Results 1  10
of
51
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
Abstract

Cited by 425 (121 self)
 Add to MetaCart
An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
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 78 (22 self)
 Add to MetaCart
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...
On levels in arrangements of lines, segments, planes, and triangles
 Geom
, 1998
"... We consider the problem of bounding the complexity of the kth level in an arrangement of n curves or surfaces, a problem dual to, and extending, the wellknown kset problem. (a) We review and simplify some old proofs in new disguise and give new proofs of the bound O(n p k + 1) for the complexity ..."
Abstract

Cited by 42 (21 self)
 Add to MetaCart
We consider the problem of bounding the complexity of the kth level in an arrangement of n curves or surfaces, a problem dual to, and extending, the wellknown kset problem. (a) We review and simplify some old proofs in new disguise and give new proofs of the bound O(n p k + 1) for the complexity of the kth level in an arrangement of n lines. (b) We derive an improved version of Lov'asz Lemma in any dimension, and use it to prove a new bound, O(n 2
A Linear Upper Bound in Extremal Theory of Sequences
, 1994
"... An extremal problem considering sequences related to DavenportSchinzel sequences is investigated in this paper. We prove that f(x k ; n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a 1 a 2 ::a m of integers such that 1) 1 a r n, 2) ..."
Abstract

Cited by 36 (12 self)
 Add to MetaCart
An extremal problem considering sequences related to DavenportSchinzel sequences is investigated in this paper. We prove that f(x k ; n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a 1 a 2 ::a m of integers such that 1) 1 a r n, 2) a r = a s ; r 6= s implies jr \Gamma sj k and 3) u contains no subsequence of the type x k (x stands for xx::x itimes).
DavenportSchinzel Theory Of Matrices
"... Let C be a configuration of 1's. We define f(n; C) to be the maximal number of 1's in a 01 matrix of size n \Theta n not having C as a subconfiguration. We consider the problem of determining the order of f(n; C) for several forbidden C's. Among others we prove that f(n; i 1 1 1 1 j ) = \Thet ..."
Abstract

Cited by 29 (1 self)
 Add to MetaCart
Let C be a configuration of 1's. We define f(n; C) to be the maximal number of 1's in a 01 matrix of size n \Theta n not having C as a subconfiguration. We consider the problem of determining the order of f(n; C) for several forbidden C's. Among others we prove that f(n; i 1 1 1 1 j ) = \Theta(ff(n)n), where ff(n) is the inverse of the Ackermann function.
Visibility with a moving point of view
 Algorithmica
, 1994
"... We investigate 3d visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering rayshooting queries for rays with origin on the ..."
Abstract

Cited by 28 (1 self)
 Add to MetaCart
We investigate 3d visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering rayshooting queries for rays with origin on the flightpath. Three progressively more specialized problems are considered: general scenes, terrains, and terrains with vertical flightpaths. 1.
A New Technique for Analyzing Substructures in Arrangements of Piecewise Linear Surfaces
, 1996
"... . We present a simple but powerful new probabilistic technique for analyzing the combinatorial complexity of various substructures in arrangements of piecewiselinear surfaces in higher dimensions. We apply the technique (a) to derive new and simpler proofs of the known bounds on the complexity of t ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
. We present a simple but powerful new probabilistic technique for analyzing the combinatorial complexity of various substructures in arrangements of piecewiselinear surfaces in higher dimensions. We apply the technique (a) to derive new and simpler proofs of the known bounds on the complexity of the lower envelope, of a single cell, or of a zone in arrangements of simplices in higher dimensions, and (b) to obtain improved bounds on the complexity of the vertical decomposition of a single cell in an arrangement of triangles in 3space, and of several other substructures in such an arrangement (the entire arrangement, all nonconvex cells, and any collection of cells). The latter results also lead to improved algorithms for computing substructures in arrangements of triangles and for translational motion planning in three dimensions. 1. Introduction The study of arrangements of curves or surfaces is an important area of research in computational and combinatorial geometry, because many...
Counting patternfree set partitions I: A generalization of Stirling numbers of the second kind
, 2000
"... A partition u of [k] = f1; 2; : : : ; kg is contained in another partition v of [l] if [l] has a ksubset on which v induces u. We are interested in counting partitions v not containing a given partition u or a given set of partitions R. This concept is related to that of forbidden permutations. ..."
Abstract

Cited by 22 (11 self)
 Add to MetaCart
A partition u of [k] = f1; 2; : : : ; kg is contained in another partition v of [l] if [l] has a ksubset on which v induces u. We are interested in counting partitions v not containing a given partition u or a given set of partitions R. This concept is related to that of forbidden permutations. A strengthening of StanleyWilf conjecture is proposed.
Voronoi Diagrams of Lines in 3Space Under Polyhedral Convex Distance Functions
 J. Algorithms
"... The combinatorial complexity of the Voronoi diagram of n lines in three dimensions under a convex distance function induced by a polytope with a constant number of edges is shown to be O(n 2 ff(n) log n), where ff is a slowly growing inverse of the Ackermann function. There are arrangements of ..."
Abstract

Cited by 20 (5 self)
 Add to MetaCart
The combinatorial complexity of the Voronoi diagram of n lines in three dimensions under a convex distance function induced by a polytope with a constant number of edges is shown to be O(n 2 ff(n) log n), where ff is a slowly growing inverse of the Ackermann function. There are arrangements of n lines where this complexity can be as large as \Omega\Gamma n 2 ff(n)). 1 Introduction Statement of result. Let P be a closed convex polytope in 3space which contains the origin. Given any point w 2 IR 3 , its distance from a line (or any other object) `, as induced by P , is d P (w; `) = inf ft 0 : (w + tP ) " ` 6= ;g ; Work by Paul Chew and Klara Kedem has been supported by AFOSR Grant AFOSR910328 and by the U.S.Israeli Binational Science Foundation. Work by Paul Chew has also been supported by ONR Grant N0001489J1946 and ARPA under ONR contract N0001488K0591. Work by Micha Sharir and Emo Welzl has been supported by the G.I.F.  the German Israeli Foundation for Sc...
The Common Exterior of Convex Polygons in the Plane
 Comput. Geom. Theory Appl
, 1997
"... We establish several combinatorial bounds on the complexity (number of vertices and edges) of the complement of the union (also known as the common exterior) of k convex polygons in the plane, with a total of n edges. We show: 1. The maximum complexity of the entire common exterior is \Theta(nff( ..."
Abstract

Cited by 20 (9 self)
 Add to MetaCart
We establish several combinatorial bounds on the complexity (number of vertices and edges) of the complement of the union (also known as the common exterior) of k convex polygons in the plane, with a total of n edges. We show: 1. The maximum complexity of the entire common exterior is \Theta(nff(k) + k 2 ). 1 2. The maximum complexity of a single cell of the common exterior is \Theta(nff(k)). 3. The complexity of m distinct cells in the common exterior is O(m 2=3 k 2=3 log 1=3 ( k 2 m )+ n log k) and can be \Omega\Gamma m 2=3 k 2=3 + nff(k)) in the worst case. 1 Introduction In this paper we establish several combinatorial bounds on the complexity of the common exterior (namely, the complement of the union) of a collection of k convex polygons in the plane, with a total of n edges. The arrangement of such a collection of polygons can be viewed as a special case of an arrangement of n segments, but we prefer to regard it as a generalization of an arrangement of k...