Michael Drmota

Systems of functional equations

The aim of this paper is to discuss the asymptotic properties of the coefficients of generating functions which satisfy a system of functional equations. It turns out that under certain general conditions these coefficients are related to the distribution of a multivariate random variable that is asymptotically normal. As an application it turns out that the distribution of the terminal symbols in context-free languages is typically asymptotically normal.

Precise minimax redundancy and regret

Recent years have seen a resurgence of interest in redundancy of lossless coding. The redundancy (regret) of universal fixed-to-variable length coding for a class of sources determines by how much the actual code length exceeds the optimal (ideal over the class) code length. In a minimax scenario one finds the best code for the worst source either in the worst case (called also maximal minimax) or on average. We first study the worst case minimax redundancy over a class of stationary ergodic sources and replace Shtarkovs bound by an exact formula. Among others, we prove that a generalized Shannon code minimizes the worst case redundancy, derive asymptotically its redundancy, and establish some general properties. This allows us to obtain precise redundancy for memoryless, Markov, and renewal sources. For example, we present the exact constant of the redundancy for memoryless and Markov sources by showing that the integer nature of coding contributes log(logm/(m-1))/logm+o(1) where m is the size of the alphabet. Then we deal with the average minimax redundancy and regret. Our approach here is orthogonal to most recent research in this area since we aspire to show that asymptotically the average minimax redundancy is equivalent to the worst case minimax redundancy for some classes of sources. After formulating some general bounds relating these two redundancies, we prove our assertion for memoryless and Markov sources. Nevertheless, we provide evidence that maximal redundancy of renewal processes does not have the same leading term as the average minimax redundancy (however, our general results show that maximal and average regrets are asymptotically equivalent).

A Bivariate Asymptotic Expansion of Coefficients of Powers of Generating Functions

The aim of this paper is to give a bivariate asymptotic expansion of the coefficient ynk = xny(x)k, where y(x) = ? ynxn has a power series expansion with non-negative coefficients yn ? 0. Such expansions are known for k/n ? a, b with a > 0. In the first part we provide two versions of full asymptotic series expansions for ynk and in the second part we show how to generalize these expansions to the case k/n ? 0, b if y(x) has an algebraic singularity of the kind y(x) = g(x) - h (x)1 - x/x0. A concluding section provides extensions to multivariate asymptotic expansions and applications to multivariate generating functions. As a byproduct, we obtain a remarkable identity for Catalan numbers.

An analytic approach to the height of binary search trees II

It is shown that all centralized absolute moments <b>E</b>|<i>H<inf>n</inf></i> − <b>E</b><i>H<inf>n</inf></i>|^α (α ≥ 0) of the height <i>H<inf>n</inf></i> of binary search trees of size <i>n</i> and of the saturation level <i>H<inf>n</inf></i>′ are bounded. The methods used rely on the analysis of a <i>retarded</i> differential equation of the form Φ′(<i>u</i>) = −α⁻²Φ(<i>u</i>/α)² with α > 1. The method can also be extended to prove the same result for the height of <i>m</i>-ary search trees. Finally the limiting behaviour of the distribution of the height of binary search trees is precisely determined.

A functional limit theorem for the profile of search trees

We study the profile X-n,X-k of random search trees including binary search trees and m-ary search trees. Our main result is a functional limit theorem of the normalized profile X-n,X-k/EXn,k for ...

Asymptotic distributions and a multivariate Darboux method in enumeration problems

Abstract Let c(x, z) = Σ cnkxnzk (cnk ⩾ 0) be a bivariate generating function satisfying a functional equation c = G(c, x, z). By using a central limit theorem of Bender it is shown that discrete random variables Xn with P[Xn = k] = cnk/(Σ cni) are asymptotically normal with mean μn ∼ μn and variance σn2 ∼ σ2n. Furthermore a bivariate asymptotic expansion for the coefficients cnk can be obtained by two different methods. After some applications to tree enumeration problems a multivariate Darboux-method is formulated.

An analytic approach to the height of binary search trees

AbstractBy using analytic tools it is shown that the expected value of the heightHn of binary search trees of sizen is asymptotically given by EHn =c logn+

Marking in combinatorial constructions: generating functions and limiting distributions

\mathcal{O}