An (n; s) Davenport-Schinzel 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 non-contiguous) 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 Davenport-Schinzel 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 near-linear bound on the maximum length of Davenport-Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.

This paper examines the extremal problem of how many 1-entries 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 n-permutations avoiding a fixed permutation and a related conjecture of Alon and Friedgut [1]. 1

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).

Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension 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.

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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 (non-geometric) 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 P-free object M may be reused in different contexts. All of our proofs begin by transcribing the individual operations of a dynamic data structure

A matrix A P t0, 1u m n is said to avoid a forbidden pattern P P t0, 1u k l if no k l submatrix of A matches P, where a 0 in P can match either a 0 or 1 in A. Let ExpP, nq be the maximum weight (i.e., number of 1s) of an n n matrix avoiding the pattern P or all patterns in the set P. The theory of forbidden submatrices subsumes many extremal problems in combinatorics and graph theory, including Davenport-Schinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turán-type subgraph avoidance problems. Forbidden submatrices have found many applications in discrete geometry and the analysis of both geometric and non-geometric algorithms. In general terms, to bound the complexity of an arrangement of objections or the running time of an algorithm, one transcribes the objects or operations as a 0-1 matrix that provably avoids some forbidden pattern or collection of patterns P. This method is useful only to the extent that ExpP, nq can be tightly bounded, for specific P s or whole classes of P s. A 0-1 matrix can be interpreted as the incidence matrix of a bipartite graph where vertices on either side of the bipartition are ordered. In 1992, Füredi and Hajnal conjectured that imposing

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A 0-1 matrix A is said to avoid a forbidden 0-1 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 Davenport-Schinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turán-type 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 non-geometric algorithms. Clearly a 0-1 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 0-1 matrices. Our second result is a simplified proof that there is an infinite antichain (with respect

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 0-1 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 σ-free 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 Davenport-Schinzel sequences, which are those whose subsequences are not isomorphic to some fixed sequence σ. (The standard Davenport-Schinzel sequences restrict σ to be of the form abab · · ·.) 1. We prove that N-shaped 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