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26
External Memory Data Structures
, 2001
"... In many massive dataset applications the data must be stored in space and query efficient data structures on external storage devices. Often the data needs to be changed dynamically. In this chapter we discuss recent advances in the development of provably worstcase efficient external memory dynami ..."
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Cited by 81 (36 self)
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In many massive dataset applications the data must be stored in space and query efficient data structures on external storage devices. Often the data needs to be changed dynamically. In this chapter we discuss recent advances in the development of provably worstcase efficient external memory dynamic data structures. We also briefly discuss some of the most popular external data structures used in practice.
Dynamic Trees and Dynamic Point Location
 In Proc. 23rd Annu. ACM Sympos. Theory Comput
, 1991
"... This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using ..."
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Cited by 44 (9 self)
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This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the linkcut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of kedge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial pointlocation in a 3dimensional convex subdivision. In addition, the interlacedtree approach is applied to online pointlo...
I/OEfficient Dynamic Planar Point Location
"... We present the first provably I/Oefficient dynamic data structure for point location in a general planar subdivision. Our structure uses O(N/B) disk blocks to store a subdivision of size N , where B is the disk block size. Queries can be answered in ... I/Os in the worstcase, and insertions and de ..."
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Cited by 29 (17 self)
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We present the first provably I/Oefficient dynamic data structure for point location in a general planar subdivision. Our structure uses O(N/B) disk blocks to store a subdivision of size N , where B is the disk block size. Queries can be answered in ... I/Os in the worstcase, and insertions and deletions can be performed in ... and ... I/Os amortized, respectively. Previously, an I/Oefficient dynamic point location structure was only known for monotone subdivisions. Part of our data structure...
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 ..."
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Cited by 22 (1 self)
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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 Unified Approach to Dynamic Point Location, Ray Shooting, and Shortest Paths in Planar Maps
, 1992
"... We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map M with 7 ~ vertices, and apply it to the development of a unified dynamic data structure that supports pointlocation, rayshooting, and shortestpath queries in M. The space requirement i ..."
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Cited by 21 (6 self)
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We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map M with 7 ~ vertices, and apply it to the development of a unified dynamic data structure that supports pointlocation, rayshooting, and shortestpath queries in M. The space requirement is O(nlog n). Pointlocation queries take time O(log 7~). Rayshooting and shortestpath queries take time O(log3 TZ) (plus O(k) time if the k edges of the shortest path are reported in addition to its length). Updates consist of insertions and deletions of vertices and edges, and take O(log3 n) time (amortized for vertex updates).
I/OEfficient Dynamic Point Location in Monotone Planar Subdivisions (Extended Abstract)
"... We present an efficient externalmemory dynamic data structure for point location in monotone planar subdivisions. Our data structure uses O(N=B) disk blocks to store a monotone subdivision of size N, where B is the size of a disk block. It supports queries in O(log2B N) I/Os (worstcase) and upda ..."
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Cited by 19 (15 self)
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We present an efficient externalmemory dynamic data structure for point location in monotone planar subdivisions. Our data structure uses O(N=B) disk blocks to store a monotone subdivision of size N, where B is the size of a disk block. It supports queries in O(log2B N) I/Os (worstcase) and updates in O(log2B N) I/Os (amortized). We also
Dynamic and I/OEfficient Algorithms for Computational Geometry and Graph Problems: Theoretical and Experimental Results
, 1995
"... As most important applications today are largescale in nature, highperformance methods are becoming indispensable. Two promising computational paradigms for largescale applications are dynamic and I/Oefficient computations. We give efficient dynamic data structures for several fundamental proble ..."
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Cited by 18 (4 self)
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As most important applications today are largescale in nature, highperformance methods are becoming indispensable. Two promising computational paradigms for largescale applications are dynamic and I/Oefficient computations. We give efficient dynamic data structures for several fundamental problems in computational geometry, including point location, ray shooting, shortest path, and minimumlink path. We also develop a collection of new techniques for designing and analyzing I/Oefficient algorithms for graph problems, and illustrate how these techniques can be applied to a wide variety of specific problems, including list ranking, Euler tour, expressiontree evaluation, leastcommon ancestors, connected and biconnected components, minimum spanning forest, ear decomposition, topological sorting, reachability, graph drawing, and visibility representation. Finally, we present an extensive experimental study comparing the practical I/O efficiency of four algorithms for the orthogonal s...
No Quadrangulation is Extremely Odd
, 1995
"... Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if a ..."
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Cited by 16 (4 self)
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Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points. If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n log n) time even in the presence of collinear points. If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. We also provide an\Omega (n log n) time lower bound for the problem. Finally, our results imply that a kangulation of a set of points can be achieved with the addition of at most k \Gamma 3 extra points within the same time bound.
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 ..."
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Cited by 14 (0 self)
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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).
Dynamization of the Trapezoid Method for Planar Point Location
, 1991
"... We present a fully dynamic data structure for point location in a monotone subdivision, based on the trapezoid method. The operations supported are insertion and deletion of vertices and edges, and horizontal translation of vertices. Let n be the current number of vertices of the subdivision. Point ..."
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Cited by 14 (4 self)
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We present a fully dynamic data structure for point location in a monotone subdivision, based on the trapezoid method. The operations supported are insertion and deletion of vertices and edges, and horizontal translation of vertices. Let n be the current number of vertices of the subdivision. Point location queries take O(log n) time, while updates take O(log2 n) time. The space requirement is O(n log n). This is the first fully dynamic point location data structure for monotone subdivisions that achieves optimal query time.