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Keywords:

  • Quicksort;
  • characteristic function;
  • density, moment generating function;
  • sorting algorithm;
  • coupling;
  • Fourier analysis;
  • Kolmogorov–Smirnov distance;
  • total variation distance;
  • integral equation;
  • numerical analysis;
  • dp-metric

Abstract

The limiting distribution of the normalized number of comparisons used by Quicksort to sort an array of n numbers is known to be the unique fixed point with zero mean of a certain distributional transformation S. We study the convergence to the limiting distribution of the sequence of distributions obtained by iterating the transformation S, beginning with a (nearly) arbitrary starting distribution. We demonstrate geometrically fast convergence for various metrics and discuss some implications for numerical calculations of the limiting Quicksort distribution. Finally, we give companion lower bounds which show that the convergence is not faster than geometric. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 376–406, 2001