## Efficient Expected-Case Algorithms for Planar Point Location (2000)

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Citations: | 13 - 4 self |

### BibTeX

@MISC{Arya00efficientexpected-case,

author = {Sunil Arya and Siu-Wing Cheng and David M. Mount and H. Ramesh},

title = {Efficient Expected-Case Algorithms for Planar Point Location},

year = {2000}

}

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### Abstract

. Planar point location is among the most fundamental search problems in computational geometry. Although this problem has been heavily studied from the perspective of worst-case query time, there has been surprisingly little theoretical work on expected-case query time. We are given an n-vertex planar polygonal subdivision S satisfying some weak assumptions (satisfied, for example, by all convex subdivisions). We are to preprocess this into a data structure so that queries can be answered efficiently. We assume that the two coordinates of each query point are generated independently by a probability distribution also satisfying some weak assumptions (satisfied, for example, by the uniform distribution). In the decision tree model of computation, it is well-known from information theory that a lower bound on the expected number of comparisons is entropy(S). We provide two data structures, one of size O(n 2 ) that can answer queries in 2 entropy(S) + O(1) expected number...

### Citations

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Citation Context ...quirement could be reduced to O(n), but their approach was rather impractical. Since then a number of more practical methods have been proposed. These include Kirkpatrick’s clever hierarchical method =-=[12]-=-, the separator method by Edelsbrunner et al. [8], the persistent search tree method by Sarnak and Tarjan [21], and the randomized incremental method by Mulmuley [20]. All of these are based on worst-... |

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Citation Context ...of the unit interval. Consider any binary search tree whose leaves correspond to the intervals. It is easy to see that the expected number of comparisons is given by the weighted external path length =-=[14]-=- of the tree, where the weight of a leaf is the probability of the query point lying in the associated interval. Let pi denote the probability of falling in the ith interval. A fundamental information... |

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Citation Context ...ber of holes, and the total boundary length of each region is bounded by a constant. We construct a hierarchical decomposition of U by building a box-decomposition tree (BD-tree) on the vertices of S =-=[4, 22]-=-. Initially, the BD-tree contains only one node which is the root. Each node represents a cell and the root represents U. We keep expanding the tree until some terminating condition is satisfied. The ... |

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Citation Context ...ent given by Vaidya leads to a construction time of O(n log n). (We mention that we can achieve the same construction time without using non-algebraic operations by building the sliding-midpoint tree =-=[16, 18]-=- instead. It can be shown that Lemma 1 holds for the fragments induced by the leaves of the sliding-midpoint tree. The query algorithm and the rest of the analysis given in this section can also be ea... |

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Citation Context ...ctical methods have been proposed. These include Kirkpatrick’s clever hierarchical method [12], the separator method by Edelsbrunner et al. [8], the persistent search tree method by Sarnak and Tarjan =-=[21]-=-, and the randomized incremental method by Mulmuley [20]. All of these are based on worst-case analyses. Recently Adamy and Seidel [1] presented an O(n) space data structure that achieves a worst-case... |

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Citation Context ...on problem, let n denote the number of vertices in the subdivision. The early work of Dobkin and Lipton [6] showed that a query time of O(log n) and space O(n 2 ) could be achieved. Lipton and Tarjan =-=[15]-=- showed that the space requirement could be reduced to O(n), but their approach was rather impractical. Since then a number of more practical methods have been proposed. These include Kirkpatrick’s cl... |

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Citation Context ...oach was rather impractical. Since then a number of more practical methods have been proposed. These include Kirkpatrick’s clever hierarchical method [12], the separator method by Edelsbrunner et al. =-=[8]-=-, the persistent search tree method by Sarnak and Tarjan [21], and the randomized incremental method by Mulmuley [20]. All of these are based on worst-case analyses. Recently Adamy and Seidel [1] pres... |

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Citation Context ...ny binary tree (and hence the expected number of comparisons) is at least the entropy of the probability distribution � � � 1 pi log . pi i (Unless otherwise stated, all logarithms are base 2.) Knuth =-=[13]-=- shows how to construct an optimum binary search tree in O(n 2 ) time using dynamic programming. Hu and Tucker [11] presented a bottom-up construction of the tree, which takes O(n log n) time, but is ... |

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Citation Context ...the plane into (bounded and unbounded) convex polygons. 3 Background For the planar point location problem, let n denote the number of vertices in the subdivision. The early work of Dobkin and Lipton =-=[6]-=- showed that a query time of O(log n) and space O(n 2 ) could be achieved. Lipton and Tarjan [15] showed that the space requirement could be reduced to O(n), but their approach was rather impractical.... |

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Citation Context ...ber of holes, and the total boundary length of each region is bounded by a constant. We construct a hierarchical decomposition of U by building a box-decomposition tree (BD-tree) on the vertices of S =-=[4, 22]-=-. Initially, the BD-tree contains only one node which is the root. Each node represents a cell and the root represents U. We keep expanding the tree until some terminating condition is satisfied. The ... |

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Citation Context ...rick’s clever hierarchical method [12], the separator method by Edelsbrunner et al. [8], the persistent search tree method by Sarnak and Tarjan [21], and the randomized incremental method by Mulmuley =-=[20]-=-. All of these are based on worst-case analyses. Recently Adamy and Seidel [1] presented an O(n) space data structure that achieves a worst-case query time of log n + 2 √ log n + O(log 1/4 n) point-li... |

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Citation Context ...f computation in which exclusive-or, integer floor, powers of 2, and integer logarithm can be computed on point coordinates, the shrink operation can be performed in O(d) time. (For example, see Bern =-=[3]-=-). Straightforward modification of the argument given by Vaidya leads to a construction time of O(n log n). (We mention that we can achieve the same construction time without using non-algebraic opera... |

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Citation Context ...ues. Their method may use Θ(n 2 ) space in the worst case. Methods using kd-trees, quad-trees, and R-trees are also popular in practice, but their analyses do not hold in the worst case. Mucke et al. =-=[19]-=- and Devroye et al. [5] have analyzed methods based on walking through subdivisions. For Delaunay triangulations of uniformly distributed data sets, these methods take expected time close to O(n 1/3 )... |

26 |
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Citation Context ...an optimum binary search tree in O(n 2 ) time using dynamic programming. Hu and Tucker [11] presented a bottom-up construction of the tree, which takes O(n log n) time, but is quite complex. Mehlhorn =-=[17]-=- gives a simple construction of a binary search tree whose weighted path length is within a constant additive factor of the entropy-based lower bound. It is eminently natural to ask whether these resu... |

23 | A note on point location in Delaunay triangulations of random points, Algorithmica 22
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Citation Context ...e Θ(n 2 ) space in the worst case. Methods using kd-trees, quad-trees, and R-trees are also popular in practice, but their analyses do not hold in the worst case. Mucke et al. [19] and Devroye et al. =-=[5]-=- have analyzed methods based on walking through subdivisions. For Delaunay triangulations of uniformly distributed data sets, these methods take expected time close to O(n 1/3 ) and O(n 1/4 ) in two a... |

20 | Methods for achieving fast query times in point location data structures
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Citation Context ...divisions. For Delaunay triangulations of uniformly distributed data sets, these methods take expected time close to O(n 1/3 ) and O(n 1/4 ) in two and three dimensions, respectively. Goodrich et al. =-=[10]-=- presented an interesting point location method, which adapts to the query distribution. Intuitively, if a cell is accessed more frequently, then the data structure is modified to ensure that the time... |

18 | Analysis of approximate nearest neighbor searching with clustered point sets. Data structures, near neighbor searches, and methodology: fifth and sixth DIMACS implementation challenges: papers related to the DIMACS challenge on dictionaries and priority q
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(Show Context)
Citation Context ...ent given by Vaidya leads to a construction time of O(n log n). (We mention that we can achieve the same construction time without using non-algebraic operations by building the sliding-midpoint tree =-=[16, 18]-=- instead. It can be shown that Lemma 1 holds for the fragments induced by the leaves of the sliding-midpoint tree. The query algorithm and the rest of the analysis given in this section can also be ea... |

17 |
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Citation Context ... theoretic lower bound.sExisting work on expected case-performance has been based on the assumption that both the subdivision and the queries satisfy certain probabilistic assumptions. Edahiro et al. =-=[7]-=- proposed a practical algorithm for planar point location based on bucketing techniques. Their method may use Θ(n 2 ) space in the worst case. Methods using kd-trees, quad-trees, and R-trees are also ... |

8 |
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Citation Context ...q, let xq and yq denote its x and y coordinate. Throughout this paper, we assume that xq and yq are two independent random variables. We denote the probability distribution function for xq by P : I → =-=[0, 1]-=- and the probability distribution function for yq by Q : J → [0, 1]. That is, P(x) is the probability that the random variable xq is less than or equal to x, and Q(y) is the probability that the rando... |

8 | Expected-case complexity of approximate nearest neighbor searching
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- 2000
(Show Context)
Citation Context ...esent another algorithm which reduces the space to O(n) and the preprocessing time to O(n log n). The expected number of point-line comparisons goes up to nearly 4entropy(S) + O(1). A lemma proved in =-=[2]-=- will be very useful. We state it in a form which is applicable in two dimensions. The result concerns with overlaying two planar subdivisions of U. One subdivision is the given planar subdivision S o... |

7 |
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(Show Context)
Citation Context ...ections between ℓ and the boundary of z in O(|z|) time by brute-force. Second, apply Jordan sorting to sort these intersections in order of their appearance on ℓ. This can also be done in O(|z|) time =-=[9]-=-. Third, start a clockwise traversal from some vertex of z within the halfplane. If we come to an intersection on ℓ, then we use the sorted list of intersections to jump to the next intersection along... |

7 |
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(Show Context)
Citation Context ...on � � � 1 pi log . pi i (Unless otherwise stated, all logarithms are base 2.) Knuth [13] shows how to construct an optimum binary search tree in O(n 2 ) time using dynamic programming. Hu and Tucker =-=[11]-=- presented a bottom-up construction of the tree, which takes O(n log n) time, but is quite complex. Mehlhorn [17] gives a simple construction of a binary search tree whose weighted path length is with... |