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For a large class of examples arising in statistical ph~ics known as attnxtiue spin @ems (e.g., the Ising model), one seeks to sample from a probability distribution T on an enormously large state space, but elementary sampling is ruled out by the infeaaibility of calculating an appropriate normalizing constant. The same difficulty arises in computer… (More)

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)

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)

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)

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)

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)

We revisit the classical QuickSort and QuickSelect algorithms , under a complexity model that fully takes into account the elementary comparisons between symbols composing the records to be processed. Our probabilistic models belong to a broad category of information sources that encompasses memoryless (i.e., independent-symbols) and Markov sources, as well… (More)

- James Allen Fill
- 2007

A well-known theorem usually attributed to Keilson states that, for an irreducible continuous-time birth-and-death chain on the nonnegative integers and any d, the passage time from state 0 to state d is distributed as a sum of d independent exponential random variables. Until now, no probabilistic proof of the theorem has been known. In this paper we use… (More)