Stochastic analysis of the merge-sort algorithm

The number L-k of comparisons performed by the Merge-Sort algorithm to sort 2(k) keys follows the recursion L-k(d) = Lk-1 + <(Lk-1)over bar> + Y-k, where Lk-1, <(Lk-1)over bar>, and Y-k are independent and <(Lk-1)over bar> is a copy of Lk-1. The contraction method for ideal metrics of Rachev and Ruschendorf [7] is appropriate to handle recursions of the above type and to show asymptotic normality of the normalized L-k. Additional challenges have to be faced if the number of keys is not a power of 2. The difficulty lies in the fact that the cases of odd and even numbers differ slightly. Therefore, exact calculations of mean and variance have to be substituted by asymptotic results which are gained with the help of a run of a C-program. But even in that case asymptotic normality can be achieved by a refinement of the contraction principle. (C) 1997 John Wiley & Sons, Inc.