James Allen Fill

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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)
Additive tree functionals represent the cost of many divide-andconquer algorithms. We derive the limiting distribution of the additive functionals induced by toll functions of the form (a) nα 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)
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)
Total path length, or search cost, for a rooted tree is defined as the sum of all root-to-node distances. Let Tn be the total path length for a random recursive tree of order n. Mahmoud (1991) showed that Wn := (Tn − E[Tn])/n converges almost surely and in L2 to a nondegenerate limiting random variable W . Here we give recurrence relations for the moments(More)
The analyses of many algorithms and data structures (such as digital search trees) for searching and sorting are based on the representation of the keys involved as bit strings and so count the number of bit comparisons. On the other hand, the standard analyses of many other algorithms (such as Quicksort) are performed in terms of the number of key(More)
We study the space requirement of m-ary search trees under the random permutation model when m ≥ 27 is fixed. Chauvin and Pouyanne have shown recently that Xn, the space requirement of an m-ary search tree on n keys, equals μ(n + 1) + 2Re [Λn2 ] + nn Reλ2 , where μ and λ2 are certain constants, Λ is a complex-valued random variable, and n → 0 a.s. and in L(More)