Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
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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. -
Dynamic Euclidean Minimum Spanning Trees and Extrema of Binary Functions
Dynamic Euclidean Minimum Spanning Trees and Extrema of Binary Functions David Eppstein Department of Information and Computer Science University of California, Irvine, CA 92717 Abstract We maintain the minimum spanning tree of a point set in the plane, sub- ject to point insertions and deletions, in amortized time O(n1/2 log2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n) per update. Our algorithm uses a novel construction, the ordered nearest neighbor path of a set of points. Our results generalize to higher dimensions, and to fully dynamic algorithms for main- taining minima of binary functions, including the diameter of a point set and the bichromatic farthest pair. 1 Introduction A dynamic geometric data structure is one that maintains the solution to some problem, defined on a geometric input such as a point set, as the input undergoes update operations such as insertions or deletions of single points. Dynamic al- gorithms have been studied for many geometric optimization problems, including closest pairs [7, 23, 25, 26], diameter [7, 26], width [4], convex hulls [15, 22], lin- ear programming [2, 9, 18], smallest k-gons [6, 11], and minimum spanning trees (MSTs) [8]. Many of these algorithms suffer under a restriction that, if deletions are allowed at all, they may only occur at certain prespecified times—the algorithms are not fully dynamic. A number of other papers have considered dynamic computational geometry problems under an average case model that assumes that among a given set of points each point is equally likely to be inserted or deleted next [10, 19, 20, 21, 24]. -
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.