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45
Problems in Computational Geometry
 Packing and Covering
, 1974
"...  reproduced, stored In a retrieval system, or transmlt'ted, In any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the author. ..."
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Cited by 453 (2 self)
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 reproduced, stored In a retrieval system, or transmlt'ted, In any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the author.
Spatial Planning: A Configuration Space Approach
, 1980
"... This paper presents algorithms fi)r computing constraints on the pusiliun of' an object due o the presence of obstacles. This problem arises in applicalion'.g xYhMi require choosing how to arrange or move obje(:[s among other objects. The basis uf the approach presenlcd here is to characterize Ihe p ..."
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Cited by 340 (1 self)
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This paper presents algorithms fi)r computing constraints on the pusiliun of' an object due o the presence of obstacles. This problem arises in applicalion'.g xYhMi require choosing how to arrange or move obje(:[s among other objects. The basis uf the approach presenlcd here is to characterize Ihe positinn and orientation of theobject of interest as a single point in a Coufiguration Space, in which each coordinate represents a degree of fi'eedom in the position. nnd/o' orientation of the object. The configurations forbidden to this object, due to the presence of obstacles, can then be characterized as regions in the Configuration Space. The paper presents algorithms for computing these Configuration Space obstacles when the objects and obstacles are polygons or polyhedra. An approximation technique fi.>r highdilnensional Configuration Space obstacles, based ou projections of obstacles slices, is described
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 55 (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.
Dynamic planar convex hull
 Proc. 43rd IEEE Sympos. Found. Comput. Sci
, 2002
"... In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage o ..."
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Cited by 51 (1 self)
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In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure.
Optimal OutputSensitive Convex Hull Algorithms in Two and Three Dimensions
, 1996
"... We present simple outputsensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worstcase optimal O(n log h) time and O(n) space, where h denotes the number of vertices of the convex hull. ..."
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Cited by 47 (6 self)
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We present simple outputsensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worstcase optimal O(n log h) time and O(n) space, where h denotes the number of vertices of the convex hull.
Primal Dividing and Dual Pruning: OutputSensitive Construction of 4d Polytopes and 3d Voronoi Diagrams
, 1997
"... In this paper, we give an algorithm for outputsensitive construction of an fface convex hull of a set of n points in general position in E 4 . Our algorithm runs in O((n + f)log 2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f ..."
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Cited by 33 (3 self)
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In this paper, we give an algorithm for outputsensitive construction of an fface convex hull of a set of n points in general position in E 4 . Our algorithm runs in O((n + f)log 2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f . By a standard lifting map, we obtain outputsensitive algorithms for the Voronoi diagram or Delaunay triangulation in E 3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the "ultimate convex hull algorithm" of Kirkpatrick and Seidel in E 2 and also leads to improved outputsensitive results on constructing convex hulls in E d for any even constant d ? 4. 1 Introduction Geometric structures induced by n points in Euclidean ddimensional space, such as the convex hull, Voronoi diagram, or Delaunay triangulation, can be of larger size than the point set that defines them. In many practical situat...
Approximately Optimal Assignment For Unequal Loss Protection
 IN PROC. INT'L CONF. IMAGE PROCESSING
, 1999
"... This paper describes an algorithm that achieves an approximately optimal assignment of forward error correction to progressive data within the unequal loss protection framework [1]. It first finds the optimal assignment under convex hull and fractional bit allocation assumptions. It then relaxes tho ..."
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Cited by 26 (3 self)
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This paper describes an algorithm that achieves an approximately optimal assignment of forward error correction to progressive data within the unequal loss protection framework [1]. It first finds the optimal assignment under convex hull and fractional bit allocation assumptions. It then relaxes those constraints to find an assignment that approximates the global optimum. The algorithm has a running time of O(hN logN ) where h is the number of points on the convex hull of the source's utilitycost curve and N is the number of packets transmitted.
Spaceefficient planar convex hull algorithms
 Proc. Latin American Theoretical Informatics
, 2002
"... A spaceefficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. We describe four spaceefficient algorithms for computing the convex hull of a planar point set. ..."
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Cited by 20 (1 self)
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A spaceefficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. We describe four spaceefficient algorithms for computing the convex hull of a planar point set.
Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
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Cited by 19 (3 self)
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We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
Fast Greedy Triangulation Algorithms
 Proc. 10th Ann. Symp. Computational Geometry
, 1994
"... this paper we present a simple, practical algorithm that computes the greedy triangulation in expected time O(n log n) and space O(n) for points uniformly distributed over any convex shape. A variant of this algorithm should also be fast for many other distributions. We first describe a surprisingly ..."
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Cited by 16 (2 self)
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this paper we present a simple, practical algorithm that computes the greedy triangulation in expected time O(n log n) and space O(n) for points uniformly distributed over any convex shape. A variant of this algorithm should also be fast for many other distributions. We first describe a surprisingly simple method for testing the compatibility of a candidate edge with edges in a partially constructed greedy triangulation. The new edge is tentatively added to the embedding of the partial GT and at most four constant time tests are done involving edges lying clockwise and counterclockwise from the candidate edge at each vertex. Even though there can be O(n) edges adjacent to one of the endpoints, we are able to show that if we can determine where in angular order the new edge falls among a subset of at most 10 of those edges then we can perform the compatibility test and if necessary update the triangulation. Our method therefore provides a \Theta(1) time edge test that requires only \Theta(1) time to update the structure, \Theta(n) time for initialization, and \Theta(n) space. This compares favorably with the previous method of Gilbert [10], which requires \Theta(log n) time for an edge test, \Theta(n log n) time for an update, \Theta(n