Results 1  10
of
16
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 ..."
Abstract

Cited by 391 (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...
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.
Product Range Spaces, Sensitive Sampling, and Derandomization
, 1993
"... We introduce the concept of a sensitive Eapproximation, and use it to derive a more efficient algorithm for computing &nets. We define and investigate product range spaces, for which we establish sampling theorems analogous to the standard finite VCdimensional case. This generalizes and si ..."
Abstract

Cited by 46 (6 self)
 Add to MetaCart
We introduce the concept of a sensitive Eapproximation, and use it to derive a more efficient algorithm for computing &nets. We define and investigate product range spaces, for which we establish sampling theorems analogous to the standard finite VCdimensional case. This generalizes and simplifies results from previous works. We derive a simpler optimal deterministic convex hull algorithm, and by extending the method to the intersection of a set of balls with the same radius, we obtain an O(n log3 n) deterministic algorithm for computing the diameter of an npoint set in 3dimensional space.
SOME UNSOLVED PROBLEMS
, 1957
"... ... gave occasion for an informal lecture in which I discussed various old and new questions on number theory, geometry and analysis. In the following list, I record these problems, with the addition of references and of a few further questions. ..."
Abstract

Cited by 29 (0 self)
 Add to MetaCart
... gave occasion for an informal lecture in which I discussed various old and new questions on number theory, geometry and analysis. In the following list, I record these problems, with the addition of references and of a few further questions.
Construction of 1d lower envelopes and applications
 In Proceedings of the ACM Symposium on Computational Geometry
, 1997
"... ..."
Randomized algorithms for geometric optimization problems
 Handbook of Randomized Computation
, 2001
"... This chapter reviews randomization algorithms developed in the last few years to solve a wide range of geometric optimization problems. We rst review a number of general techniques, including randomized binary search, randomized linearprogramming algorithms, and random sampling. Next, we describe s ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
This chapter reviews randomization algorithms developed in the last few years to solve a wide range of geometric optimization problems. We rst review a number of general techniques, including randomized binary search, randomized linearprogramming algorithms, and random sampling. Next, we describe several applications of these and other techniques, including facility location, proximity problems, statistical estimators, nearest neighbor searching, and Euclidean TSP.
Intersection of UnitBalls and Diameter of a Point Set in R³
"... We describe an algorithm for computing the intersection of n balls of equal radius in R³ which runs in time O(n lg² n). The algorithm can be parallelized so that the comparisons that involve the radius of the balls are performed in O(lg³ n) batches. Using parametric search, these algorithms are used ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
We describe an algorithm for computing the intersection of n balls of equal radius in R³ which runs in time O(n lg² n). The algorithm can be parallelized so that the comparisons that involve the radius of the balls are performed in O(lg³ n) batches. Using parametric search, these algorithms are used to obtain an algorithm for computing the diameter of a set of n points in R³ (the maximum distance between any pair) which runs in time O(n lg^5 n). The algorithms are deterministic and elementary; this is in contrast with the running time O(n log n) in both cases that can be achieved using randomization [3], and the running times O(n lg n) and O(n lg³ n) using deterministic geometric sampling [2, 1].
Geometric Representations of Graphs
 IN PAUL ERDÖS, PROC. CONF
, 1999
"... The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, c ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, construction methods for geometric representations, and their applications in proofs and algorithms.
The Union of Unit Balls Has Quadratic Complexity, Even If They All Contain the Origin
, 1999
"... We provide a lower bound construction showing that the union of unit balls in R³ has quadratic complexity, even if they all contain the origin. This settles a conjecture of Sharir. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
We provide a lower bound construction showing that the union of unit balls in R³ has quadratic complexity, even if they all contain the origin. This settles a conjecture of Sharir.
Unit distances and diameters in euclidean spaces, Discrete and
, 2008
"... Abstract. We show that the maximum number of unit distances or of diameters in a set of n points in ddimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large, depending on d. As a corollary we determine the exact maximum number of un ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. We show that the maximum number of unit distances or of diameters in a set of n points in ddimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large, depending on d. As a corollary we determine the exact maximum number of unit distances for all even d ≥ 6, and the exact maximum number of diameters for all d≥4, for all n sufficiently large, depending on d. 1.