Planar realizations of nonlinear Davenport-Schinzel sequences by segments
Discrete & Computational Geometry
Sharp upper and lower bounds on the length of general Davenport-Schinzel Sequences
Journal of Combinatorial Theory Series A
The maximum number of unit distances in a convex n-gon
Journal of Combinatorial Theory Series A
An extremal problem on sparse 0-1 matrices
SIAM Journal on Discrete Mathematics
Davenport-Schnizel theory of matrices
Discrete Mathematics
Generalized Davenport-Schinzel sequences with linear upper bound
Discrete Mathematics - Topological, algebraical and combinatorial structures; Froli´k's memorial volume
Fat Triangles Determine Linearly Many Holes
SIAM Journal on Computing
Extremal functions for sequences
Discrete Mathematics - Special issue: selected papers in honour of Paul Erdo&huml;s on the occasion of his 80th birthday
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
A near-linear algorithm for the planar segment center problem
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Graph Drawings with no k Pairwise Crossing Edges
GD '97 Proceedings of the 5th International Symposium on Graph Drawing
Excluded permutation matrices and the Stanley-Wilf conjecture
Journal of Combinatorial Theory Series A
On 0-1 matrices and small excluded submatrices
Journal of Combinatorial Theory Series A
Splay trees, Davenport-Schinzel sequences, and the deque conjecture
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Note: On linear forbidden submatrices
Journal of Combinatorial Theory Series A
Note: Extremal functions of forbidden double permutation matrices
Journal of Combinatorial Theory Series A
Improved bounds and new techniques for Davenport--Schinzel sequences and their generalizations
Journal of the ACM (JACM)
On nonlinear forbidden 0-1 matrices: a refutation of a Füredi-Hajnal conjecture
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Applications of forbidden 0-1 matrices to search tree and path compression-based data structures
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Origins of Nonlinearity in Davenport-Schinzel Sequences
SIAM Journal on Discrete Mathematics
Improved bound for the union of fat triangles
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
GD'11 Proceedings of the 19th international conference on Graph Drawing
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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 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...) 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 k=O(1) mutually crossing edges. We prove tight Θ(nα(n)) bounds on sequences avoiding both ababab and all M-shaped sequences of the form ab ... yzzy ... baab ... yzzy ... ba. A consequence of this result is that the complexity of the union of n δ-fat triangles is O(n log*n α(n)), which improves, slightly, a recent bound of Ezra, Aronov, and Sharir. Here α is the inverse-Ackermann function. We give a complete characterization of 3-letter linear and nonlinear forbidden subsequences without repetitions. Specifically, a repetition-free forbidden subsequence is nonlinear (Ω(nα(n))) if and only if contains ababa, abcacbc, or its reversal; all others are linear. Many of our results are obtained by reinterpreting (forbidden) sequences as (forbidden) 0-1 matrices, which can alternatively be thought of as point sets with integer coordinates. By considering a dual sequence/matrix representation we can then apply techniques from both domains in tandem. For example, some of our results use a new composition operation on 0-1 matrices called grafting, which has no exact counterpart in the domain of sequences.