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# Sources of Superlinearity in Davenport-Schinzel Sequences

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DagSemProc.08081.4.pdf
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• 14 pages

## Cite As

Seth Pettie. Sources of Superlinearity in Davenport-Schinzel Sequences. In Data Structures. Dagstuhl Seminar Proceedings, Volume 8081, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)
https://doi.org/10.4230/DagSemProc.08081.4

## Abstract

A {em generalized} Davenport-Schinzel sequence is one over a finite alphabet that contains no subsequences isomorphic to a fixed {em forbidden subsequence}. One of the fundamental problems in this area is bounding (asymptotically) the maximum length of such sequences. Following Klazar, let \$Ex(sigma,n)\$ be the maximum length of a sequence over an alphabet of size \$n\$ avoiding subsequences isomorphic to \$sigma\$. It has been proved that for every \$sigma\$, \$Ex(sigma,n)\$ is either linear or very close to linear; in particular it is \$O(n2^{alpha(n)^{O(1)}})\$, where \$alpha\$ is the inverse-Ackermann function and \$O(1)\$ depends on \$sigma\$. However, very little is known about the properties of \$sigma\$ that induce superlinearity of \$Ex(sigma,n)\$. In this paper we exhibit an infinite family of independent superlinear forbidden subsequences. To be specific, we show that there are 17 {em prototypical} superlinear forbidden subsequences, some of which can be made arbitrarily long through a simple padding operation. Perhaps the most novel part of our constructions is a new succinct code for representing superlinear forbidden subsequences.
##### Keywords
• Davenport-Schinzel Sequences
• lower envelopes
• splay trees

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