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23
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 418 (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.
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 22 (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.
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 at m ..."
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Cited by 19 (11 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.
Bounding the Number of Geometric Permutations Induced by kTransversals
, 1994
"... We prove that a suitably separated family of n compact convex sets in R d can be met by kflat transversals in at most O(k) d 2 " 2 k+1 \Gamma 2 k '` n k + 1 "k(d\Gammak) or, for fixed k and d, O(n k(k+1)(d\Gammak) ) different order types. This is the first nontrivial upper bound for ..."
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Cited by 17 (5 self)
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We prove that a suitably separated family of n compact convex sets in R d can be met by kflat transversals in at most O(k) d 2 " 2 k+1 \Gamma 2 k '` n k + 1 "k(d\Gammak) or, for fixed k and d, O(n k(k+1)(d\Gammak) ) different order types. This is the first nontrivial upper bound for 1 ! k ! d \Gamma 1, and generalizes (asymptotically) the best upper bounds known for line transversals in R d , d ? 2. Introduction Let A be a family of n compact convex sets in R d . A line transversal of the family A is a line that intersects every member of A. If the sets in A are City College, City University of New York, New York, NY 10031, U.S.A. (jegcc@cunyvm.cuny.edu). Supported in part by NSF grant DMS9322475, NSA grant MDA90495H1012, and PSCCUNY grant 665343. y Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A. (pollack@geometry.nyu.edu). Supported in part by NSF grants DMS9400293, CCR9402640, and CCR9424398. z Ohio Stat...
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 divideandconq ..."
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Cited by 15 (4 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 15 (3 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].
Generalized DavenportSchinzel sequences: results, problems, and applications
, 1994
"... We survey in detail... ..."
Progress in Geometric Transversal Theory
 Advances in Discrete and Computational Geometry
, 2001
"... Let A be a family of convex sets in R d . A line transversal to A is a line which intersects every member of A. More generally, a ktranversal to A is an ane subspace of dimension k which intersects every member of A. ..."
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Cited by 13 (2 self)
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Let A be a family of convex sets in R d . A line transversal to A is a line which intersects every member of A. More generally, a ktranversal to A is an ane subspace of dimension k which intersects every member of A.
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 is the to ..."
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Cited by 12 (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.
A Tight Bound on the Number of Geometric Permutations of Convex Fat Objects in R^d
 Geom
, 1999
"... We show that the maximum number of geometric permutations of a set of n fat pairwise disjoint convex objects in R d is O(n d\Gamma1 ). This generalizes the bound of \Theta(n d\Gamma1 ) obtained by Smorodinsky et al. [4] on the number of geometric permutations of n pairwise disjoint balls. D ..."
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Cited by 9 (0 self)
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We show that the maximum number of geometric permutations of a set of n fat pairwise disjoint convex objects in R d is O(n d\Gamma1 ). This generalizes the bound of \Theta(n d\Gamma1 ) obtained by Smorodinsky et al. [4] on the number of geometric permutations of n pairwise disjoint balls. Department of Mathematics and Computer Science, BenGurion University of the Negev, BeerSheva 84105, Israel. Email: matya@cs.bgu.ac.il. Work by this author has been supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. y Tata Institute of Fundamental Research, Mumbai, INDIA. Email: krv@tcs.tifr.res.in. Part of this work was done when this author was at DIMACS, Rutgers University. 1 1 Introduction Let A be a family of n pairwise disjoint convex sets in R d . An oriented line ` is said to be a transversal of A if it intersects all the convex sets in A. Any tranversal ` of A induces an ordering of A, which is the order in which ` meets the...