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99
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 78 (7 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.
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 67 (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.
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 64 (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.
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 39 (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 33 (4 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.
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 28 (4 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.
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 24 (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.
Zonotope/Hyperplane Intersection for Hybrid Systems Reachability Analysis
 HSCC’08, to appear
, 2008
"... In this paper, we are concerned with the problem of computing the reachable sets of hybrid systems with (possibly high dimensional) linear continuous dynamics and guards defined by switching hyperplanes. For the reachability analysis of the continuous dynamics, we use an efficient approximation algo ..."
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Cited by 23 (0 self)
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In this paper, we are concerned with the problem of computing the reachable sets of hybrid systems with (possibly high dimensional) linear continuous dynamics and guards defined by switching hyperplanes. For the reachability analysis of the continuous dynamics, we use an efficient approximation algorithm based on zonotopes. In order to use this technique for the analysis of hybrid systems, we must also deal with the discrete transitions in a satisfactory (i.e. scalable and accurate) way. For that purpose, we need to approximate the intersection of the continuous reachable sets with the guards enabling the discrete transitions. The main contribution of this paper is a novel algorithm for computing efficiently a tight overapproximation of the intersection of (possibly highorder) zonotopes with a hyperplane. We show the accuracy and the scalability of our approach by considering two examples of reachability analysis of hybrid systems.