## On the Exact Worst Case Query Complexity of Planar Point Location (1998)

### Cached

### Download Links

- [www-tcs.cs.uni-sb.de]
- [www.inf.ethz.ch]
- [www.cs.ubc.ca]
- DBLP

### Other Repositories/Bibliography

Venue: | IN PROCEEDINGS OF THE NINTH ANNUAL ACM-SIAM SYMPOSIUM ON DISCRETE ALGORITHMS |

Citations: | 14 - 0 self |

### BibTeX

@INPROCEEDINGS{Adamy98onthe,

author = {Udo Adamy and Raimund Seidel},

title = {On the Exact Worst Case Query Complexity of Planar Point Location},

booktitle = {IN PROCEEDINGS OF THE NINTH ANNUAL ACM-SIAM SYMPOSIUM ON DISCRETE ALGORITHMS},

year = {1998},

pages = {609--618},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

What is the smallest constant c so that the planar point location queries can be answered in c log 2 n + o(log n) steps (i.e. point-line comparisons) in the worst case? In SODA 97 Goodrich, Orletsky, and Ramaiyer [6] showed that c = 2 is possible using linear space and conjectured this to be optimal. We disprove this conjecture and show that c = 1 can be achieved. Moreoever by giving upper and lower bounds we show that without space restrictions the worst case query complexity of planar point location differs from log 2 n + 2 p log 2 n at most by an additive factor of (1=2)log 2 log 2 n +O(1). For the case of linear space we show the query complexity to be bounded by log 2 n + 2 p log 2 n +O(log 1=4 n).

### Citations

256 |
Optimal search in planar subdivisions
- Kirkpatrick
- 1983
(Show Context)
Citation Context ...Informatik, Univ. des Saarlandes, Postfach 151150, D-66041 Saarbrucken, GERMANY. seidel@cs.uni-sb.de A fair number of such "practical" methods have been found since. Kirkpatrick's hierarchic=-=al method [7]-=- made the beginning; the method of Cole [2] based on searching similar lists followed; the separator based method of Edelsbrunner, Guibas, and Stolfi [5], the method of Sarnak and Tarjan [14] based on... |

173 |
Planar point location using persistent search trees
- Sarnak, Tarjan
- 1986
(Show Context)
Citation Context ...al method [7] made the beginning; the method of Cole [2] based on searching similar lists followed; the separator based method of Edelsbrunner, Guibas, and Stolfi [5], the method of Sarnak and Tarjan =-=[14]-=- based on persistent search trees, and the randomized methods of Mulmuley [12] and Seidel [15]. All these methods are more or less easy to explain and to implement and they allow queries to be answere... |

163 |
Tarjan, Applications of a planar separator theorem
- Lipton, E
- 1980
(Show Context)
Citation Context ...dges in the subdivision. The question whether such a logarithmic query time could be attained using just linear space was one of the first stepping stones in computational geometry. Lipton and Tarjan =-=[10]-=- answered this question in the affirmative, giving a rather involved construction based on their planar separator theorem. However, their answer was satisfactory only from a theoretical point of view.... |

158 |
Optimal point location in a monotone subdivision
- Edelsbrunner, Guibas, et al.
- 1986
(Show Context)
Citation Context ...found since. Kirkpatrick's hierarchical method [7] made the beginning; the method of Cole [2] based on searching similar lists followed; the separator based method of Edelsbrunner, Guibas, and Stolfi =-=[5]-=-, the method of Sarnak and Tarjan [14] based on persistent search trees, and the randomized methods of Mulmuley [12] and Seidel [15]. All these methods are more or less easy to explain and to implemen... |

100 | A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Computational Geometry: Theory and Applications
- Seidel
- 1991
(Show Context)
Citation Context ...owed; the separator based method of Edelsbrunner, Guibas, and Stolfi [5], the method of Sarnak and Tarjan [14] based on persistent search trees, and the randomized methods of Mulmuley [12] and Seidel =-=[15]-=-. All these methods are more or less easy to explain and to implement and they allow queries to be answered in c \Delta log 2 n + o(log n) steps 1 for some reasonably small c, with the record c = 3 gi... |

93 |
Multidimensional searching problems
- Dobkin, Lipton
- 1976
(Show Context)
Citation Context ...ision of the plane one is to construct a data structure so that queries of the form "which polygon of the subdivision contains a given query point" can be answered quickly. Early on Dobkin a=-=nd Lipton [3]-=- showed that a query time of O(log n) is achievable, although at the expense of O(n 2 ) space. Here n is the number of edges in the subdivision. The question whether such a logarithmic query time coul... |

82 |
A fast planar partition algorithm
- Mulmuley
- 1990
(Show Context)
Citation Context ...milar lists followed; the separator based method of Edelsbrunner, Guibas, and Stolfi [5], the method of Sarnak and Tarjan [14] based on persistent search trees, and the randomized methods of Mulmuley =-=[12]-=- and Seidel [15]. All these methods are more or less easy to explain and to implement and they allow queries to be answered in c \Delta log 2 n + o(log n) steps 1 for some reasonably small c, with the... |

61 |
Location of a Point in a Planar Subdivision and its Applications
- Lee, Preparata
- 1977
(Show Context)
Citation Context ...ethods of Mulmuley [12] and Seidel [15]. The query process of most other planar point location methods can be modelled by trapezoidal search graphs. Examples are the chain method of Lee and Preparata =-=[9]-=-, the interval tree based method of Cheng and Janardan [1], and the brute force method of comparing every segment in S to the query point q. The most prominent example of an approach not modelled by t... |

49 |
Searching and storing similar lists
- Cole
- 1986
(Show Context)
Citation Context ...151150, D-66041 Saarbrucken, GERMANY. seidel@cs.uni-sb.de A fair number of such "practical" methods have been found since. Kirkpatrick's hierarchical method [7] made the beginning; the metho=-=d of Cole [2]-=- based on searching similar lists followed; the separator based method of Edelsbrunner, Guibas, and Stolfi [5], the method of Sarnak and Tarjan [14] based on persistent search trees, and the randomize... |

33 |
A New Approach to Planar Point Location
- Preparata
- 1981
(Show Context)
Citation Context ...eprocessing to O(n log n) while maintaining a query time of log 2 n + 2 p log 2 n + O(log 1=4 n). Our basic method uses only simple tools, namely a variation on Preparata's so-called trapezoid method =-=[13]-=- and weighted search trees. In the next section we present some fundamentals and briefly outline a method by which one can easily see why the 2log 2 n bound on the query time can be beaten and also ho... |

31 | Sorting and Searching, Volume 1 of Data Structures and Algorithms - Mehlhorn - 1984 |

26 |
New Results on Dynamic Planar Point Location
- Cheng, Janardan
- 1990
(Show Context)
Citation Context ... of most other planar point location methods can be modelled by trapezoidal search graphs. Examples are the chain method of Lee and Preparata [9], the interval tree based method of Cheng and Janardan =-=[1]-=-, and the brute force method of comparing every segment in S to the query point q. The most prominent example of an approach not modelled by trapezoidal search graphs is Kirkpatrick's hierarchical met... |

21 | Methods for achieving fast query times in point location data structures
- Goodrich, Orletsky, et al.
- 1997
(Show Context)
Citation Context ...mallest constant c so that the planar point location queries can be answered in c log 2 n + o(log n) steps (i.e. point-line comparisons) in the worst case? In SODA 97 Goodrich, Orletsky, and Ramaiyer =-=[6]-=- showed that c = 2 is possible using linear space and conjectured this to be optimal. We disprove this conjecture and show that c = 1 can be achieved. Moreoever by giving upper and lower bounds we sho... |

9 |
Alphabetic minimax trees
- Kirkpatrick, Klawe
- 1985
(Show Context)
Citation Context ...r n+1 = +1. Associate with each interval I i a positive weight w i , and let W = P i w i . We will make extensive use of the following weighted search tree lemma (for a proof see e.g. [11, p. 180] or =-=[8]-=-): Lemma 2.1. For any weighted sequence as described above there is a search tree T with interior nodes corresponding to the r i 's (1sisn) and leaves corresponding to the intervals I i 's (0sisn), so... |

4 |
A new point-location algorithm and its practical efficiency: comparison with existing algorithms
- Asano
- 1984
(Show Context)
Citation Context ...minent example of an approach not modelled by trapezoidal search graphs is Kirkpatrick's hierarchical method [7], since it makes comparisons of q against segments that are not in S. Bucketing methods =-=[4]-=- are also not modelled by trapezoidal search graphs. Every search method that can be modelled by a trapezoidal search graph can clearly also be modelled by a trapezoidal search graph that is a tree. I... |

3 |
Parallel evaluation of division-free arithmetic expressions
- Kosaraju
- 1986
(Show Context)
Citation Context ...lgebraic expressions. We discovered this connection after Ian Munro had pointed out to us that upper and lower bounds of the form log n+ \Theta( p log n) had also appeared in the work of Rao Kosaraju =-=[9]-=-. Suppose E is a fully parenthesized arithmetic expression involving only variables and the binary operations of multiplication and addition. Let us identify expression E with its natural binary tree ... |