Improved Lower Bounds On the Length of Davenport Schinzel Sequences

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Improved Lower Bounds On the Length of Davenport Schinzel Sequences
Micha Sharir
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The inductive step in [HS] first -7- constructs T(k, m — 1) and an associated path compression scheme on it by temporarily ignoring the last leaf in each base-tree T. Then it constructs T{k—l, Ci^(m — l)) and an associated compression scheme by using as a base tree the tree T* obtained from T{k, m — l) by discarding all nodes other than its root and the Ci^{m-l) roots of the copies of the base tree T in T{k, m — 1). Finally, the tree r(A:-l, Q(m- 1)) and the Q_i(Q(m-l)) copies of T{k, m—1), tog...ether with their associated compression schemes, are properly merged together in the manner described in [HS] to yield a compression scheme on a tree T(k, m) having the desired properties.
(More specifically, this merging is performed as follows. T{k, m) is obtained from T{k — \, Ci^{m — V)) by replacing each copy of its base tree T* by a copy of T{k, m-\) in which each base tree T has now m leaves. The combined path compression scheme on T{k, m) is defined as follows. For a leaf / which is not the last (i.


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