Results 11  20
of
51
Methods for Achieving Fast Query Times in Point Location Data Structures
, 1997
"... Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data struc ..."
Abstract

Cited by 22 (1 self)
 Add to MetaCart
(Show Context)
Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data structures can be constructed so as to achieve O(log n) query times. But applications, such as those common in geographic information systems, motivate a reexamination of this problem with the goal of improving query times further while also simplifying the methods needed to achieve such query times. In this paper we perform such a reexamination, focusing on the issues that arise in three different classes of pointlocation query sequences: ffl sequences that are reasonably uniform spatially and temporally (in which case the constant factors in the query times become critical), ffl sequences that are nonuniform spatially or temporally (in which case one desires data structures that adapt to s...
A simple entropybased algorithm for planar point location
 In Proceedings of the Twelfth Annual ACMSIAM Symposium on Discrete Algorithms
, 2001
"... Abstract Given a planar polygonal subdivision S, point location involves preprocessing this subdivisioninto a data structure so that given any query point q, the cell of the subdivision containing qcan be determined efficiently. Suppose that for each cell z in the subdivision, the probability pz tha ..."
Abstract

Cited by 22 (4 self)
 Add to MetaCart
Abstract Given a planar polygonal subdivision S, point location involves preprocessing this subdivisioninto a data structure so that given any query point q, the cell of the subdivision containing qcan be determined efficiently. Suppose that for each cell z in the subdivision, the probability pz that a query point lies within this cell is also given. The goal is to design the data structureto minimize the average search time. This problem has been considered before, but existing
Purely Functional Representations of Catenable Sorted Lists.
 In Proceedings of the 28th Annual ACM Symposium on Theory of Computing
, 1996
"... The power of purely functional programming in the construction of data structures has received much attention, not only because functional languages have many desirable properties, but because structures built purely functionally are automatically fully persistent: any and all versions of a structur ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
The power of purely functional programming in the construction of data structures has received much attention, not only because functional languages have many desirable properties, but because structures built purely functionally are automatically fully persistent: any and all versions of a structure can coexist indefinitely. Recent results illustrate the surprising power of pure functionality. One such result was the development of a representation of doubleended queues with catenation that supports all operations, including catenation, in worstcase constant time [19].
Nearly Optimal ExpectedCase Planar Point Location
"... We consider the planar point location problem from the perspective of expected search time. We are given a planar polygonal subdivision S and for each polygon of the subdivision the probability that a query point lies within this polygon. The goal is to compute a search structure to determine which ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
(Show Context)
We consider the planar point location problem from the perspective of expected search time. We are given a planar polygonal subdivision S and for each polygon of the subdivision the probability that a query point lies within this polygon. The goal is to compute a search structure to determine which cell of the subdivision contains a given query point, so as to minimize the expected search time. This is a generalization of the classical problem of computing an optimal binary search tree for onedimensional keys. In the onedimensional case it has long been known that the entropy H of the distribution is the dominant term in the lower bound on the expectedcase search time, and further there exist search trees achieving expected search times of at most H + 2. Prior to this work, there has been no known structure for planar point location with an expected search time better than 2H, and this result required strong assumptions on the nature of the query point distribution. Here we present a data structure whose expected search time is nearly equal to the entropy lower bound, namely H + o(H). The result holds for any polygonal subdivision in which the number of sides of each of the polygonal cells is bounded, and there are no assumptions on the query distribution within each cell. We extend these results to subdivisions with convex cells, assuming a uniform query distribution within each cell.
On the Exact Worst Case Query Complexity of Planar Point Location
 IN PROCEEDINGS OF THE NINTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 1998
"... 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. pointline 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 ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
(Show Context)
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. pointline 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).
Purely Functional, RealTime Deques with Catenation
 Journal of the ACM
, 1999
"... We describe an efficient, purely functional implementation of deques with catenation. In addition to being an intriguing problem in its own right, finding a purely functional implementation of catenable deques is required to add certain sophisticated programming constructs to functional programming ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
(Show Context)
We describe an efficient, purely functional implementation of deques with catenation. In addition to being an intriguing problem in its own right, finding a purely functional implementation of catenable deques is required to add certain sophisticated programming constructs to functional programming languages. Our solution has a worstcase running time of O(1) for each push, pop, inject, eject and catenation. The best previously known solution has an O(log k) time bound for the k deque operation. Our solution is not only faster but simpler. A key idea used in our result is an algorithmic technique related to the redundant digital representations used to avoid carry propagation in binary counting.
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 ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
(Show Context)
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...
EntropyPreserving Cuttings and SpaceEfficient Planar Point Location
 In Proceedings of the Twelfth Annual ACMSIAM Symposium on Discrete Algorithms
, 2001
"... Point location is the problem of preprocessing a planar polygonal subdivision S into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point. Given the probabilities pz that the query point lies within each cell z 2 S, a natural question is ho ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
(Show Context)
Point location is the problem of preprocessing a planar polygonal subdivision S into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point. Given the probabilities pz that the query point lies within each cell z 2 S, a natural question is how to design such a structure so as to minimize the expectedcase query time. The entropy H of the probability distribution is the dominant term in the lower bound on the expectedcase search time. Clearly the number of edges n of the subdivision is a lower bound on the space required. There is no known approach that simultaneously achieves the goals of H + o(H) query time and O(n) space. In this paper we introduce entropypreserving cuttings and show how to use them to achieve query time H+o(H), using only O(n log n) space. 1 Introduction Planar point location is an important problem in computational geometry. We are given a polygonal subdivision S consisting of n edges, and the goal is ...
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., &QUOT;PERSISTENCY IN COMPUTATIONAL GEOMETRY,&QUOT; PROC. 7TH CANADIAN CONF. COMP. GEOMETRY, QUEBEC
, 1995
"... ..."
Stabbing and Ray Shooting in 3 Dimensional Space
, 1994
"... In this paper we consider the following problems: given a set T of triangles in 3space, with jT j = n, a) answer the query " given a line l, does l stab the set of triangles?" (query problem). b) find whether a stabbing line exists for the set of triangles (existence problem). c) Given ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
In this paper we consider the following problems: given a set T of triangles in 3space, with jT j = n, a) answer the query " given a line l, does l stab the set of triangles?" (query problem). b) find whether a stabbing line exists for the set of triangles (existence problem). c) Given a ray ae, which is the first triangle in T hit by ae? The following results are shown. 1. There is an \Omega\Gamma n 3 ) lower bound on the descriptive complexity of the set of all stabbers for a set of triangles. 2. The existence problem for triangles on a set of planes with g different plane inclinations can be solved in O(g 2 n 2 log n) time (Theorem 2). 3. The query problem is solvable in quasiquadratic O(n 2+ffl ) preprocessing and storage and logarithmic O(log n) query time (Theorem 4). 4. All stabbing results for triangles extend, with the same asymptotic bounds, to sets of convex polyhedra with total complexity n. 5. Using O(n 3+ffl ) preprocessing time and storage we can det...