### BibTeX

@MISC{Agarwal96rangesearching,

author = {Pankaj K. Agarwal},

title = { Range Searching},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a range-searching problem. A typical range-searching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.

### Citations

2435 |
The Design and Analysis of Computer Algorithms
- Aho, Hopcroft, et al.
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(Show Context)
Citation Context ...ntial, which makes them unsuitable for large values of d. We assume that each memory cell can store log n bits. The upper bounds will be given on pointer-machine or RAM models, which are described in =-=[15, 107]-=-. The main difference between the two models is that on the pointer machine a memory cell can be accessed only through a series of pointers while in the RAM model any memory cell can be accessed in co... |

2220 | R-trees: A dynamic index structure for spatial searching - Guttman - 1984 |

1989 |
Robot Motion Planning
- Latombe
- 1990
(Show Context)
Citation Context ...2 m is a region K i ` R k . If B and the obstacles are semialgebraic sets, then each K i is also a semialgebraic set. A placement p of B is free if and only if p does not intersect any of K i 's. See =-=[76]-=- for a survey of known results on the collision-detection problem and [11, 37, 38] for a few other applications of point intersection-searching structures. SEGMENT INTERSECTION SEARCHING Preprocess a ... |

1763 |
Computational Geometry: An Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...s numerous applications, orthogonal range searching has been studied extensively for the last 25 years. A survey of earlier results can be found in the books by Mehlhorn [88] and Preparata and Shamos =-=[99]-=-. In this section we review the more recent data structures and the lower bounds. GLOSSARY EPM A pointer machine with + operation. APM A pointer machine with basic arithmetic and shift operations. Fai... |

1115 | Multidimensional binary search trees used for associative searching - Bentley - 1975 |

995 |
Computer Graphics : Principles and Practice
- Foley
- 1990
(Show Context)
Citation Context ...ray-shooting data structures. ffl Preparata and Shamos [99]: A text book on basic topics in computational geometry. Chapters 2 includes earlier results on orthogonal range searching. ffl Foley et al. =-=[58]-=-: A text book on graphics. Discusses practical data structures for ray tracing and intersection searching. ffl de Berg [50]: A monograph on ray shooting and related problems. ffl Schwarzkopf [103]: Th... |

972 | Computational Geometry Algorithms and Applications - Berg, Cheong, et al. - 2008 |

696 | Data cube: A relational aggregation operator generalizing group-by, cross-tab, and sub-totals - Gray, Chaudhuri, et al. - 1997 |

604 |
Data Structures and Network Algorithms
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(Show Context)
Citation Context ...ntial, which makes them unsuitable for large values of d. We assume that each memory cell can store log n bits. The upper bounds will be given on pointer-machine or RAM models, which are described in =-=[15, 107]-=-. The main difference between the two models is that on the pointer machine a memory cell can be accessed only through a series of pointers while in the RAM model any memory cell can be accessed in co... |

585 | An algorithm for finding best matches in logarithmic expected time
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- 1977
(Show Context)
Citation Context ... be counted. Its query time is O(log n+1=ffi d\Gamma1 ). Overmars and van der Stappen [97] developed fast data structures for the special case in which the ranges are `fat' and have bounded size. See =-=[62, 70]-=- for some other `heuristic based' data structures. We conclude this subsection by noting that better bounds can be obtained for the halfspace range-reporting problem, using the so-called filtering sea... |

563 | Multi- dimensional Access Methods - Gaede, Gunther - 1998 |

383 | The grid file: an adaptable, symmetric multikey file structure - Nievergelt, Hinterberger, et al. - 1984 |

286 |
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- Mulmuley
- 1994
(Show Context)
Citation Context ... (1 + o(1))(d=") log 1=". The "-nets have turned out to be a powerful tool in developing divide-and-conquer algorithms for several geometric problems and in learning theory; see the boo=-=ks by Mulmuley [94]-=- and Anthony and Biggs [17]. Building on the theory developed by Haussler and Welzl, Welzl [111] proved that one can construct a spanning path of S of O(n 1\Gamma1=d log n) stabbing number; the bound ... |

267 | Quad trees a data structure for retrieval on composite keys - Finkel, Bentley - 1974 |

260 | Epsilon-nets and simplex range queries
- Haussler, Welzl
- 1987
(Show Context)
Citation Context ...g query in time O(n :792 + k). After a few initial improvements and extensions on Willard's data structure [55, 56, 49], a major breakthrough in simplex range searching was made by Haussler and Welzl =-=[68]-=-. They formulated the range searching in an abstract setting and, using elegant probabilistic methods, gave a randomized algorithm to construct a linear-size partition tree with O(n ff ) query time, w... |

251 | Geometric range searching and its relatives - Agrawal, Erickson - 1999 |

233 |
A combinatorial problem in geometry
- Erdős, Szekeres
- 1935
(Show Context)
Citation Context ... trees with low stabbing number. Matousek and Welzl [81] gave an entirely different algorithm for the halfspace range-counting problem in the plane, using a combinatorial result of Erdos and Szekeres =-=[57]-=-. The query time of their data structure is O( p n log n), and it uses O(n) space and O(n 3=2 ) preprocessing time. If subtractions are allowed, their algorithm can be extended to the triangle range-c... |

232 | Applying parallel computation algorithms in the design of serial algorithms
- Megiddo
- 1983
(Show Context)
Citation Context ...ave also been developed using ray-shooting data structures. A general approach to the ray-shooting problem, using segment intersectiondetection structures and Megiddo's parametric searching technique =-=[87]-=-, was proposed by Agarwal and Matousek [5]. The basic idea of their approach is as follows. Suppose we have a segment intersection-detection data structure for S, based on partition trees. Let ae be a... |

181 | The hb-tree: a multiattribute indexing method with good guaranteed performance - Lomet, Salzberg - 1990 |

176 | Priority Search Trees - McCreight - 1985 |

169 | Indexing Moving Points - Agarwal, Arge, et al. - 2000 |

154 | L.J.: Fractional cascading: I. A data structuring technique
- Chazelle, Guibas
- 1986
(Show Context)
Citation Context ... b i ]. Otherwise, we recursively visit both children of v. The query time of this procedure is O(log d n + k), which can be improved to O(log d\Gamma1 n+ k), using the fractional-cascading technique =-=[40, 77]-=-. A range tree can also answer a range- counting query in time O(log d\Gamma1 n). The best-known data structures for orthogonal range searching are by Chazelle [25, 27], who used compressed range tree... |

149 |
Data structures for range searching
- Bentley, Friedman
- 1979
(Show Context)
Citation Context ...HING In the d-dimensional orthogonal range searching, the ranges are d-rectangles, each of the form Q d i=1 [a i ; b i ], where a i ; b i 2 R. This is an abstraction of the `multikey ' searching; see =-=[21, 115]-=-. For example, the points of S may correspond to employees of a company, each coordinate corresponding to a key such as age, salary, experience, etc. The queries of the form --- report all employees b... |

132 | A functional approach to data structures and its use in multidimensional searching
- Chazelle
- 1988
(Show Context)
Citation Context ... fractional-cascading technique [40, 77]. A range tree can also answer a range- counting query in time O(log d\Gamma1 n). The best-known data structures for orthogonal range searching are by Chazelle =-=[25, 27]-=-, who used compressed range trees and other techniques (such as filtering search) to improve the storage and query time. His results in the plane, under various models of computation, are summarized i... |

129 |
The Design of Dynamic Data Structures
- Overmars
- 1983
(Show Context)
Citation Context ...he preprocessing time of the data structure. If we allow only insertions (i.e., a point cannot be deleted from the structure), the static data structure can be modified, using the standard techniques =-=[22, 95]-=-, so that a point can be inserted in time O(P (n) log n=n) and a query can be answered in time O(Q(n) log n), where Q(n) is the query time of the original static data structure; in some cases the loga... |

127 | Ray shooting and parametric search
- Agarwal, Matouˇsek
- 1993
(Show Context)
Citation Context ... S(n) Q(n) Source Notes Simple polygon n (k + 1) log n [69] Reporting d = 2 Segments m n= p m [8, 47] Counting Circles n 2+" log n [13] Counting Circular arcs m n=m 1=3 [13] Counting Planes m n=m=-= 1=3 [5] Coun-=-ting d = 3 Triangles m n=m 1=4 [6] Counting Spheres m n=m 1=4 [6] Counting Spheres n 3+" (k + 1) log 2 n [2] Reporting A special case of segment intersection searching, in which the objects are h... |

127 | Data Structures and Algorithms: 3. Multidimensional Searching and Computational Geometry - Mehlhorn - 1984 |

123 |
Multidimensional divide-and-conquer
- Bentley
- 1975
(Show Context)
Citation Context ...+) is a faithful semigroup, but (f0; 1g; \Phi) is not a faithful semigroup. UPPER BOUNDS Most of the recent orthogonal range-searching data structures are based on range trees , introduced by Bentley =-=[20]-=-. For d = 1, the range tree of S is an array storing S in a nondecreasing order. For d ? 1, let S 1 be the sequence of x-coordinates of points in S sorted in a nondecreasing order. The range tree of S... |

110 |
Heuristics for ray tracing using space subdivision
- MacDonald, Booth
- 1990
(Show Context)
Citation Context ...that does not intersect any object in S. Hershberger and Suri [69] showed that a triangulation with O(log n) query time can be constructed when S is the boundary of a simple polygon in the plane. See =-=[3, 91, 54, 78]-=- and the references therein for other ray-shooting results using this approach. Agarwal et al. [3] proved worst-case bounds for many cases on the number of cells in the subdivision that a line can int... |

107 | Filtering search: A new approach to query-answering
- Chazelle
- 1986
(Show Context)
Citation Context ... fractional-cascading technique [40, 77]. A range tree can also answer a range- counting query in time O(log d\Gamma1 n). The best-known data structures for orthogonal range searching are by Chazelle =-=[25, 27]-=-, who used compressed range trees and other techniques (such as filtering search) to improve the storage and query time. His results in the plane, under various models of computation, are summarized i... |

106 | Determining the Separation of Preprocessed Polyhedra - A Unified Approach
- Dobkin, Kirkpatrick
- 1990
(Show Context)
Citation Context ...that does not intersect any object in S. Hershberger and Suri [69] showed that a triangulation with O(log n) query time can be constructed when S is the boundary of a simple polygon in the plane. See =-=[3, 91, 54, 78]-=- and the references therein for other ray-shooting results using this approach. Agarwal et al. [3] proved worst-case bounds for many cases on the number of cells in the subdivision that a line can int... |

105 |
Computational Learning Theory
- Anthony, Biggs
- 1992
(Show Context)
Citation Context ...he "-nets have turned out to be a powerful tool in developing divide-and-conquer algorithms for several geometric problems and in learning theory; see the books by Mulmuley [94] and Anthony and B=-=iggs [17]-=-. Building on the theory developed by Haussler and Welzl, Welzl [111] proved that one can construct a spanning path of S of O(n 1\Gamma1=d log n) stabbing number; the bound was improved by Chazelle an... |

104 |
Decomposable searching problems I: Staticto-dynamic transformation
- Bentley, Saxe
- 1980
(Show Context)
Citation Context ...he preprocessing time of the data structure. If we allow only insertions (i.e., a point cannot be deleted from the structure), the static data structure can be modified, using the standard techniques =-=[22, 95]-=-, so that a point can be inserted in time O(P (n) log n=n) and a query can be answered in time O(Q(n) log n), where Q(n) is the query time of the original static data structure; in some cases the loga... |

102 | The Discrepancy Method: Randomness and Complexity - Chazelle - 2000 |

93 |
The power of geometric duality
- Chazelle, Guibas, et al.
- 1985
(Show Context)
Citation Context ...lfspace range-counting problem. Using a standard duality transform, this problem can be reduced to the Range Searching 11 TABLE 4 Halfspace range-searching. d S(n) Q(n) Source Notes d = 2 n log n + k =-=[41]-=- Reporting d = 3 n log n log n + k [14] Reporting d = 3 n log n [52] Emptiness d ? 3 n log log n n 1\Gamma1=bd=2c log c n [79] Reporting d ? 3 n n 1\Gamma1=d 2 O(log n) [79] Emptiness following proble... |

90 | Kinetic data structures: a state of the art report - Guibas - 1998 |

89 | Applications of parametric searching in geometric optimization
- Agarwal, Sharir, et al.
- 1994
(Show Context)
Citation Context ... then each K i is also a semialgebraic set. A placement p of B is free if and only if p does not intersect any of K i 's. See [76] for a survey of known results on the collision-detection problem and =-=[11, 37, 38]-=- for a few other applications of point intersection-searching structures. SEGMENT INTERSECTION SEARCHING Preprocess a set of objects in R d into a data structure so that all the objects of S intersect... |

85 | Approximate range searching
- Arya, Mount
(Show Context)
Citation Context ...lex range-searching data structure is only n 1=d factor faster than the naive method, researchers have developed practical data structures that work well most of the time. For example, Arya and Mount =-=[18]-=- have developed a linear-size data structure for answering approximate rangecounting queries, in the sense that the points lying within distance ffi \Delta diam(\Delta) distance of the boundary of the... |

85 |
A linear algorithm for determining the separation of convex polyhedra
- Dobkin, Kirkpatrick
- 1985
(Show Context)
Citation Context ...ons n (s 2 + n) log s p s log s log n [9, 69] s convex polygons sn log s log s log n [9] d = 2 Segments m n= p m [8, 47] Circlular arcs n n=m 1=3 [13] Disjoint arcs n p n [13] convex polytope n log n =-=[53] c-oriente-=-d polytopes n log n [51] s convex polytopes s 2 n 2+" log 2 n [10] d = 3 Halfplanes m n= p m [5] Terrain m n= p m [5, 39] Triangles m n=m 1=4 [6] Spheres n 3+" log 2 n [2] Hyperplanes m n=m ... |

84 |
A pedestrian approach to ray shooting: Shoot a ray, take a walk
- Hershberger, Suri
(Show Context)
Citation Context ...ve omitted polylogarithmic terms from the query-search time whenever it is of the form n=m ff . TABLE 10 Segment intersection searching d Objects S(n) Q(n) Source Notes Simple polygon n (k + 1) log n =-=[69] Repo-=-rting d = 2 Segments m n= p m [8, 47] Counting Circles n 2+" log n [13] Counting Circular arcs m n=m 1=3 [13] Counting Planes m n=m 1=3 [5] Counting d = 3 Triangles m n=m 1=4 [6] Counting Spheres... |

83 | Dynamic Algorithms in Computational Geometry
- Chiang, Tamassia
- 1991
(Show Context)
Citation Context ...arching. ffl Chazelle [32]: A general survey of recent developments in computational geometry. It contains most of the references on simplex and semialgebraic range searching. ffl Chiang and Tamassia =-=[48]-=-: A survey of dynamic data structures. ffl Goodman et al. [63]: A survey of stabbing problems and related topics. ffl Matousek [86]: A comprehensive survey of simplex range searching and related topic... |

83 | Cutting hyperplanes for divide-and-conquer - Chazelle - 1991 |

80 | Range searching with semi-algebraic sets - Agarwal, Matousek - 1994 |

79 |
The P-range tree: a new data structure for range searching in secondard memory
- Subramanian, Ramaswamy
- 1995
(Show Context)
Citation Context ... for the planar case. Table 8 summarizes the known results on secondary-memory structures for orthogonal range searching; here fi(n) = log log log B n. The data structure by Subramanian and Ramaswamy =-=[106]-=- for 3-sided queries supports insertion/deletion of a point in time O(log B n + (log B n) 2 =B). Extending the lower-bound proof by Chazelle [43], they also proved that any secondary-memory data struc... |

79 | External memory data structures - Arge - 2002 |

79 | On TwoDimensional Indexability and Optimal Range Search Indexing - Arge, Samoladas, et al. - 1999 |

70 |
A general approach to D-dimensional geometric queries
- Yao, Yao
- 1985
(Show Context)
Citation Context ...n n 1\Gamma 1 2d\Gamma3 +" [6] Partition tree Tarski cell n n 1\Gamma 1s+" [6] Linearization One approach to answer \Gamma f -range queries is to use linearization, originally proposed by Ya=-=o and Yao [118]-=-. We represent the polynomial f(x; a) in the form f(x; a) = / 0 (a) + / 1 (a)' 1 (x) + \Delta \Delta \Delta + / k (a)' k (x) where ' 1 ; : : : ; ' k ; / 0 ; : : : ; / k are real functions. A point x 2... |

69 |
Quasi-optimal upper bounds of simplex range searching and new zone theorems
- Chazelle, Sharir, et al.
- 1990
(Show Context)
Citation Context ...tinct if they do not contain the same subset of S). Since there are \Theta(n d(d+1) ) combinatorially distinct simplices, such an approach will require\Omega\Gamma n d(d+1) ) storage. Chazelle et al. =-=[44] showed th-=-at the size can be reduced to O(n d+" ), for any " ? 0, using a multi-level data structure. The space bound can be reduced to O(n d ) by increasing the query time to O(log d+1 n) [85] . Half... |

66 |
Analytical Geometry of Three Dimensions
- Sommerville
- 1947
(Show Context)
Citation Context ...in R 3 admit a linearization of dimension 9. One of the most widely used linearization in computational geometry is the so-called Plucker coordinates , which map a line in R 3 to a point in R 5 ; see =-=[39, 105]-=- for more details on Plucker coordinates. Agarwal and Matousek [6] have also proposed another approach to answer \Gamma f - range queries by extending Theorem 4 to Tarski cells and by constructing par... |

64 | Lower bounds for orthogonal range searching: I. The reporting case
- Chazelle
- 1990
(Show Context)
Citation Context ...a structure, using m units of storage, is \Omega\Gamma/151 n= log(2m=n)) d\Gamma1 ). In fact, Chazelle's lower bound holds even for the average-case complexity. A rather surprising result of Chazelle =-=[29]-=- shows that the size of any data structure on a 6 Pankaj K. Agarwal pointer machine that answers a d-dimensional range-reporting query in O(log c n+k) time, for any constant c, is \Omega\Gamma n(log n... |