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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 \ ..."
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Cited by 420 (116 self)
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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 ..."
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Cited by 81 (20 self)
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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...
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) ..."
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Cited by 35 (12 self)
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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 ..."
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Cited by 29 (1 self)
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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 ..."
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Cited by 28 (1 self)
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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 ..."
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Cited by 25 (3 self)
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. 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. ..."
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Cited by 23 (11 self)
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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.
Generalized DavenportSchinzel Sequences
, 1993
"... The extremal function Ex(u; n) (introduced in the theory of DavenportSchinzel sequences in other notation) denotes for a fixed finite alternating sequence u = ababa : : : the maximum length of a finite sequence v over n symbols with no immediate repetition which does not contain u. Here (following ..."
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Cited by 20 (4 self)
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The extremal function Ex(u; n) (introduced in the theory of DavenportSchinzel sequences in other notation) denotes for a fixed finite alternating sequence u = ababa : : : the maximum length of a finite sequence v over n symbols with no immediate repetition which does not contain u. Here (following the idea of J. Nesetril) we generalize this concept for arbitrary sequence u. We summarize the already known properties of Ex(u; n) and we present also two new theorems which give good upper bounds on Ex(u; n) for u consisting of (two) smaller subsequences u i provided we have good upper bounds on Ex(u i ; n). We use these theorems to describe a wide class of sequences u ("linear sequences") for which Ex(u; n) = O(n). Both theorems are used for obtaining new superlinear upper bounds as well. We partially characterize linear sequences over three symbols. We also present several problems about Ex(u; n).
On the Combinatorial and Topological Complexity of a Single Cell
, 1998
"... The problem of bounding the combinatorial complexity of a single connected component (a single cell) of the complement of a set of n geometric objects in R k of constant description complexity is an important problem in computational geometry which has attracted much attention over the past decad ..."
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Cited by 20 (7 self)
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The problem of bounding the combinatorial complexity of a single connected component (a single cell) of the complement of a set of n geometric objects in R k of constant description complexity is an important problem in computational geometry which has attracted much attention over the past decade. It has been conjectured that the combinatorial complexity of a single cell is bounded by a function much closer to O(n k 1 ) rather than O(n k ) which is the bound for the combinatorial complexity of the whole arrangement. Till now, this was known to be true only for k 3 and only for some special cases in higher dimensions. A classic result in real algebraic geometry due to OleinikPetrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semialgebraic sets. However, till now no better bounds were known if we restricted attention to a single connected component of a basic semialgebraic set. In this paper, we show how these two problems...
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 ..."
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Cited by 20 (5 self)
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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...