(\frac{1}{r}\) -cutting of size O(r 3/2).As an application related to multi-point location in three-space, we present tight bounds on the cost of spanning trees across barriers. Given n points and a finite set of disjoint disk barriers in ℝ3, the points can be connected with a straight line spanning tree such that every disk is stabbed by at most \(O(\sqrt{n})\) edges of the tree. If the barriers are axis-aligned rectangles, then there is a straight line spanning tree such that every rectangle is stabbed by O(n 1/3) edges. Both bounds are best possible.

Exploring fingerprint matching through a topology-based perspective

In the recent decade, the rapid development of biometric technologies for security applications spurred increased interest in both academic research community and industrial sectors. Still, the fingerprint recognition remains one of the core methodologies for person identification. Known since the time of the Babylonians, they used their thumbprints on clay tablets to verify their identity around 2,000 B.C. While the approach remains a reliable, efficient and a commonly accepted biometric, the methodology develops rapidly. This paper presents a survey of topology-based techniques for fingerprint recognition, and describes an original fingerprint alignment and matching scheme based on two powerful computational geometry data structures: the Voronoi diagram and the Delaunay triangulation.

COLLISION DETECTION OPTIMIZATION IN A MULTI-PARTICLE SYSTEM

Collision detection optimization in an event-driven simulation of a multi-particle system is one of the crucial tasks, determining the efficiency of the simulation. We present the event-driven simulation algorithm that employs dynamic computational geometry data structures as a tool for collision detection optimization (CDO). The first successful application of the dynamic generalized Voronoi diagram method for collision detection optimization in a system of moving particles is discussed. A comprehensive comparision of four kinetic data structures in d-dimensional space, performed in a framework of an event-driven simulation of a granular-type materials system, is supported by the experimental results.

Exact Size of Binary Space Partitionings and Improved Rectangle Tiling Algorithms

We prove the following upper and lower bounds on the exact size of binary space partition (BSP) trees for a set of n isothetic rectangles in the plane: An upper bound of 3n-1 in general, and an upper bound of 2n-1 if the rectangles tile the underlying space. This improves the upper bounds of 4n in [V. Hai Nguyen and P. Widmayer, Binary Space Partitions for Sets of Hyperrectangles, Lecture Notes in Comput. Sci. 1023, Springer-Verlag, Berlin, 1995; F. d'Amore and P. G. Franciosa, Inform. Process. Lett., 44 (1992), pp. 255--259]. A BSP satisfying the upper bounds can be constructed in O(n log n) time. A worst-case lower bound of 2n-o(n) in general, and $\frac{3n}{2}-o(n)$ if the rectangles form a tiling. The BSP tree is one of the most popular data structures in computational geometry, and hence even small factor improvements of $\frac{4}{3}$ or 2 on the previously known upper bounds that we show improve the performances of applications relying on the BSP tree. As an illustration, we present improved approximation algorithms for certain dual rectangle tiling problems using our upper bounds on the size of the BSP trees.