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18
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 479 (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.
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 121 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
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 90 (20 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...
Vertical decomposition of shallow levels in 3dimensional arrangements and its applications
 SIAM J. Comput
, 1999
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 71 (14 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Computing the Smallest kEnclosing Circle and Related Problems
, 1999
"... We present an efficient algorithm for solving the "smallest kenclosing circle " ( kSC) problem: Given a set of n points in the plane and an integer k ^ n, find the smallest disk containing k of the points. We present two solutions. When using O(nk) storage, the problem can be so ..."
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Cited by 19 (7 self)
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We present an efficient algorithm for solving the &quot;smallest kenclosing circle &quot; ( kSC) problem: Given a set of n points in the plane and an integer k ^ n, find the smallest disk containing k of the points. We present two solutions. When using O(nk) storage, the problem can be solved in time O(nk log2 n). When only O(n log n) storage is allowed, the running time is O(nk log2 n log nk). This problem
Tight bounds on the maximum size of a set of permutations with bounded VCdimension
 In Proc. Symposium on Discrete Algorithms
, 2012
"... The VCdimension of a family P of npermutations is the largest integer k such that the set of restrictions of the permutations in P on some ktuple of positions is the set of all k! permutation patterns. Let rk(n) be the maximum size of a set of npermutations with VCdimension k. Raz showed that r2 ..."
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Cited by 11 (2 self)
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The VCdimension of a family P of npermutations is the largest integer k such that the set of restrictions of the permutations in P on some ktuple of positions is the set of all k! permutation patterns. Let rk(n) be the maximum size of a set of npermutations with VCdimension k. Raz showed that r2(n) grows exponentially in n. We show that r3(n) = 2 Θ(nlogα(n)) and for every t ≥ 1, we have r2t+2(n) = 2 Θ(nα(n)t) and r2t+3(n) = 2 O(nα(n)t logα(n)) We also study the maximum number pk(n) of 1entries in an n × n (0,1)matrix with no (k + 1)tuple of columns containing all (k+1)permutation matrices. We determine that p3(n) = Θ(nα(n)) and p2t+2(n) = n2 (1/t!)α(n)t ±O(α(n) t−1) for every t ≥ 1. We also show that for every positive s there is a slowly growing function ζs(m) (for example ζs(m) = 2 O(α(s−3)/2 (m)) for every odd s ≥ 5) satisfying the following. For all positive integers m,n,B and every m×n (0,1)matrix M with ζs(m)Bn 1entries, the rows of M can be partitioned into s intervals so that some ⌊Bn/m⌋tuple of columns contains at least B 1entries in each of the intervals. 1
On the Structure and Composition of Forbidden Sequences, with Geometric Applications
, 2010
"... Forbidden substructure theorems have proved to be among of the most versatile tools in bounding the complexity of geometric objects and the running time of geometric algorithms. To apply them one typically transcribes an algorithm execution or geometric object as a sequence over some alphabet or a 0 ..."
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Cited by 10 (3 self)
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Forbidden substructure theorems have proved to be among of the most versatile tools in bounding the complexity of geometric objects and the running time of geometric algorithms. To apply them one typically transcribes an algorithm execution or geometric object as a sequence over some alphabet or a 01 matrix, proves that this object avoids some subsequence or submatrix σ, then uses an off the shelf bound on the maximum size of such a σfree object. As a historical trend, expanding our library of forbidden substructure theorems has led to better bounds and simpler analyses of the complexity of geometric objects. We establish new and tight bounds on the maximum length of generalized DavenportSchinzel sequences, which are those whose subsequences are not isomorphic to some fixed sequence σ. (The standard DavenportSchinzel sequences restrict σ to be of the form abab · · ·.) 1. We prove that Nshaped forbidden subsequences (of the form abc · · · xyzyx · · · cbabc · · · xyz) have a linear extremal function. Our proof dramatically improves an earlier one of Klazar and Valtr in the leading constants and overall simplicity. This result tightens the (astronomical) leading constants in Valtr’s O(n log n) bound on geometric graphs without
Geographic maxflow and mincut under a circular disk failure model
 Massachusetts Institute of Technology, EE, Tech. Rep
, 2012
"... Abstract—Failures in fiberoptic networks may be caused by natural disasters, such as floods or earthquakes, as well as other events, such as an Electromagnetic Pulse (EMP) attack. These events occur in specific geographical locations, therefore the geography of the network determines the effect of ..."
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Cited by 9 (0 self)
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Abstract—Failures in fiberoptic networks may be caused by natural disasters, such as floods or earthquakes, as well as other events, such as an Electromagnetic Pulse (EMP) attack. These events occur in specific geographical locations, therefore the geography of the network determines the effect of failure events on the network’s connectivity and capacity. In this paper we consider a generalization of the mincut and maxflow problems under a geographic failure model. Specifically, we consider the problem of finding the minimum number of failures, modeled as circular disks, to disconnect a pair of nodes and the maximum number of failure disjoint paths between pairs of nodes. This model applies to the scenario where an adversary is attacking the network multiple times with intention to reduce its connectivity. We present a polynomial time algorithm to solve the geographic mincut problem and develop an ILP formulation, an exact algorithm, and a heuristic algorithm for the geographic maxflow problem. I.
Degrees of Nonlinearity in Forbidden 01 Matrix Problems
"... A 01 matrix A is said to avoid a forbidden 01 matrix (or pattern) P if no submatrix of A matches P, where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of DavenportSchinzel sequen ..."
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Cited by 7 (4 self)
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A 01 matrix A is said to avoid a forbidden 01 matrix (or pattern) P if no submatrix of A matches P, where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of DavenportSchinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turántype subgraph avoidance problems. In addition, forbidden matrix theory has proved to be a powerful tool in discrete geometry and the analysis of both geometric and nongeometric algorithms. Clearly a 01 matrix can be interpreted as the incidence matrix of a bipartite graph in which vertices on each side of the partition are ordered. Füredi and Hajnal conjectured that if P corresponds to an acyclic graph then the maximum weight (number of 1s) in an n × n matrix avoiding P is O(n log n). Our first result is a refutation of this conjecture. We exhibiting n × n matrices with weight Θ(n log n log log n) that avoid a relatively small acyclic matrix. The matrices are constructed via two complementary composition operations for 01 matrices. Our second result is a simplified proof that there is an infinite antichain (with respect
Origins of nonlinearity in DavenportSchinzel sequences
, 2009
"... A generalized DavenportSchinzel sequence is one over a finite alphabet that excludes subsequences isomorphic to a fixed forbidden subsequence. The fundamental problem in this area is bounding the maximum length of such sequences. Following Klazar, we let Expσ, nq be the maximum length of a sequence ..."
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Cited by 7 (6 self)
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A generalized DavenportSchinzel sequence is one over a finite alphabet that excludes subsequences isomorphic to a fixed forbidden subsequence. The fundamental problem in this area is bounding the maximum length of such sequences. Following Klazar, we let Expσ, nq be the maximum length of a sequence over an alphabet of size n excluding subsequences isomorphic to σ. It has been proved that for every σ, Expσ, nq is either linear or very close to linear. In particular it is Opn2 αpnqOp1q q, where α is the inverseAckermann function and Op1q depends on σ. In much the same way that the complete graphs K5 and K3,3 represent the minimal causes of nonplanarity, there must exist a set ΦNonlin of minimal nonlinear forbidden subsequences. Very little is known about the size or membership of ΦNonlin. In this paper we construct an infinite antichain of nonlinear forbidden subsequences which, we argue, strongly supports the conjecture that ΦNonlin is itself infinite. Perhaps the most novel contribution of this paper is a succinct, humanly readable code for expressing the structure of forbidden subsequences.