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17
Range Searching
, 1996
"... 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 rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , an ..."
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Cited by 70 (1 self)
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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 rangesearching problem. A typical rangesearching 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.
Confluently Persistent Deques via DataStructural Bootstrapping
 J. of Algorithms
, 1993
"... We introduce datastructural bootstrapping, a technique to design data structures recursively, and use it to design confluently persistent deques. Our data structure requires O(log 3 k) worstcase time and space per deletion, where k is the total number of deque operations, and constant worstcase t ..."
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Cited by 15 (4 self)
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We introduce datastructural bootstrapping, a technique to design data structures recursively, and use it to design confluently persistent deques. Our data structure requires O(log 3 k) worstcase time and space per deletion, where k is the total number of deque operations, and constant worstcase time and space for other operations. Further, the data structure allows a purely functional implementation, with no side effects. This improves a previous result of Driscoll, Sleator, and Tarjan. 1 An extended abstract of this paper was presented at the 4th ACMSIAM Symposium on Discrete Algorithms, 1993. 2 Supported by a Fannie and John Hertz Foundation fellowship, National Science Foundation Grant No. CCR8920505, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) under NSFSTC8809648. 3 Also affiliated with NEC Research Institute, 4 Independence Way, Princeton, NJ 08540. Research at Princeton University partially supported by the National Science Foundatio...
Partially Persistent Data Structures of Bounded Degree with Constant Update Time
 Nordic Journal of Computing
, 1996
"... The problem of making bounded indegree and outdegree data structures partially persistent is considered. The node copying method of Driscoll et al. is extended so that updates can be performed in worstcase constant time on the pointer machine model. Previously it was only known to be possible in ..."
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Cited by 13 (3 self)
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The problem of making bounded indegree and outdegree data structures partially persistent is considered. The node copying method of Driscoll et al. is extended so that updates can be performed in worstcase constant time on the pointer machine model. Previously it was only known to be possible in amortised constant time [2]. The result is presented in terms of a new strategy for Dietz and Raman's dynamic two player pebble game on graphs. It is shown how to implement the strategy and the upper bound on the required number of pebbles is improved from 2b+2d+O( p b) to d+2b, where b is the bound of the indegree and d the bound of the outdegree. We also give a lower bound that shows that the number of pebbles depends on the outdegree d. This work was partially supported by the ESPRIT II Basic Research Actions Program of the EC under contract no. 7141 (project ALCOM II). y Basic Research in Computer Science, Centre of the Danish National Research Foundation. Introduction This...
Persistent data structures
 IN HANDBOOK ON DATA STRUCTURES AND APPLICATIONS, CRC PRESS 2001, DINESH MEHTA AND SARTAJ SAHNI (EDITORS) BOROUJERDI, A., AND MORET, B.M.E., "PERSISTENCY IN COMPUTATIONAL GEOMETRY," PROC. 7TH CANADIAN CONF. COMP. GEOMETRY, QUEBEC
, 1995
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Pattern Matching in Dynamic Texts
, 2000
"... Pattern matching is the problem of nding all occurrences of a pattern in a text. In a dynamic setting the problem is to support pattern matching in a text which can be manipulated online, i.e., the usual situation in text editing. We present a data structure that supports insertions and deletions ..."
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Cited by 6 (0 self)
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Pattern matching is the problem of nding all occurrences of a pattern in a text. In a dynamic setting the problem is to support pattern matching in a text which can be manipulated online, i.e., the usual situation in text editing. We present a data structure that supports insertions and deletions of characters and movements of arbitrary large blocks within a text in O(log 2 n log log n log n) time per operation. Furthermore a search for a pattern P in the text is supported in time O(log n log log n + occ + jP j), where occ is the number of occurrences to be reported. An ingredient in our solution to the above main result is a data structure for the dynamic string equality problem introduced by Mehlhorn, Sundar and Uhrig. As a secondary result we give almost quadratic better time bounds for this problem which in addition to keeping polylogarithmic factors low for our main result also improves the complexity for several other problems.
Fullydynamic orthogonal range reporting on RAM
, 2003
"... In a natural variant of the comparison model, we show that there exists a constant ! < 1 such that the fullydynamic ddimensional orthogonal range reporting problem for d 2 can be solved in time O(log n) for updates and time O((log n= log log n) + r) for queries. Here n is the number of p ..."
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Cited by 6 (2 self)
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In a natural variant of the comparison model, we show that there exists a constant ! < 1 such that the fullydynamic ddimensional orthogonal range reporting problem for d 2 can be solved in time O(log n) for updates and time O((log n= log log n) + r) for queries. Here n is the number of points stored and r is the number of points reported. The space usage is n). In the standard comparison model the result holds for d 3.
Persistent Predecessor Search and Orthogonal Point Location on the Word RAM
"... We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of int ..."
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Cited by 6 (3 self)
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We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in {1,..., U} under an arbitrary sequence of n insertions and deletions, with O(log log U) expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linearspace structures and improves previous nearO((log log U) 2) methods. The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(log log U) worstcase query time and linear space. This result is again optimal in U for linearspace structures and improves the previous O((log log U) 2) method by de Berg, Snoeyink, and van Kreveld (1992). The same result also holds for higherdimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions ≥ 3 on the RAM. Our key technique is an interesting new vanEmdeBoas–style recursion that alternates between two strategies, both quite simple.
Threedimensional layers of maxima
 Algorithmica
"... Abstract. We present an O(n log n)time algorithm to solve the threedimensional layersofmaxima problem, an improvement over the prior O(n log n log log n)time solution. A previous claimed O(n log n)time solution due to Atallah, Goodrich, and Ramaiyer [SCG’94] has technical flaws. Our algorithm i ..."
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Cited by 5 (0 self)
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Abstract. We present an O(n log n)time algorithm to solve the threedimensional layersofmaxima problem, an improvement over the prior O(n log n log log n)time solution. A previous claimed O(n log n)time solution due to Atallah, Goodrich, and Ramaiyer [SCG’94] has technical flaws. Our algorithm is based on a common framework underlying previous work, but to implement it we devise a new data structure to solve a special case of dynamic planar point location in a staircase subdivision. Our data structure itself relies on a new extension to dynamic fractional cascading that allows vertices of high degree in the control graph. 1
Optimal dynamic vertical ray shooting in rectilinear planar subdivisions
"... Optimal dynamic vertical ray shooting in rectilinear planar subdivisions. In this paper we consider the dynamic vertical ray shooting problem, that is the task of maintaining a dynamic set S of n non intersecting horizontal line segments in the plane subject to a query that reports the first segment ..."
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Cited by 4 (0 self)
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Optimal dynamic vertical ray shooting in rectilinear planar subdivisions. In this paper we consider the dynamic vertical ray shooting problem, that is the task of maintaining a dynamic set S of n non intersecting horizontal line segments in the plane subject to a query that reports the first segment in S intersecting a vertical ray from a query point. We develop a linearsize structure that supports queries, insertions and deletions in O(log n) worstcase time. Our structure works in the comparison model and uses a RAM.
SpaceEfficient Dynamic Orthogonal Point Location, Segment Intersection, and Range Reporting
, 2008
"... We describe an asymptotically optimal datastructure for dynamic point location for horizontal segments. For n linesegments, queries take O(log n) time, updates take O(log n) amortized time and the data structure uses O(n) space. This is the first structure for the problem that is optimal in space ..."
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Cited by 4 (2 self)
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We describe an asymptotically optimal datastructure for dynamic point location for horizontal segments. For n linesegments, queries take O(log n) time, updates take O(log n) amortized time and the data structure uses O(n) space. This is the first structure for the problem that is optimal in space and time (modulo the possibility of removing amortization). We also describe dynamic data structures for orthogonal range reporting and orthogonal intersection reporting. In both data structures for n points (segments) updates take O(log n) amortized time, queries take O(log n+k log n / log log n) time, and the structures use O(n) space, where k is the size of the output. The model of computation is the unit cost RAM.