# Davenport-Schinzel Sequences and their Geometric Applications

## Abstract

Davenport Schinzel sequences are sequences that do not contain forbidden alternating subsequences of certain length. They are a powerful combinatorial tool applicable in contexts which involve the calculation of the pointwise maximum or minimum of a collection of (univariate) continuous functions, and have thus many applications in computational geometry and related areas. We review recent progress in the theory of Davenport Schinzel sequences, and present some of their geometric applications. These applications include efficient algorithms for the following geometric problems: (i) Preprocessing of a 2-D polyhedral terrain so as to support fast ray shooting queries from a fixed point. (ii) Determining whether two disjoint interlocking simple polygons can be separated from one another by a sequence of translations. (iii) Determining whether a given convex polygon can be translated and rotated so as to fit into another given polygonal region. (iv) Motion planning for a convex polygon in the plane amidst po ygonal barriers. (v) We also present some results on the complexity of the pointwise minimum or maximum of bivariate functions, and describe some of their applications.

### Keywords

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