Results 1  10
of
66
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
Abstract

Cited by 89 (20 self)
 Add to MetaCart
(Show Context)
The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
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 75 (13 self)
 Add to MetaCart
(Show Context)
. 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...
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 ..."
Abstract

Cited by 64 (1 self)
 Add to MetaCart
(Show Context)
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.
An experimental analysis of selfadjusting computation
 In Proceedings of the ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI
, 2006
"... Selfadjusting computation uses a combination of dynamic dependence graphs and memoization to efficiently update the output of a program as the input changes incrementally or dynamically over time. Related work showed various theoretical results, indicating that the approach can be effective for a r ..."
Abstract

Cited by 48 (23 self)
 Add to MetaCart
(Show Context)
Selfadjusting computation uses a combination of dynamic dependence graphs and memoization to efficiently update the output of a program as the input changes incrementally or dynamically over time. Related work showed various theoretical results, indicating that the approach can be effective for a reasonably broad range of applications. In this article, we describe algorithms and implementation techniques to realize selfadjusting computation and present an experimental evaluation of the proposed approach on a variety of applications, ranging from simple list primitives to more sophisticated computational geometry algorithms. The results of the experiments show that the approach is effective in practice, often offering orders of magnitude speedup from recomputing the output from scratch. We believe this is the first experimental evidence that incremental computation of any type is effective in practice for a reasonably broad set of applications.
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 ..."
Abstract

Cited by 37 (3 self)
 Add to MetaCart
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...
New Lower Bounds for Convex Hull Problems in Odd Dimensions
 SIAM J. Comput
, 1996
"... We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result fo ..."
Abstract

Cited by 35 (7 self)
 Add to MetaCart
(Show Context)
We show that in the worst case, &Omega;(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasisimplicial nvertex polytope with &Omega;(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that ddimensional convex hulls can have &Omega;(n bd=2c ) facets, the previously best lower bound for these problems is only &Omega;(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is &lceil;d/2&rceil;hard, in the in the sense of Gajentaan and Overmars.
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. ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
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.
Towards InPlace Geometric Algorithms and Data Structures
 In Proceedings of the Twentieth ACM Symposium on Computational Geometry
, 2003
"... For many geometric problems, there are ecient algorithms that surprisingly use very little extra space other than the given array holding the input. For many geometric query problems, there are ecient data structures that need no extra space at all other than an array holding a permutation of the ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
For many geometric problems, there are ecient algorithms that surprisingly use very little extra space other than the given array holding the input. For many geometric query problems, there are ecient data structures that need no extra space at all other than an array holding a permutation of the input. In this paper, we obtain the rst such spaceeconomical solutions for a number of fundamental problems, including threedimensional convex hulls, twodimensional Delaunay triangulations, xeddimensional range queries, and xeddimensional nearest neighbor queries.
Instanceoptimal geometric algorithms
"... ... in 2d and 3d, and offline point location in 2d. We prove the existence of an algorithm A for computing 2d or 3d convex hulls that is optimal for every point set in the following sense: for every set S of n points and for every algorithm A ′ in a certain class A, the maximum running time of ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
... in 2d and 3d, and offline point location in 2d. We prove the existence of an algorithm A for computing 2d or 3d convex hulls that is optimal for every point set in the following sense: for every set S of n points and for every algorithm A ′ in a certain class A, the maximum running time of A on input 〈s1,..., sn〉 is at most a constant factor times the maximum running time of A ′ on 〈s1,..., sn〉, where the maximum is taken over all permutations 〈s1,..., sn 〉 of S. In fact, we can establish a stronger property: for every S and A ′ , the maximum running time of A is at most a constant factor times the average running time of A ′ over all permutations of S. We call algorithms satisfying these properties instanceoptimal in the orderoblivious and randomorder setting. Such instanceoptimal algorithms simultaneously subsume outputsensitive algorithms and distributiondependent averagecase algorithms, and all algorithms that do not take advantage of the order of the input or that assume the input is given in a random order. The class A under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments. To establish an instancespecific lower bound, we deviate from traditional Ben–Orstyle proofs and adopt an interesting adversary argument. For 2d convex hulls, we prove that a version of the well known algorithm by Kirkpatrick and Seidel (1986) or Chan, Snoeyink, and Yap (1995) already attains this lower bound. For 3d convex hulls, we propose a new algorithm. To demonstrate the potential of the concept, we further obtain instanceoptimal results for a few other standard problems in computational geometry, such as maxima in 2d and 3d, orthogonal line segment intersection in 2d, finding bichromatic L∞close pairs in 2d, offline orthogonal range searching in 2d, offline dominance reporting in 2d and 3d, offline halfspace range reporting 1.
OutputSensitive Algorithms for Computing NearestNeighbour Decision Boundaries
"... Given a set R of red points and a set B of blue points, the nearestneighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R B comes from R (respectively, B). ..."
Abstract

Cited by 14 (0 self)
 Add to MetaCart
(Show Context)
Given a set R of red points and a set B of blue points, the nearestneighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R B comes from R (respectively, B).