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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 471 (115 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 73 (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 61 (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 ..."
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Cited by 36 (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
A Positive Fraction ErdősSzekeres Theorem
, 1998
"... We prove a fractional version of the Erdős–Szekeres theorem: for any k there is a constant ck> 0 such that any sufficiently large finite set X ⊂ R2 contains k subsets Y1,...,Yk, each of size ≥ ckX, such that every set {y1,...,yk}with yi ∈ Yi is in convex position. The main tool is a lemma stati ..."
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Cited by 24 (5 self)
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We prove a fractional version of the Erdős–Szekeres theorem: for any k there is a constant ck> 0 such that any sufficiently large finite set X ⊂ R2 contains k subsets Y1,...,Yk, each of size ≥ ckX, such that every set {y1,...,yk}with yi ∈ Yi is in convex position. The main tool is a lemma stating that any finite set X ⊂ Rd contains “large” subsets Y1,...,Yk such that all sets {y1,...,yk}with yi ∈ Yi have the same geometric (order) type. We also prove several related results (e.g., the positive fraction Radon theorem, the positive fraction Tverberg theorem).
Hadwiger and Hellytype theorems for disjoint unit spheres in R³
 IN PROC. 20TH ANN. SYMP. ON COMPUTATIONAL GEOMETRY
, 2005
"... Let S be an ordered set of disjoint unit spheres in R³. 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 ..."
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Cited by 22 (14 self)
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Let S be an ordered set of disjoint unit spheres in R³. 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.
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 21 (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
 J. Combin. Theory Ser. A
, 1996
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A Geometric Optimization Approach to Detecting and Intercepting Dynamic Targets
"... Abstract — A methodology is developed to deploy a mobile sensor network for the purpose of detecting and capturing mobile targets in the plane. The sensingpursuit problem considered in this paper is analogous to the Marco Polo game, in which the pursuer must capture multiple mobile targets that are ..."
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Cited by 17 (9 self)
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Abstract — A methodology is developed to deploy a mobile sensor network for the purpose of detecting and capturing mobile targets in the plane. The sensingpursuit problem considered in this paper is analogous to the Marco Polo game, in which the pursuer must capture multiple mobile targets that are sensed intermittently, and with very limited information. In this paper, the mobile sensor network consists of a set of robotic sensors that must track and capture mobile targets based on the information obtained through cooperative detections. Since the sensors are installed on robotic platforms and have limited range, the geometry of the platforms and of the sensors fieldofview play a key role in obstacle avoidance and target detection. Thus, a new cell decomposition approach is presented to formulate the probability of detection and the cost of operating the robots based on the geometric properties of the network. Numerical simulations verify the validity and flexibility of our methodology. I.