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35
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 421 (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.
Solving geometric problems with the rotating calipers
, 1983
"... Shamos [1] recently showed that the diameter of a convex nsided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several se ..."
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Cited by 112 (14 self)
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Shamos [1] recently showed that the diameter of a convex nsided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be used on several convex polygons simultaneously. We then show that these generalizations allow us to obtain simple O(n) algorithms for solving a variety of problems defined on convex polygons. Such problems include (1) finding the minimumarea rectangle enclosing a polygon, (2) computing the maximum distance between two polygons, (3) performing the vectorsum of two polygons, (4) merging polygons in a convex hull finding algorithms, and (5) finding the critical support lines between two polygons. Finding the critical support lines, in turn, leads to obtaining solutions to several additional problems concerned with visibility, collision, avoidance, range fitting, linear separability, and computing the Grenander distance between sets. 1.
On the convex layers of a planar set
 IEEE Transactions on Information Theory
, 1985
"... AbstractLet S be a set of n points in the Euclidean plane. The convex layers of S are the convex polygons obtained by iterating on the following procedure: compute the convex hull of S and remove its vertices from S. This process of peeling a planar point set is central in the study of robust estim ..."
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Cited by 56 (1 self)
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AbstractLet S be a set of n points in the Euclidean plane. The convex layers of S are the convex polygons obtained by iterating on the following procedure: compute the convex hull of S and remove its vertices from S. This process of peeling a planar point set is central in the study of robust estimators in statistics. It also provides valuable information on the morphology of a set of sites and has proven to be an efficient preconditioning for range search problems. An optimal algorithm is described for computing the convex layers of S. The algorithm runs in O ( n log n) time and requires O(n) space. Also addressed is the problem of determining the depth of a query point within the convex layers of S, i.e., the number of layers that enclose the query point. This is essentially a planar point location problem, for which optimal solutions are therefore known. Taking advantage of structural properties of the problem, however, a much simpler optimal solution is derived. L I.
An Optimal Algorithm for Determining the Visibility of a Polygon from an Edge
 IEEE Transactions on Computers
, 1981
"... In many computer applications areas such as graphics, automated cartography, image processing, and robotics the notion of visibility among objects modeled as polygons is a recurring theme. This paper is concerned with the visibility of a simple polygon from one of its edges. Three natural definition ..."
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Cited by 43 (10 self)
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In many computer applications areas such as graphics, automated cartography, image processing, and robotics the notion of visibility among objects modeled as polygons is a recurring theme. This paper is concerned with the visibility of a simple polygon from one of its edges. Three natural definitions of the visibility of a polygon from an edge are presented. The following computational problem is considered. Given an n sided simple polygon, is the polygon visible from a specified edge? An O(n), and thus optimal, algorithm is exhibited for determining edge visibility under any of the three definitions. The paper closes with an interesting characterization of visibility and some open problems in this area. Index Terms  Algorithms, computational complexity, computational geometry, computer graphics, hidden line problems, image processing, robotics, simple polygon, visibility. 1. Introduction The notion of visibility in geometric objects is one that appears in many applications: the ...
Triangulation and shapecomplexity
 ACM Trans. Graph
, 1984
"... This paper describes a new method for triangulating a simple nsided polygon. The algorithm runs in time O(n log s), with s _< n. The quantity s measures the sinuosity of the polygon, that is, the number of times the boundary alternates between complete spirals of opposite orientation. The value of ..."
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Cited by 41 (1 self)
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This paper describes a new method for triangulating a simple nsided polygon. The algorithm runs in time O(n log s), with s _< n. The quantity s measures the sinuosity of the polygon, that is, the number of times the boundary alternates between complete spirals of opposite orientation. The value of s is in practice a very small constant, even for extremely winding polygons. Our algorithm is the first method whose performance is linear in the number of vertices, up to within a factor that depends only on the shapecomplexity of the polygon. Informally, this notion of shapecomplexity measures how entangled a polygon is, and is thus highly independent of the number of vertices. A practical advantage of the algorithm is that it does not require sorting or the use of any balanced tree structure. Aside from the notion of sinuosity, we are also able to characterize a large class of polygons for which the algorithm can be proven to run in O(n log log n) time. The algorithm has been implemented, tested, and empirical evidence has confirmed its theoretical claim to efficiency.
Movable Separability of Sets
 Computational Geometry
, 1985
"... Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane to polyhedra in three dimensions, without allowing collisions betwee ..."
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Cited by 39 (4 self)
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Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane to polyhedra in three dimensions, without allowing collisions between the objects. One class of such problems considers the separability of sets of objects under different kinds of motions and various definitions of separation. This paper surveys this new area of research in a tutorial fashion, present new results, and provides a list of open problems and suggestions for further research. Key Words and Phrases: sofa problem, polygons, polyhedra, movable separability, visibility hulls, hidden lines, hidden surfaces, algorithms, complexity, computational geometry, spatial planning, collision avoidance, robotics, artificial intelligence. CR Categories: 3.36, 3.63, 5.25. 5.32. 5.5 * Research supported by NSERC Grant no. A9293 and FCAR Grant no.EQ1678.  2  ...
Algorithms for Cluster Busting in Anchored Graph Drawing
 Journal of Graph Algorithms and Applications
, 1998
"... Given a graph G and a drawing or layout of G, it is sometimes desirable to alter or adjust the layout. The challenging aspect of designing layout adjustment algorithms is to maintain a user's mental picture of the original layout. We present a new approach to layout adjustment called cluster busting ..."
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Cited by 34 (0 self)
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Given a graph G and a drawing or layout of G, it is sometimes desirable to alter or adjust the layout. The challenging aspect of designing layout adjustment algorithms is to maintain a user's mental picture of the original layout. We present a new approach to layout adjustment called cluster busting in anchored graph drawing. We then give two algorithms as examples of this approach. The goals of cluster busting in anchored graph drawing are to more evenly distribute the nodes of the graph in a drawing window while maintaining the user's mental picture of the original drawing. We present simple and eÆcient iterative heuristics to accomplish these goals. We formally de ne some measures of distribution and similarity and give empirical results based on these measures to quantify our methods. The theoretical analysis of our heuristics presents a formidable challenge, thus justifying our empirical analysis. Communicated by G. Di Battista: submitted April 1996; revised March 1998. Research...
Computing Simple Circuits from a Set of Line Segments . . .
, 1987
"... Given a collection of line segments in the plane we would like to connect the segments by their endpoints to construct a simple circuit. (A simple circuit is the boundary of a simple polygon.) However, there are collections of line segments where this cannot be done. In this note it is proved that d ..."
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Cited by 26 (1 self)
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Given a collection of line segments in the plane we would like to connect the segments by their endpoints to construct a simple circuit. (A simple circuit is the boundary of a simple polygon.) However, there are collections of line segments where this cannot be done. In this note it is proved that deciding whether a set of line segments admits a simple circuit is NPcomplete. Deciding whether a set of horizontal line segments can be connected with horizontal and vertical line segments to construct an orthogonal simple circuit is also shown to be NPcomplete.
The expected size of some graphs in computational geometry
 COMPUTERS & MATHEMATICS WITH APPLICATIONS
, 1988
"... We consider n independent points with a common but arbitrary density fin R a. Two points (X~, Xj) are joined by an edge when a certain set S(X~, Xj) does not contain any other data points. The expected number E(N) of edges in the graph depends upon n, f and the definition of S. Examples include rec ..."
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Cited by 24 (4 self)
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We consider n independent points with a common but arbitrary density fin R a. Two points (X~, Xj) are joined by an edge when a certain set S(X~, Xj) does not contain any other data points. The expected number E(N) of edges in the graph depends upon n, f and the definition of S. Examples include rectangles, spheres and loons; these lead to the graph of all dominance pairs, the Gabriel graph and the relative neighborhood graph, respectively. Other graphs covered by our analysis include the nearest neighbor graph and the directional nearest neighbor graph. In all cases, we obtain asymptotic lower bounds that do not depend uponf(and are hence useful in all applications involving these graphs, since we usually do not know f). For sparse graphs, exact asymptotic constants are obtained for E(N) that are valid for all densities.
Proximity Graphs for Nearest Neighbor Decision Rules: Recent Progress
 Progress”, Proceedings of the 34 th Symposium on the INTERFACE
, 2002
"... In the typical nonparametric approach to pattern classification, random data (the training set of patterns) are collected and used to design a decision rule (classifier). One of the most well known such rules is the knearestneighbor decision rule (also known as instancebased learning, and lazy le ..."
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Cited by 22 (0 self)
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In the typical nonparametric approach to pattern classification, random data (the training set of patterns) are collected and used to design a decision rule (classifier). One of the most well known such rules is the knearestneighbor decision rule (also known as instancebased learning, and lazy learning) in which an unknown pattern is classified into the majority class among its k nearest neighbors in the training set. Several questions related to this rule have received considerable attention over the years. Such questions include the following. How can the storage of the training set be reduced without degrading the performance of the decision rule? How should the reduced training set be selected to represent the different classes? How large should k be? How should the value of k be chosen? Should all k neighbors be equally weighted when used to decide the class of an unknown pattern? If not, how should the weights be chosen? Should all the features (attributes) we weighted equally and if not how should the feature weights be chosen? What distance metric should be used? How can the rule be made robust to overlapping classes or noise present in the training data? How can the rule be made invariant to scaling of the measurements? Geometric proximity graphs such as Voronoi diagrams and their many relatives provide elegant solutions to most of these problems. After a brief and nonexhaustive review of some of the classical canonical approaches to solving these problems, the methods that use proximity graphs are discussed, some new observations are made, and avenues for further research are proposed.