## Davenport-Schinzel Sequences and Their Geometric Applications (1998)

Citations: | 421 - 120 self |

### BibTeX

@MISC{Agarwal98davenport-schinzelsequences,

author = {Pankaj K. Agarwal and Micha Sharir},

title = {Davenport-Schinzel Sequences and Their Geometric Applications},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

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.