Results 1  10
of
150
Applications of Random Sampling in Computational Geometry, II
 Discrete Comput. Geom
, 1995
"... We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms ..."
Abstract

Cited by 389 (12 self)
 Add to MetaCart
We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divideandconquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of n points in E d in O(n log n) expected time for d = 3, and O(n bd=2c ) expected time for d ? 3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set of n...
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
Abstract

Cited by 189 (0 self)
 Add to MetaCart
This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
A Subexponential Bound for Linear Programming
 ALGORITHMICA
, 1996
"... We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected min{O(d 2 2 d n),e 2 d ln(n / d)+O ( d+ln n)} time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorith ..."
Abstract

Cited by 166 (16 self)
 Add to MetaCart
We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected min{O(d 2 2 d n),e 2 d ln(n / d)+O ( d+ln n)} time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorithm computes the lexicographically smallest nonnegative point satisfying n given linear inequalities in d variables. The expectation is over the internal randomizations performed by the algorithm, and holds for any input. In conjunction with Clarkson’s linear programming algorithm, this gives an expected bound of O(d 2 n + e O( √ d ln d) The algorithm is presented in an abstract framework, which facilitates its application to several other related problems like computing the smallest enclosing ball (smallest volume enclosing ellipsoid) of n points in dspace, computing the distance of two nvertex (or nfacet) polytopes in dspace, and others. The subexponential running time can also be established for some of these problems (this relies on some recent results due to Gärtner).
A Fast Algorithm for Incremental Distance Calculation
 In IEEE International Conference on Robotics and Automation
, 1991
"... A simple and efficient algorithm for finding the closest points between two convex polyhedra is described here. Data from numerous experiments tested on a broad set of convex polyhedra on ! 3 show that the running time is roughly constant for finding closest points when nearest points are approxim ..."
Abstract

Cited by 154 (4 self)
 Add to MetaCart
A simple and efficient algorithm for finding the closest points between two convex polyhedra is described here. Data from numerous experiments tested on a broad set of convex polyhedra on ! 3 show that the running time is roughly constant for finding closest points when nearest points are approximately known and is linear in total number of vertices if no special initialization is done. This algorithm can be used for collision detection, computation of the distance between two polyhedra in threedimensional space, and other robotics problems. It forms the heart of the motion planning algorithm of [1]. 1 Introduction In this paper we present a simple method for finding and tracking the closest points on a pair of convex polyhedra. The method is generally applicable, but is especially well suited to repetitive distance calculation as the objects move in a sequence of small, discrete steps. The method works by finding and maintaining the pair of closest features (vertex, edge, or face)...
Efficient Collision Detection for Animation and Robotics
, 1993
"... We present efficient algorithms for collision detection and contact determination between geometric models, described by linear or curved boundaries, undergoing rigid motion. The heart of our collision detection algorithm is a simple and fast incremental method to compute the distance between two ..."
Abstract

Cited by 109 (19 self)
 Add to MetaCart
We present efficient algorithms for collision detection and contact determination between geometric models, described by linear or curved boundaries, undergoing rigid motion. The heart of our collision detection algorithm is a simple and fast incremental method to compute the distance between two convex polyhedra. It utilizes convexity to establish some local applicability criteria for verifying the closest features. A preprocessing procedure is used to subdivide each feature's neighboring features to a constant size and thus guarantee expected constant running time for each test. The expected constant time performance is an attribute from exploiting the geometric coherence and locality. Let n be the total number of features, the expected run time is between O( p n) and O(n) ...
Las Vegas algorithms for linear and integer programming when the dimension is small
 J. ACM
, 1995
"... Abstract. This paper gives an algcmthm for solving linear programming problems. For a problem with tz constraints and d variables, the algorithm requires an expected O(d’n) + (log n)o(d)d’’+(’(’) + o(dJA log n) arithmetic operations, as rz ~ ~. The constant factors do not depend on d. Also, an algor ..."
Abstract

Cited by 105 (2 self)
 Add to MetaCart
Abstract. This paper gives an algcmthm for solving linear programming problems. For a problem with tz constraints and d variables, the algorithm requires an expected O(d’n) + (log n)o(d)d’’+(’(’) + o(dJA log n) arithmetic operations, as rz ~ ~. The constant factors do not depend on d. Also, an algorlthm N gwen for integer hnear programmmg. Let p bound the number of bits required to specify the ratmnal numbers defmmg an input constraint or the ob~ective function vector. Let n and d be as before. Then, the algorithm requires expected 0(2d dn + S~dm In n) + dc)’d) ~ in H operations on numbers with O(1~p bits d ~ ~ ~z + ~, where the constant factors do not depend on d or p. The expectations are with respect to the random choices made by the algorithms, and the bounds hold for any gwen input. The techmque can be extended to other convex programming problems. For example, m algorlthm for finding the smallest sphere enclosing a set of /z points m Ed has the same t]me bound
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
Abstract

Cited by 93 (12 self)
 Add to MetaCart
We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
On LinearTime Deterministic Algorithms for Optimization Problems in Fixed Dimension
, 1992
"... We show that with recently developed derandomization techniques, one can convert Clarkson's randomized algorithm for linear programming in fixed dimension into a lineartime deterministic one. The constant of proportionality is d O(d) , which is better than for previously known such algorithms. We s ..."
Abstract

Cited by 92 (10 self)
 Add to MetaCart
We show that with recently developed derandomization techniques, one can convert Clarkson's randomized algorithm for linear programming in fixed dimension into a lineartime deterministic one. The constant of proportionality is d O(d) , which is better than for previously known such algorithms. We show that the algorithm works in a fairly general abstract setting, which allows us to solve various other problems (such as finding the maximum volume ellipsoid inscribed into the intersection of n halfspaces) in linear time.
Applications of parametric searching in geometric optimization
 J. Algorithms
, 1994
"... z Sivan Toledo x ..."
OutputSensitive Results on Convex Hulls, Extreme Points, and Related Problems
, 1996
"... . We use known data structures for rayshooting and linearprogramming queries to derive new outputsensitive results on convex hulls, extreme points, and related problems. We show that the f face convex hull of an npoint set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) ..."
Abstract

Cited by 66 (13 self)
 Add to MetaCart
. We use known data structures for rayshooting and linearprogramming queries to derive new outputsensitive results on convex hulls, extreme points, and related problems. We show that the f face convex hull of an npoint set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) 11/(#d/2#+1) log O(1) n) time; this is optimal if f = O(n 1/#d/2# / log K n) for some sufficiently large constant K . We also show that the h extreme points of P can be computed in O(n log O(1) h + (nh) 11/(#d/2#+1) log O(1) n) time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers of P in O(n 2# ) time for any constant #<2/(#d/2# 2 + 1). Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms the input is assumed to be in general position. 1. Introduction Let P be a set of n points in ddimen...