• 2001
The limiting distribution of the normalized number of comparisons used by Quick-sort 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(More)
• 2004
Additive tree functionals represent the cost of many divide-and-conquer algorithms. We derive the limiting distribution of the additive func-tionals induced by toll functions of the form (a) n α when α > 0 and (b) log n (the so-called shape functional) on uniformly distributed binary search trees, sometimes called Catalan trees. The Gaussian law obtained in(More)
• 2005
We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m-ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so-called shape functional fall under this framework. The(More)
• 2012
Using a recursive approach, we obtain a simple exact expression for the L 2-distance from the limit in Régnier's [5] classical limit theorem for the number of key comparisons required by QuickSort. A previous study by Fill and Janson [1] using a similar approach found that the d2-distance is of order between n −1 log n and n −1/2 , and another by Neininger(More)
• 3
• 2000
Using Fourier analysis, we prove that the limiting distribution of the standardized random number of comparisons used by Quicksort to sort an array of n numbers has an everywhere positive and infinitely differentiable density f , and that each derivative f (k) enjoys superpolynomial decay at ±∞. In particular, each f (k) is bounded. Our method is(More)
• 2
• 2002
The number of comparisons X n used by Quicksort to sort an array of n distinct numbers has mean µ n of order n log n and standard deviation of order n. Using different methods, Régnier and Rösler each showed that the normalized variate Y n := (X n −µ n)/n converges in distribution, say to Y ; the distribution of Y can be characterized as the unique fixed(More)
We explore and relate two notions of monotonicity, stochastic and realizable, for a system of probability measures on a common nite partially ordered set (poset) S when the measures are indexed by another poset A. We give counterexamples to show that the two notions are not always equivalent, but for various large classes of S we also present conditions on(More)