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14
Crossing patterns of semialgebraic sets
 J. Combin. Theory Ser. A
, 2005
"... We prove that, for every family F of n semialgebraic sets in R d of constant description complexity, there exist a positive constant ε that depends on the maximum complexity of the elements of F, and two subfamilies F1, F2 ⊆ F with at least εn elements each, such that either every element of F1 int ..."
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Cited by 30 (10 self)
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We prove that, for every family F of n semialgebraic sets in R d of constant description complexity, there exist a positive constant ε that depends on the maximum complexity of the elements of F, and two subfamilies F1, F2 ⊆ F with at least εn elements each, such that either every element of F1 intersects all elements of F2 or no element of F1 intersects any element of F2. This implies the existence of another constant δ such that F has a subset F ′ ⊆ F with n δ elements, so that either every pair of elements of F ′ intersect each other or the elements of F ′ are pairwise disjoint. The same results hold when the intersection relation is replaced by any other semialgebraic relation. We apply these results to settle several problems in discrete geometry and in Ramsey theory. 1
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 (9 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.
Edges and Switches, Tunnels and Bridges
"... Abstract. Edge casing is a wellknown method to improve the readability of drawings of nonplanar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than othe ..."
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Cited by 4 (3 self)
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Abstract. Edge casing is a wellknown method to improve the readability of drawings of nonplanar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a “good ” cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NPhardness results. 1
Iterative universal rigidity
, 2014
"... A bar framework determined by a finite graph G and configuration p = (p1,...,pn) in Rd is universally rigid if it is rigid in any RD ⊃ Rd. We provide a characterization of universally rigidity for any graph G and any configuration p in terms of a sequence of affine subsets of the space of configurat ..."
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Cited by 2 (0 self)
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A bar framework determined by a finite graph G and configuration p = (p1,...,pn) in Rd is universally rigid if it is rigid in any RD ⊃ Rd. We provide a characterization of universally rigidity for any graph G and any configuration p in terms of a sequence of affine subsets of the space of configurations. This corresponds to a facial reduction process for closed finite dimensional convex cones.
Cutting triangular cycles of lines in space
 Proc. 35th Annu. ACM Sympos. Theory Comput
, 2003
"... We show that n lines in 3space can be cut into O(n 2−1/69 log 16/69 n) pieces, such that all depth cycles defined by triples of lines are eliminated. This partially resolves a longstanding open problem in computational geometry, motivated by hiddensurface removal in computer graphics. 1 ..."
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Cited by 1 (1 self)
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We show that n lines in 3space can be cut into O(n 2−1/69 log 16/69 n) pieces, such that all depth cycles defined by triples of lines are eliminated. This partially resolves a longstanding open problem in computational geometry, motivated by hiddensurface removal in computer graphics. 1
Spindle configurations of skew lines
, 2002
"... Abstract. We simplify slightly the exposition of some known invariants for configurations of skew lines and use them to define a natural partition of the lines in a skew configuration. Finally, we describe an algorithm constructing a spindle in a given switching class, provided such a spindle exists ..."
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Abstract. We simplify slightly the exposition of some known invariants for configurations of skew lines and use them to define a natural partition of the lines in a skew configuration. Finally, we describe an algorithm constructing a spindle in a given switching class, provided such a spindle exists. 1.