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Dense Point Sets Have Sparse Delaunay Triangulations
"... Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms ..."
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Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since threedimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worstcase running time \Omega (n2). However, this behavior is almost never observed in practice except for highlycontrived inputs. For all practical purposes, threedimensional Delaunay triangulations appear to have linear complexity. This frustrating
Planar Point Location For Large Data Sets: To Seek Or Not To Seek
, 2000
"... . We present an algorithm for external memory planar point location that is both effective and easy to implement. The base algorithm is an external memory variant of the bucket method by Edahiro, Kokubo and Asano that is combined with Lee and Yang's batched internal memory algorithm for planar p ..."
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Cited by 7 (1 self)
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. We present an algorithm for external memory planar point location that is both effective and easy to implement. The base algorithm is an external memory variant of the bucket method by Edahiro, Kokubo and Asano that is combined with Lee and Yang's batched internal memory algorithm for planar point location. Although our algorithm is not optimal in terms of its worstcase behavior, weshow its efficiency for both batched and singleshot queries by experiments with realworld data. The experiments show that the algorithm benefits from its mainly sequential disk access pattern and significantly outperforms the fastest algorithm for internal memory. 1 Introduction The wellknown problem of planar point location consists of determining the region of a planar subdivision that contains a given query point. We assume that a planar subdivision is given by N line segments, and that each segment is labeled with the names of the two regions it separates. In this setting, a point locati...
DISTRIBUTIONSENSITIVE POINT LOCATION IN CONVEX SUBDIVISIONS
"... A data structure is presented for point location in convex planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. ..."
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Cited by 4 (3 self)
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A data structure is presented for point location in convex planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal.
Succinct Geometric Indexes Supporting Point Location Queries
"... We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that support geometric queries in optimal time, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence. Our first and main result is a succi ..."
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We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that support geometric queries in optimal time, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time1. We also design three variants of this index. The first supports point location using lg n +2 √ lg n + O(lg 1/4 n) pointline comparisons. The second supports point location in o(lg n) time when the coordinates are integers bounded by U. The last variant can answer point location queries in O(H +1) expected time, where H is the entropy of the query distribution. These results match the query efficiency of previous point location structures that occupy O(n) words or O(n lg n) bits, while saving drastic amounts of space. We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in O(lg 2 n) time. 1
InPlace 2d Nearest Neighbor Search
, 2007
"... Abstract We revisit a classic problem in computational geometry: preprocessing a planar npoint set to answer nearest neighbor queries. In SoCG 2004, Br"onnimann, Chan, and Chen showed that it is possible to design an efficient data structure that takes no extra space at all other than the inpu ..."
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Abstract We revisit a classic problem in computational geometry: preprocessing a planar npoint set to answer nearest neighbor queries. In SoCG 2004, Br"onnimann, Chan, and Chen showed that it is possible to design an efficient data structure that takes no extra space at all other than the input array holding a permutation of the points. The best query time known for such "inplace data structures " is O(log 2 n). In this paper, we break the O(log 2 n) barrier by providing a method that answers nearest neighbor queries in time O((log n) log3=2 2 log log n) = O(log
BIASED RANGE TREES
, 2008
"... ABSTRACT. A data structure, called a biased range tree, is presented that preprocesses a set S of n points in 2 and a query distribution D for 2sided orthogonal range counting queries. The expected query time for this data structure, when queries are drawn according to D, matches, to within a const ..."
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ABSTRACT. A data structure, called a biased range tree, is presented that preprocesses a set S of n points in 2 and a query distribution D for 2sided orthogonal range counting queries. The expected query time for this data structure, when queries are drawn according to D, matches, to within a constant factor, that of the optimal decision tree for S and D. The memory and preprocessing requirements of the data structure are O(n log n). 1
ENTROPY, TRIANGULATION, AND POINT LOCATION IN PLANAR SUBDIVISIONS
, 2009
"... A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connecte ..."
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A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of pointline comparisons performed by this data structure, when the queries are distributed according to D, is ˜ H + O ( ˜ H2/3 + 1) where ˜ H = ˜ H(G, D) is a lower bound on the expected number of pointline comparisons performed by any linear decision tree for point location in G under the query distribution D. The preprocessing algorithm runs in O(n log n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has nearminimum entropy.
8. Point Location
"... Our goal in the following is to provide the missing bit needed to solve the post o problem optimally. ce Theorem 8.1 Given a triangulation T for a set P R2 of n points, one can build in O(n) time an O(n) size data structure that allows for any query point q 2 conv(P) to nd in O(log n) time the trian ..."
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Our goal in the following is to provide the missing bit needed to solve the post o problem optimally. ce Theorem 8.1 Given a triangulation T for a set P R2 of n points, one can build in O(n) time an O(n) size data structure that allows for any query point q 2 conv(P) to nd in O(log n) time the triangle from T containing q. Corollary 8.2 (Nearest Neighbor Search) Given a set P R2 of n points, one can build in O(n log n) time an O(n) size data structure that allows for any query point q 2 conv(P) to nd in O(log n) time the nearest neighbor of q among the points from P. Proof. First construct the Voronoi Diagram V of P in O(n log n) time. It has exactly n faces, all of which are convex polygons. Any convex polygon can easily be triangulated in time linear in its number of edges ( = number of vertices). As V has at most 3n − 6 edges and every edge appears in exactly two faces, V can be triangulated in O(n) time overall. Label each of the resulting triangles with the point from p, whose Voronoi region contains it, and apply the data structure from Theorem 8.1. 8.1 Kirkpatrick’s Hierarchy Idea: Construct a hierarchy T0,...,Th of triangulations, such that
Chapter 10 Voronoi Diagrams
"... Suppose there are n post offices p1,...pn in a city. Someone who is located at a position q within the city would like to know which post office is closest to him. Modeling the city as a planar region, we think of p1,...pn and q as points in the plane. Denote the set of post offices by P = {p1,...pn ..."
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Suppose there are n post offices p1,...pn in a city. Someone who is located at a position q within the city would like to know which post office is closest to him. Modeling the city as a planar region, we think of p1,...pn and q as points in the plane. Denote the set of post offices by P = {p1,...pn}. Figure 10.1: Closest post offices for various query points. While the locations of post offices are known and do not change so frequently, we do not know in advance for which—possibly many—query locations the closest post office is to be found. Therefore, our long term goal is to come up with a data structure on top of P that allows to answer any possible query efficiently. The basic idea is to apply a socalled locus approach: we partition the query space into regions on which is the answer is the same. In our case, this amounts to partition the plane into regions such that for all points within a region the same point from P is closest (among all points from P). As a warmup, consider the problem for two post offices pi, pj ∈ P. For which query locations is the answer pi rather than pj? This region is bounded by the bisector of pi and pj, that is, the set of points which have the same distance to both points. 106 CG 2012 10.2. Voronoi Diagram Proposition 10.1 For any two distinct points inR d the bisector is a hyperplane, that is, inR 2 it is a line. Proof. Let p = (p1,...,pd) and q = (q1,...,qd) be two points inR d. The bisector of p and q consists of those points x = (x1,...,xd) for which p−x  = q−x  ⇐ ⇒ p−x  2 = q−x  2 ⇐ ⇒ p  2 −q  2 = 2(p−q) ⊤ x. As p and q are distinct, this is the equation of a hyperplane. pj pi H(pi, pj) Figure 10.2: The bisector of two points. Denote by H(pi, pj) the closed halfspace bounded by the bisector of pi and pj that contains pi. InR 2, H(pi, pj) is a halfplane; see Figure 10.2. Exercise 10.2 a) What is the bisector of a line ℓ and a point p ∈R 2 \ ℓ, that is, the set of all points x ∈R 2 with p−x  = p−ℓ  ( = minq∈ℓp−q)? b) For two points p ̸ = q ∈R 2, what is the region that contains all points whose distance to p is exactly twice their distance to q? 10.2 Voronoi Diagram In the following we work with a set P = {p1,..., pn} of points inR 2. Definition 10.3 (Voronoi cell) For pi ∈ P denote the Voronoi cell VP(i) of pi by VP(i): = { q ∈R 2 ∣ q−pi  � q−p  for all p ∈ P}.