Davenport-Schinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a Davenport-Schinzel sequence composed of n symbols is 6(noc(n»), where t1.(n)is the functional inverse of Ackermann's function, and is thus very slowly increasing to infinity. This is achieved by establishing an equivalence between such sequences and generalized path compression schemes on rooted trees, and then by analyzing these schemes.
In particular, the upper (lower) envelope of a set of n line segments is a piecewise-linear function that consists of Θ(nα(n)) linear segments REF in the worst case.
Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes
Combinatorica
Mathematics Computer Science