On the structure and composition of forbidden sequences, with geometric applications

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.