Salvatore Certo, Anh Pham, et al.
Quantum Machine Intelligence
The 1935 result of Erdos and Szekeres that any sequence of ≥ n 2 + 1 real numbers contains a monotonic subsequence of ≥ n + 1 terms has stimulated extensive further research, including a paper of J. B. Kruskal that defined an extension of monotonicity for higher dimensions. This paper provides a proof of a weakened form of Kruskal's conjecture for 2-dimensional Euclidean space by showing that there exist sequences of n points in the plane for which the longest monotonic subsequences have length ≤ n1/2 + 3. Weaker results are obtained for higher dimensions. When points are selected at random from reasonable distributions, the average length of the longest monotonic subsequence is shown to be ∼2n1/2 as n → ∞ for each dimension.
Salvatore Certo, Anh Pham, et al.
Quantum Machine Intelligence
Shu Tezuka
WSC 1991
A.R. Conn, Nick Gould, et al.
Mathematics of Computation
Daniel J. Costello Jr., Pierre R. Chevillat, et al.
ISIT 1997