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26
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 78 (22 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Excluded permutation matrices and the StanleyWilf conjecture
 J. Combin. Theory Ser. A
, 2004
"... This paper examines the extremal problem of how many 1entries an n × n 0–1 matrix can have that avoids a certain fixed submatrix P. For any permutation matrix P we prove a linear bound, settling a conjecture of Zoltán Füredi and Péter Hajnal [8]. Due to the work of Martin Klazar [12], this also set ..."
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Cited by 77 (3 self)
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This paper examines the extremal problem of how many 1entries an n × n 0–1 matrix can have that avoids a certain fixed submatrix P. For any permutation matrix P we prove a linear bound, settling a conjecture of Zoltán Füredi and Péter Hajnal [8]. Due to the work of Martin Klazar [12], this also settles the conjecture of Stanley and Wilf on the number of npermutations avoiding a fixed permutation and a related conjecture of Alon and Friedgut [1]. 1
Counting patternfree set partitions. II: Noncrossing and other hypergraphs
 J. Combin
, 2000
"... A (multi)hypergraph H with vertices in N contains a permutation p = a 1 a 2: : : a k of 1; 2; : : : ; k if one can reduce H by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to H p = (fi; k + a i g: i = 1; : : : ; k). We formulate six co ..."
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Cited by 20 (9 self)
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A (multi)hypergraph H with vertices in N contains a permutation p = a 1 a 2: : : a k of 1; 2; : : : ; k if one can reduce H by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to H p = (fi; k + a i g: i = 1; : : : ; k). We formulate six conjectures stating that if H has n vertices and does not contain p then the size of H is O(n) and the number of such Hs is O(c n). The latter part generalizes the StanleyWilf conjecture on permutations. Using generalized DavenportSchinzel sequences, we prove the conjectures with weaker bounds O(nfi(n)) and O(fi(n) n), where fi(n) ! 1 very slowly. We prove the conjectures fully if p first increases and then decreases or if p
A NearLinear Algorithm for the Planar Segment Center Problem
, 1996
"... Let P be a set of n points in the plane and let e be a segment of fixed length. The segment center problem is to find a placement of e (allowing translation and rotation) which minimizes the maximum euclidean distance from e to the points of P. We present an algorithm that solves the problem in tim ..."
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Cited by 16 (7 self)
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Let P be a set of n points in the plane and let e be a segment of fixed length. The segment center problem is to find a placement of e (allowing translation and rotation) which minimizes the maximum euclidean distance from e to the points of P. We present an algorithm that solves the problem in time O(n1+"), for any " ? 0, improving the previous solution of Agarwal et al. [3] by nearly a factor of O(n).
Geometric Graph Theory
, 1999
"... A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications. ..."
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Cited by 13 (0 self)
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A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications.
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 13 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Geometric Representations of Graphs
 IN PAUL ERDÖS, PROC. CONF
, 1999
"... The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, c ..."
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Cited by 8 (0 self)
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The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, construction methods for geometric representations, and their applications in proofs and algorithms.
The maximum number of times the same distance can occur among the vertices of a convex ngon is O(n log n)
, 2001
"... We present a short proof of Füredi's theorem [F] stated in the title. ..."
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Cited by 6 (3 self)
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We present a short proof of Füredi's theorem [F] stated in the title.
Applications of forbidden 01 matrices to search tree and path compression based data structures
, 2009
"... In this paper we improve, reprove, and simplify a variety of theorems concerning the performance of data structures based on path compression and search trees. We apply a technique very familiar to computational geometers but still foreign to many researchers in (nongeometric) algorithms and data s ..."
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Cited by 6 (5 self)
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In this paper we improve, reprove, and simplify a variety of theorems concerning the performance of data structures based on path compression and search trees. We apply a technique very familiar to computational geometers but still foreign to many researchers in (nongeometric) algorithms and data structures, namely, to bound the complexity of an object via its forbidden substructures. To analyze an algorithm or data structure in the forbidden substructure framework one proceeds in three discrete steps. First, one transcribes the behavior of the algorithm as some combinatorial object M; for example, M may be a graph, sequence, permutation, matrix, set system, or tree. (The size of M should ideally be linear in the running time.) Second, one shows that M excludes some forbidden substructure P, and third, one bounds the size of any object avoiding this substructure. The power of this framework derives from the fact that M lies in a more pristine environment and that upper bounds on the size of a Pfree object M may be reused in different contexts. All of our proofs begin by transcribing the individual operations of a dynamic data structure