Dynamic Euclidean Minimum Spanning Trees and Extrema of Binary Functions
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32 PROXIMITY ALGORITHMS Joseph S
32 PROXIMITY ALGORITHMS Joseph S. B. Mitchell and Wolfgang Mulzer INTRODUCTION The notion of distance is fundamental to many aspects of computational geometry. A classic approach to characterize the distance properties of planar (and high- dimensional) point sets that has been studied since the early 1980s uses proximity graphs (Section 32.1). Proximity graphs are geometric graphs in which two vertices p, q are connected by an edge (p; q) if and only if a certain exclusion region for p; q contains no points from the vertex set. Depending on the specific exclusion region, many variants of proximity graphs can be defined, such as relative neighborhood graphs, Delaunay triangulations, β-skeletons, empty-strip graphs, etc. Since prox- imity graphs encode interesting information on the intrinsic structure of the point set, they have found many applications. From an algorithmic point of view, it is extremely useful to have a compact representation of the distance structure of a point set. The well-separated pair decomposition (WSPD) offers one way to achieve this (Section 32.2). WSPDs have numerous algorithmic applications, and the no- tion generalizes to certain non-Euclidean metrics. Furthermore, several variants of the WSPD have been developed to address its shortcomings, e.g., semi-separated pair decompositions and (α; β)-pair decompositions. Geometric spanners provide another means to approximate the complete Euclidean metric (Section 32.3). Here, the distance function is approximated by the shortest path distance in a sparse geo- metric graph. There are four basic constructions for geometric spanners: the greedy spanner, the Yao graph, the Θ-graph and the WSPD-spanner. -
Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Chan, Timothy M., and Mihai Pa#trascu. “Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time.” SIAM Journal on Computing 39.2 (2009) : 703. 2009 Society for Industrial and Applied Mathematics As Published http://dx.doi.org/10.1137/07068669x Publisher Society for Industrial and Applied Mathematics (SIAM).. Version Final published version Citable link http://hdl.handle.net/1721.1/64948 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. SIAM J. COMPUT. c 2009 Society for Industrial and Applied Mathematics Vol. 39, No. 2, pp. 703–729 TRANSDICHOTOMOUS RESULTS IN COMPUTATIONAL GEOMETRY, I: POINT LOCATION IN SUBLOGARITHMIC TIME∗ † ˇ ‡ TIMOTHY M. CHAN AND MIHAI PATRAS¸CU w Abstract. Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 ,we present a linear-space data structure that can answer point-location queries in O(min{lg n/ lg lg n, lg U/lg lg U}) time on the unit-cost random access machine (RAM) with word size w. This is the first result to beat the standard Θ(lg n) bound for infinite precision models. As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a three-dimensional (3D) point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. -
32 PROXIMITY ALGORITHMS Joseph S
32 PROXIMITY ALGORITHMS Joseph S. B. Mitchell and Wolfgang Mulzer INTRODUCTION The notion of distance is fundamental to many aspects of computational geometry. A classic approach to characterize the distance properties of planar (and high- dimensional) point sets that has been studied since the early 1980s uses proximity graphs (Section 32.1). Proximity graphs are geometric graphs in which two vertices p, q are connected by an edge (p; q) if and only if a certain exclusion region for p; q contains no points from the vertex set. Depending on the specific exclusion region, many variants of proximity graphs can be defined, such as relative neighborhood graphs, Delaunay triangulations, β-skeletons, empty-strip graphs, etc. Since prox- imity graphs encode interesting information on the intrinsic structure of the point set, they have found many applications. From an algorithmic point of view, it is extremely useful to have a compact representation of the distance structure of a point set. The well-separated pair decomposition (WSPD) offers one way to achieve this (Section 32.2). WSPDs have numerous algorithmic applications, and the no- tion generalizes to certain non-Euclidean metrics. Furthermore, several variants of the WSPD have been developed to address its shortcomings, e.g., semi-separated pair decompositions and (α; β)-pair decompositions. Geometric spanners provide another means to approximate the complete Euclidean metric (Section 32.3). Here, the distance function is approximated by the shortest path distance in a sparse geo- metric graph. There are four basic constructions for geometric spanners: the greedy spanner, the Yao graph, the Θ-graph and the WSPD-spanner. -
Voronoi Diagrams
Voronoi Diagrams Franz Aurenhammer Rolf Klein1 Institut f¨ur Grundlagen der FernUniversit¨at Hagen Informationsverarbeitung Praktische Informatik VI Technische Universit¨at Graz Elberfelder Straße 95 Klosterwiesgasse 32/2 D-58084 Hagen, Germany A-8010 Graz, Austria 1Partially supported by the Deutsche Forschungsgemeinschaft, grant Kl 655 2-2. Contents 1 Introduction 1 2 Definitions and elementary properties 2 3 Algorithms 6 3.1 A lower bound ............................... 7 3.2 Incremental construction ......................... 8 3.3 Divide & conquer ............................. 12 3.4 Sweep ................................... 14 3.5 Lifting to 3-space ............................. 16 4 Generalizations and structural properties 18 4.1 Characterization of Voronoi diagrams .................. 18 4.2 Optimization properties of Delaunay triangulations .......... 21 4.3 Higher dimensions, power diagrams, and order-k diagrams ...... 25 4.3.1 Voronoi diagrams and Delaunay tesselations in 3-space .... 25 4.3.2 Power diagrams and convex hulls ................ 27 4.3.3 Higher-order Voronoi diagrams and arrangements ....... 31 4.4 Generalized sites ............................. 34 4.4.1 Line segment Voronoi diagram and medial axis ......... 35 4.4.2 Straight skeletons ......................... 38 4.4.3 Convex polygons ......................... 40 4.4.4 Constrained Voronoi diagrams and Delaunay triangulations .. 41 4.5 Generalized spaces and distances .................... 43 4.5.1 Generalized spaces ........................ 43 4.5.2 Convex distance functions .................... 46 4.5.3 Nice metrics ............................ 51 4.6 General Voronoi diagrams ........................ 55 5 Geometric applications 59 5.1 Distance problems ............................ 59 5.1.1 Post office ............................. 59 5.1.2 Nearest neighbors and the closest pair ............. 61 5.1.3 Largest empty and smallest enclosing circle ........... 64 5.2 Subgraphs of Delaunay triangulations ................. -
Practical Construction of K-Nearest Neighbor Graphs in Metric Spaces ⋆
Practical Construction of k-Nearest Neighbor Graphs in Metric Spaces ⋆ Rodrigo Paredes1, Edgar Ch´avez2, Karina Figueroa1,2, and Gonzalo Navarro1 1 Center for Web Research, Dept. of Computer Science, University of Chile. 2 Escuela de Ciencias F´ısico-Matem´aticas, Univ. Michoacana, Mexico. {raparede,kfiguero,gnavarro}@dcc.uchile.cl, [email protected] Abstract. Let U be a set of elements and d a distance function defined among them. Let NNk(u) be the k elements in U−{u} having the small- est distance to u. The k-nearest neighbor graph (knng) is a weighted directed graph G(U,E) such that E = {(u, v), v ∈ NNk(u)}. Several knng construction algorithms are known, but they are not suitable to general metric spaces. We present a general methodology to construct knngs that exploits several features of metric spaces. Experiments sug- 1.27 gest that it yields costs of the form c1n distance computations for low 1.90 and medium dimensional spaces, and c2n for high dimensional ones. 1 Introduction Let U be a set of elements and d a distance function defined among them. Let NNk(u) be the k elements in U −{u} having the smallest distance to u according to the function d. The k-nearest neighbor graph (knng) is a weighted directed graph G(U, E) connecting each element to its k-nearest neighbors, thus E = {(u, v), v ∈ NNk(u)}. Building the knng is a direct generalization of the all- nearest-neighbor (ann) problem, so ann corresponds to the 1nng construction problem. knngs are central in many applications: cluster and outlier detection [14, 4], VLSI design, spin glass and other physical process simulations [6], pattern recognition [12], query or document recommendation systems [3], and others.