Introduction to algorithms
Descending subsequences of random permutations
Journal of Combinatorial Theory Series A
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We develop computationally feasible algorithms to numerically investigate the asymptotic behavior of the length Hd(n) of a maximal chain (longest totally ordered subset) of a set of n points drawn from a uniform distribution on the d-dimensional unit cube Vd = [0, 1]d. For d ≥ 2, it is known that cd(n) = Hd(n)/n1/d converges in probability to a constant cd e, with limd→∞ cd = e. For d = 2, the problem has been extensively studied, and, it is known that c2 = 2. Monte Carlo simulations coupled with the standard dynamic programming algorithm for obtaining the length of a maximal chain do not yield computationally feasible experiments. We show that Hd(n) can be estimated by considering only the chains that are close to the diagonal of the cube and develop efficient algorithms for obtaining the maximal chain in this region of the cube. We use the improved algorithm together with a linearity conjecture regarding the asymptotic behavior of cd(n) to obtain even faster convergence to cd. We present experimental simulations to demonstrate our results and produce new estimates of cd for d ∈ {3,...,6}.