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Longest Increasing Subsequences of Randomly Chosen Multi-Row Arrays

  • MARCOS KIWI (a1) and JOSÉ A. SOTO (a2)

Abstract

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 α n is a set of indices i 1 < i 2 < ⋅⋅⋅ < i such that a i 1 < a i 2 < ⋅⋅⋅ < a i and b i 1 < b i 2 < ⋅⋅⋅ < b i . We are interested in the statistics of the length of a longest increasing subsequence of α n 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.

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[1] Aldous, D. and Diaconis, P. (1999) Longest increasing subsequences: From patience sorting to the Baik–Deift–Johansson theorem. Bull. Amer. Math. Soc. 36 413432.
[2] Arlotto, A., Chen, R. W., Shepp, L. A. and Steele, J. M. (2011) Online selection of alternating subsequences from a random sample. J. Appl. Prob. 48 11141132.
[3] Arlotto, A. and Steele, J. M. (2011) Optimal sequential selection of a unimodal subsequence of a random sequence. Combin. Probab. Comput. 20 799814.
[4] Arlotto, A. and Steele, J. M. (2014) Optimal sequential selection of an alternating subsequence: A central limit theorem. Adv. Appl. Probab. 46 536559.
[5] Babaioff, M., Immorlica, N., Kempe, D. and Kleinberg, R. (2007) A knapsack secretary problem with applications. In Proc. 10th APPROX and 11th RANDOM, pp. 16–28.
[6] Babaioff, M., Immorlica, N. and Kleinberg, R. (2007) Matroids, secretary problems, and online mechanisms. In Proc. 18th SODA, pp. 434–443.
[7] Baik, J., Deift, P. and Johansson, K. (1999) On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 11191178.
[8] Baik, J. and Rains, E. (2001) Symmetrized random permutations. In Random Matrix Models and their Applications (Bleher, P. M. and Its, A. R., eds), Vol. 40 of Mathematical Sciences Research Institute Publications, Cambridge University Press, pp. 119.
[9] Baryshnikov, Y. and Gnedin, A. V. (2000) Sequential selection of an increasing sequence from a multidimensional random sample. Ann. Appl. Probab. 10 258267.
[10] Bollobás, B. and Brightwell, B. (1992) The height of a random partial order: Concentration of measure. Ann. Probab. 2 10091018.
[11] Bollobás, B. and Winkler, P. (1988) The longest chain among random points in Euclidean space. Proc. Amer. Math. Soc. 103 347353.
[12] Boshuizen, F. A. and Kertz, R. P. (1999) Smallest-fit selection of random sizes under a sum constraint: Weak convergence and moment comparisons. Adv. Appl. Probab. 31 178198.
[13] Bruss, F. T. and Delbaen, F. (2001) Optimal rules for the sequential selection of monotone subsequences of maximum expected length. Stoch. Proc. Appl. 96 313342.
[14] Bruss, F. T. and Delbaen, F. (2004) A central limit theorem for the optimal selection process for monotone subsequences of maximum expected length. Stoch. Proc. Appl. 114 287311.
[15] Bruss, F. T. and Robertson, J. B. (1991) `Wald's Lemma' for sums of order statistics of i.i.d. random variables. Adv. Appl. Probab. 23 612623.
[16] Chvátal, V. and Sankoff, D. (1975) Longest common subsequences of two random sequences. J. Appl. Probab. 12 306315.
[17] Coffman, E. G., Flatto, L. and Weber, R. R. (1987) Optimal selection of stochastic intervals under a sum constraint. Adv. Appl. Probab. 19 454473.
[18] Dinitz, M. (2013) Recent advances on the matroid secretary problem. SIGACT News 44 126142.
[19] Dynkin, E. B. (1963) The optimum choice of the instant for stopping a Markov process. Sov. Math. Doklady 4 627629.
[20] Gilbert, J. P. and Mosteller, F. (1966) Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61 3573.
[21] Gnedin, A. V. (2000) A note on sequential selection of permutations. Combin. Probab. Comput. 9 1317.
[22] Hardy, G., Littlewood, J. E. and Pólya, G. (1952) Inequalities, second edition, Cambridge University Press.
[23] Janson, S., Łuczak, T. and Rucinski, A. (2000) Random Graphs, Wiley.
[24] Kiwi, M. (2006) A concentration bound for the longest increasing subsequence of a randomly chosen involution. Discrete Appl. Math. 154 18161823.
[25] Kiwi, M., Loebl, M. and Matoušek, J. (2005) Expected length of the longest common subsequence for large alphabets. Adv. Math. 197 480498.
[26] Odlyzko, A. and Rains, E. (1998) On longest increasing subsequences in random permutations. Technical report, AT&T Labs.
[27] Rhee, W. and Talagrand, M. (1991) A note on the selection of random variables under a sum constraint. J. Appl. Probab. 28 919923.
[28] Samuels, S. M. and Steele, J. M. (1981) Optimal sequential selection of a monotone sequence from a random sample. Ann. Appl. Probab. 9 937947.
[29] Seppäläinen, T. (1997) Increasing sequences of independent points on the planar lattice. Ann. Appl. Probab. 7 886898.
[30] Stanley, R. (2002) Recent progress in algebraic combinatorics. Bull. Amer. Math. Soc. 40 5568.

Keywords

Longest Increasing Subsequences of Randomly Chosen Multi-Row Arrays

  • MARCOS KIWI (a1) and JOSÉ A. SOTO (a2)

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