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34
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 479 (121 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.
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 26 (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.
Bounding the number of geometric permutations induced by ktransversals
 J. Combin. Theory Ser. A
, 1996
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Transversals to line segments in threedimensional space
 DISCRETE AND COMPUTATIONAL GEOMETRY
, 2005
"... We completely describe the structure of the connected components of transversals to a collection of n line segments in R³. Generically, the set of transversal to four segments consist of zero or two lines. We catalog the nongeneric cases and show that n >= 3 arbitrary line segments in R³ admit a ..."
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Cited by 20 (10 self)
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We completely describe the structure of the connected components of transversals to a collection of n line segments in R³. Generically, the set of transversal to four segments consist of zero or two lines. We catalog the nongeneric cases and show that n >= 3 arbitrary line segments in R³ admit at most n connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of n on the number of geometric permutations of line segments in R³.
Generalized DavenportSchinzel sequences: results, problems, and applications
, 1994
"... We survey in detail... ..."
The Overlay of Lower Envelopes and its Applications
, 1995
"... ... "), for any " ? 0. This result has several applications: (i) a nearquadratic upper bound on the complexity of the region in 3 space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an efficient and simple divid ..."
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Cited by 16 (5 self)
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... "), for any " ? 0. This result has several applications: (i) a nearquadratic upper bound on the complexity of the region in 3 space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an efficient and simple divideandconquer algorithm for constructing lower envelopes in three dimensions; and (iii) a nearquadratic upper bound on the complexity of the space of all plane transversals of a collection of simplyshaped convex sets in three dimensions.
Sharp Bounds on Geometric Permutations of Pairwise Disjoint Balls in R^d
, 1999
"... We prove that the maximum number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls in IR d , is \Theta(n d\Gamma1 ). This improves substantially the upper bound of O(n 2d\Gamma2 ) known for general convex sets [9]. We show that the maximum nu ..."
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Cited by 16 (4 self)
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We prove that the maximum number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls in IR d , is \Theta(n d\Gamma1 ). This improves substantially the upper bound of O(n 2d\Gamma2 ) known for general convex sets [9]. We show that the maximum number of geometric permutations of a sufficiently large collection of pairwise disjoint unit discs in the plane is 2, improving the previous upper bound of 3 given in [5].
Finding Stabbing Lines in 3Space
, 1992
"... A line intersecting all polyhedra in a set B is called a "stabber" for the set B. This paper addresses some combinatorial and algorithmic questions about the set S(B) of all lines stabbing B. We prove that the combinatorial complexity of S(B) has an O(n 3 2 c p log n ) upper bound, where n ..."
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Cited by 14 (3 self)
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A line intersecting all polyhedra in a set B is called a "stabber" for the set B. This paper addresses some combinatorial and algorithmic questions about the set S(B) of all lines stabbing B. We prove that the combinatorial complexity of S(B) has an O(n 3 2 c p log n ) upper bound, where n is the total number of facets in B, and c a suitable constant. This bound is almost tight. Within the same time bound it is possible to determine if a stabbing line exists and to find one.
Betti Number Bounds, Applications and Algorithms
 Current Trends in Combinatorial and Computational Geometry: Papers from the Special Program at MSRI, MSRI Publications Volume 52
, 2005
"... Abstract. Topological complexity of semialgebraic sets in R k has been studied by many researchers over the past fifty years. An important measure of the topological complexity are the Betti numbers. Quantitative bounds on the Betti numbers of a semialgebraic set in terms of various parameters (such ..."
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Cited by 13 (8 self)
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Abstract. Topological complexity of semialgebraic sets in R k has been studied by many researchers over the past fifty years. An important measure of the topological complexity are the Betti numbers. Quantitative bounds on the Betti numbers of a semialgebraic set in terms of various parameters (such as the number and the degrees of the polynomials defining it, the dimension of the set etc.) have proved useful in several applications in theoretical computer science and discrete geometry. The main goal of this survey paper is to provide an up to date account of the known bounds on the Betti numbers of semialgebraic sets in terms of various parameters, sketch briefly some of the applications, and also survey what is known about the complexity of algorithms for computing them. 1.