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128
The quadtree and related hierarchical data structures
 ACM Computing Surveys
, 1984
"... A tutorial survey is presented of the quadtree and related hierarchical data structures. They are based on the principle of recursive decomposition. The emphasis is on the representation of data used in applications in image processing, computer graphics, geographic information systems, and robotics ..."
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Cited by 422 (11 self)
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A tutorial survey is presented of the quadtree and related hierarchical data structures. They are based on the principle of recursive decomposition. The emphasis is on the representation of data used in applications in image processing, computer graphics, geographic information systems, and robotics. There is a greater emphasis on region data (i.e., twodimensional shapes) and to a lesser extent on point, curvilinear, and threedimensional data. A number of operations in which such data structures find use are examined in greater detail.
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. ..."
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Cited by 139 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...
A separator theorem for graphs with an excluded minor and its applications
 IN PROCEEDINGS OF THE 22ND ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1990
"... Let G be an nvertex graph with nonnegative weights whose sum is 1 assigned to its vertices, and with no minor isomorphic to a given hvertex graph H. We prove that there is a set X of no more than h 3/2 n 1/2 vertices of G whose deletion creates a graph in which the total weight of every connected ..."
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Cited by 95 (1 self)
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Let G be an nvertex graph with nonnegative weights whose sum is 1 assigned to its vertices, and with no minor isomorphic to a given hvertex graph H. We prove that there is a set X of no more than h 3/2 n 1/2 vertices of G whose deletion creates a graph in which the total weight of every connected component is at most 1/2. This extends significantly a wellknown theorem of Lipton and Tarjan for planar graphs. We exhibit an algorithm which finds, given an nvertex graph G with weights as above and an hvertex graph H, either such a set X or a minor of G isomorphic to H. The algorithm runs in time O(h 1/2 n 1/2 m), where m is the number of edges of G plus the number of its vertices. Our results supply extensions of the many known applications of the LiptonTarjan separator theorem from the class of planar graphs (or that of graphs with bounded genus) to any class of graphs with an excluded minor. For example, it follows that for any fixed graph H, given a graph G with n vertices and with no Hminor one can approximate the size of the maximum independent set of G up to a relative error of 1 / √ log n in polynomial time, find that size exactly and find the chromatic number of G in time 2 O( √ n) and solve any sparse system of n linear equations in n unknowns whose sparsity structure 0 corresponds to G in time O(n 3/2). We also describe a combinatorial application of our result which relates the treewidth of a graph to the maximum size of a Khminor in it.
Optimal expectedtime algorithms for closest point problems
 ACM Transactions of Mathematical Software
, 1980
"... Geometric closest potnt problems deal with the proxLmity relationships in kdimensional point sets. Examples of closest point problems include building minimum spanning trees, nearest neighbor searching, and triangulation constructmn Shamos and Hoey [17] have shown how the Voronoi dtagram can be use ..."
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Cited by 88 (0 self)
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Geometric closest potnt problems deal with the proxLmity relationships in kdimensional point sets. Examples of closest point problems include building minimum spanning trees, nearest neighbor searching, and triangulation constructmn Shamos and Hoey [17] have shown how the Voronoi dtagram can be used to solve a number of planar closest point problems in optimal worst case tune. In this paper we extend thmr work by giving optimal expected.trine algorithms for solving a number of closest point problems in kspace, including nearest neighbor searching, finding all nearest neighbors, and computing planar minimum spanning trees. In addition to establishing theoretical bounds, the algorithms in this paper can be implemented to solve practical problems very efficiently. Key Words and Phrases ' computational geometry, closest point problems, minunum spanning trees, nearest neighbor searching, optimal algorithms, probabfllstm analysis of algorithms, Voronoi diagrams CR Categories: 3.74, 5 25, 5.31, 5.32 1.
Diameter and Treewidth in MinorClosed Graph Families
, 1999
"... It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a ..."
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Cited by 84 (3 self)
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It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minorclosed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to boundedgenus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph families.
Special Purpose Parallel Computing
 Lectures on Parallel Computation
, 1993
"... A vast amount of work has been done in recent years on the design, analysis, implementation and verification of special purpose parallel computing systems. This paper presents a survey of various aspects of this work. A long, but by no means complete, bibliography is given. 1. Introduction Turing ..."
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Cited by 77 (5 self)
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A vast amount of work has been done in recent years on the design, analysis, implementation and verification of special purpose parallel computing systems. This paper presents a survey of various aspects of this work. A long, but by no means complete, bibliography is given. 1. Introduction Turing [365] demonstrated that, in principle, a single general purpose sequential machine could be designed which would be capable of efficiently performing any computation which could be performed by a special purpose sequential machine. The importance of this universality result for subsequent practical developments in computing cannot be overstated. It showed that, for a given computational problem, the additional efficiency advantages which could be gained by designing a special purpose sequential machine for that problem would not be great. Around 1944, von Neumann produced a proposal [66, 389] for a general purpose storedprogram sequential computer which captured the fundamental principles of...
Nearest Neighbors In HighDimensional Spaces
, 2004
"... In this chapter we consider the following problem: given a set P of points in a highdimensional space, construct a data structure which given any query point q nds the point in P closest to q. This problem, called nearest neighbor search is of significant importance to several areas of computer sci ..."
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Cited by 76 (2 self)
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In this chapter we consider the following problem: given a set P of points in a highdimensional space, construct a data structure which given any query point q nds the point in P closest to q. This problem, called nearest neighbor search is of significant importance to several areas of computer science, including pattern recognition, searching in multimedial data, vector compression [GG91], computational statistics [DW82], and data mining. Many of these applications involve data sets which are very large (e.g., a database containing Web documents could contain over one billion documents). Moreover, the dimensionality of the points is usually large as well (e.g., in the order of a few hundred). Therefore, it is crucial to design algorithms which scale well with the database size as well as with the dimension. The nearestneighbor problem is an example of a large class of proximity problems, which, roughly speaking, are problems whose definitions involve the notion of...
The Vertex Separation And Search Number Of A Graph
"... We relate two concepts in graph theory and algorithmic complexity, namely the search number and the vertex separation of a graph. Let s (G ) denote the search number and vs (G ) denote the vertex separation of a connected, undirected graph G . We show that vs (G ) s (G ) vs (G ) + 2 and we give a ..."
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Cited by 74 (1 self)
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We relate two concepts in graph theory and algorithmic complexity, namely the search number and the vertex separation of a graph. Let s (G ) denote the search number and vs (G ) denote the vertex separation of a connected, undirected graph G . We show that vs (G ) s (G ) vs (G ) + 2 and we give a simple transformation from G to G such that vs (G ) = s (G ). We characterize those trees having a given vertex separation and describe the smallest such trees. We also note that there exist trees for which the difference between search number and vertex separation is indeed 2. We give algorithms that, for any tree T , compute vs (T ) in linear time and compute an optimal layout with respect to vertex separation in time O (n log n ). Vertex separation has previously been related to progressive black/white pebble demand and has been shown to be identical to a variant of search number, node search number, and to path width, which has been related directly to gate matrix layout cost. All these...
Parameterized Complexity: Exponential SpeedUp for Planar Graph Problems
 in Electronic Colloquium on Computational Complexity (ECCC
, 2001
"... A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniqu ..."
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Cited by 62 (21 self)
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A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c p k for a large variety of planar graph problems. The key to this type of algorithm is what we call the "Layerwise Separation Property" of a planar graph problem. Problems having this property include planar vertex cover, planar independent set, and planar dominating set.