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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 425 (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.
Range Searching
, 1996
"... Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , an ..."
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Cited by 70 (1 self)
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Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.
Hellytype theorems and generalized linear programming
 DISCRETE COMPUT. GEOM
, 1994
"... This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems which we call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use the ..."
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Cited by 60 (0 self)
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This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems which we call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use these results to explore the class GLP and show new applications to geometric optimization, and also to prove Helly theorems. In general, a GLP is a set...
HellyType Theorems and Geometric Transversals
 Handbook of Discrete and Computational Geometry, chapter 4
, 1997
"... INTRODUCTION A geometric transversal is an affine subspace of R d , such as a point, line, plane or hyperplane, which intersects every member of a family of convex sets. Eduard Helly's celebrated theorem gives conditions for the members of a family of convex sets to have a point in common, i.e., ..."
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Cited by 35 (3 self)
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INTRODUCTION A geometric transversal is an affine subspace of R d , such as a point, line, plane or hyperplane, which intersects every member of a family of convex sets. Eduard Helly's celebrated theorem gives conditions for the members of a family of convex sets to have a point in common, i.e., a point transversal. In Section 1 we highlight some of the more notable theorems related to Helly's Theorem and point transversals. Section 2 is devoted to geometric transversal theory. 4.1 HELLYTYPE THEOREMS In 1913, Eduard Helly proved the following theorem: Theorem 1 (Helly's Theorem) Let A be a finite family of at least d + 1 convex sets in R d . If every d + 1 members of A have a point in common, then there is a point common to all the members of A. The theorem also holds for infinite families
Hadwiger and Hellytype theorems for disjoint unit spheres
 in R 3 , in Proc. 20th Ann. Symp. on Computational Geometry, 2005
"... Let S be an ordered set of disjoint unit spheres in R 3. We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for ever ..."
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Cited by 23 (15 self)
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Let S be an ordered set of disjoint unit spheres in R 3. We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every subset of at most 11 spheres from S implies the existence of a line transversal for S. Categories and Subject Descriptors: F.2.2 [Nonnumerical
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 21 (13 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.
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.