Experimental Evaluation of the Height of a Random Set of Points in a d-Dimensional Cube

Abstract

We develop computationally feasible algorithms to numerically investigate the asymptotic behavior of the length H d (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 c d (n) = H d (n)/n 1/d converges in probability to a constant c d < e, with limd→∞ c d = 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 H d (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 c d (n) to obtain even faster convergence to c d . We present experimental simulations to demonstrate our results and produce new estimates of c d for d ∈ {3,..., 6}.