Helmut Prodinger

Mellin transforms and asymptotics: digital sums

Abstract Arithmetic functions related to number representation systems exhibit various periodicity phenomena. For instance, a well-known theorem of Delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function. We show that such periodicity phenomena can be analyzed rather systematically using classical tools from analytic number theory, namely the Mellin—Perron formulae. This approach yields naturally the Fourier series involved in the expansions of a variety of digital sums related to number representation systems.

How to select a loser

Abstract N people select a loser by flipping coins. Recursively, the 0-party continues until the loser is found. Among other things, it is shown that this process stops on the average after about log2 N steps. Nevertheless, this very plausible result requires rather advanced methods.

Spanning tree formulas and chebyshev polynomials

AbstractThe Kirchhoff Matrix Tree Theorem provides an efficient algorithm for determiningt(G), the number of spanning trees of any graphG, in terms of a determinant. However for many special classes of graphs, one can avoid the evaluation of a determinant, as there are simple, explicit formulas that give the value oft(G). In this work we show that many of these formulas can be simply derived from known properties of Chebyshev polynomials. This is demonstrated for wheels, fans, ladders, Moebius ladders, and squares of cycles. The method is then used to derive a new spanning tree formula for the complete prismRn(m) =Km ×Cn. It is shown that

Combinatorics of geometrically distributed random variables: left-to-right maxima

2^{\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\left( {1 - \frac{1}{{r - 1}} + o\left( 1 \right)} \right)}

Analysis of Hoare's FIND algorithm with median-of-three partition

whereTn(x) is thenth order Chebyshev polynomial of the first kind.

A result in order statistics related to probabilistic counting

Abstract This study provides a detailed analysis of a function which Knuth discovered to play a central role in the analysis of hashing with linear probing. The function, named after Knuth Q ( n ), is related to several of Ramanujans investigations. It surfaces in the analysis of a variety of algorithms and discrete probability problems including hashing, the birthday paradox, random mapping statistics, the “rho” method for integer factorization, union-find algorithms, optimum caching, and the study of memory conflicts. A process related to the complex asymptotic methods of singularity analysis and saddle point integrals permits to precisely quantify the behaviour of the Q ( n ) function. In this way, tight bounds are derived. They answer a question of Knuth ( The Art of Computer Programming , Vol. 1, 1968, [Ex. 1.2.11.3.13]), itself a rephrasing of earlier questions of Ramanujan in 1911–1913.

On Carlitz Compositions

Abstract Assume that the numbers x1, …, xn are the output of n independent geometrically distributed random variables. The number xi is a left-to-right maximum if it is greater (or equal, for a variation) than x1, …, xi−1. A precise average case analysis is performed for the parameter ‘number of left-to-right maxima’. The methods include generating functions and a technique from complex analysis, called Rices method. Some additional results are also given.

The path length of random skip lists

In this paper we give exact and asymptotic analysis for variance of the external path length in a symmetric digital trie. This problem was open up to now. We prove that for the binary symmetric trie the variance is asymptotically equal to 4.35…·n+nf(log2n) where n is the number of stored records and f(x) is a periodic function with a very small amplitude.