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Longest Increasing Subsequences of Randomly Chosen Multi-Row Arrays

Published online by Cambridge University Press:  02 October 2014

MARCOS KIWI and
JOSÉ A. SOTO
MARCOS KIWI
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático UMI 2807, Universidad de Chile (e-mail: mk@dim.uchile.cl)
JOSÉ A. SOTO
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático UMI 2807, Universidad de Chile (e-mail: jsoto@dim.uchile.cl)
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Abstract

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A two-row array of integers

\[\alpha_{n}= \begin{pmatrix}a_1 & a_2 & \cdots & a_n\\ b_1 & b_2 & \cdots & b_n \end{pmatrix}\]
is said to be in lexicographic order if its columns are in lexicographic order (where character significance decreases from top to bottom, i.e., either ak < a k+1, or bk ≤ b k+1 when ak = a k+1). A length ℓ (strictly) increasing subsequence of αn is a set of indices i 1 < i 2 < ⋅⋅⋅ < i ℓ such that a i1 < a i2 < ⋅⋅⋅ < a iℓ and b i1 < b i2 < ⋅⋅⋅ < b iℓ . We are interested in the statistics of the length of a longest increasing subsequence of αn chosen according to ${\cal D}$ n , for different families of distributions ${\cal D} = ({\cal D}_{n})_{n\in\NN}$ , and when n goes to infinity. This general framework encompasses well-studied problems such as the so-called longest increasing subsequence problem, the longest common subsequence problem, and problems concerning directed bond percolation models, among others. We define several natural families of different distributions and characterize the asymptotic behaviour of the length of a longest increasing subsequence chosen according to them. In particular, we consider generalizations to d-row arrays as well as symmetry-restricted two-row arrays.

Keywords

60C05

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 1: Honouring the Memory of Philippe Flajolet - Part 3 , January 2015, pp. 254 - 293
Copyright
Copyright © Cambridge University Press 2014 

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