Showing papers by "Michael Drmota published in 2012"

A precise analysis of Cuckoo hashing

Abstract: Cuckoo hashing was introduced by Pagh and Rodler in 2001. Its main feature is that it provides constant worst-case search time. The aim of this article is to present a precise average case analysis of Cuckoo hashing. In particular, we determine the probability that Cuckoo hashing produces no conflicts and give an upper bound for the construction time, that is linear in the size of the table. The analysis rests on a generating function approach to the so called Cuckoo Graph, a random bipartite graph, and an application of a double saddle point method to obtain asymptotic expansions. Furthermore, we provide some results concerning the structure of these kinds of random graphs. Our results extend the analysis of Devroye and Morin [2003]. Additionally, we provide numerical results confirming the mathematical analysis.

Generalized Thue-Morse sequences of squares

Abstract: We consider compact group generalizations T(n) of the Thue-Morse sequence and prove that the subsequence T(n2) is uniformly distributed with respect to a measure gv that is absolutely continuous with respect to the Haar measure. The proof is based on a proper generalization of the Fourier based method of Mauduit and Rivat in their study of the sum-of-digits function of squares to group representations.

On the discrepancy of some generalized kakutani's sequences of partitions

Abstract: In this paper we study a class of generalized Kakutani's sequences of partitions of (0;1), constructed by using the technique of successive refinements. Our main focus is to derive bounds for the discrepancy of these sequences. The approach that we use is based on a tree representation of the sequence of parti- tions which is precisely the parsing tree generated by Khodak's coding algorithm. With the help of this technique we derive (partly up to a logarithmic factor) optimal upper bound in the so-called rational case. The upper bounds in the irra- tional case that we obtain are weaker, since they heavily depend on Diophantine approximation properties of a certain irrational number. Finally, we present an application of these results to a class of fractals.

A central limit theorem for the number of degree- k vertices in random maps

Abstract: We prove that the number of vertices of given degree in random planar maps satisfies a central limit theorem with mean and variance that are asymptotically linear in the number of edges. The proof relies on an analytic version of the quadratic method and singularity analysis of multivariate generating functions.

Mutual information for a deletion channel

Abstract: We study the binary deletion channel where each input bit is independently deleted according to a fixed probability. We relate the conditional probability distribution of the output of the deletion channel given the input to the hidden pattern matching problem. This yields a new characterization of the mutual information between the input and output of the deletion channel. Through this characterization we are able to comment on the the deletion channel capacity, in particular for deletion probabilities approaching 0 and 1.

Infinite Systems of Functional Equations and Gaussian Limiting Distributions

Abstract: In this paper infinite systems of functional equations in finitely or infinitely many random variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating function of the system tend to a finite or infinite dimensional limiting distribution.

The maximum degree of random planar graphs

Abstract: Let Pn denote a graph drawn uniformly at random from the class of all simple planar graphs with n vertices. We show that the maximum degree of a vertex in Pn is with probability 1 − o(1) asymptotically equal to c log n, where c a 2.529 is determined explicitly. A similar result is also true for random 2-connected planar graphs.Our analysis combines two orthogonal methods that complement each other. First, in order to obtain the upper bound, we resort to exact methods, i.e., to generating functions and analytic combinatorics. This allows us to obtain fairly precise asymptotic estimates for the expected number of vertices of any given degree in Pn. On the other hand, for the lower bound we use Boltzmann sampling. In particular, by tracing the execution of an adequate algorithm that generates a random planar graph, we are able to explicitly find vertices of sufficiently high degree in Pn.

Extremal Statistics on Non-Crossing Configurations

Abstract: We obtain several properties of extremal statistics in non-crossing configurations with n vertices. We prove that the maximum degree and the largest component are of logarithmic order, and the diameter is of order $\sqrt{n}$. The proofs are based on singularity analysis, an application of the first and second moment method and on the analysis of iterated functions.