Guy Louchard

Boltzmann Samplers for the Random Generation of Combinatorial Structures

This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class – an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.

Kac's formula, levy's local time and brownian excursion

Kacs formula for Brownian functionals and Levys local time decomposition are shown to be useful tools in analysing Brownian excursion properties. These tools are applied to maximum, local time and area distributions. Some curious connections between some of these distributions are explained by simple The original purpose of this paper was to find a workable expression for the transform of the Brownian excursion area. We wanted to use two well-known results: Kacs formula for Brownian functionals and Levys local time decompos- ition. We actually found that these two powerful tools lead also to alternative and sometimes simpler proofs for other results on Brownian excursion prob- abilities: maximum and local time distributions. We were also able to derive a useful general result on Brownian excursion symmetric additive functionals. Finally, we also wanted to understand some curious relations that had been observed by earlier authors: connections between hitting-time densities and maximum distributions. Using simple probabilistic arguments and a formula for Jacobis third 0 function, we can explain these connections in direct terms. The paper is organized as follows. In Section 1, we summarize basic notations and known results. Sections 2 and 3 are short surveys of Kacs and Levys results. These tools are applied in Sections 4 and 5 to Brownian excursion maximum and local time. Section 6 contains the main results of the paper: a useful new form for the transform of the Brownian excursion area density and a simple relation for functionals based on a symmetric positive function. Section 7 explains those curious relations alluded to earlier between Brownian excursion probabilities. 1. Basic notations and known results

The brownian excursion area: a numerical analysis

Abstract Using an explicit expression for the Laplace transform of the Brownian excursion area generating function, a numerical analysis gives moments, density, and distribution function. An asymptotic formula is given for small argument.

Phase transition for parking blocks, Brownian excursion and coalescence

In this paper, we consider hashing with linear probing for a hashing table with m places, n items (n > m), and l = m - n empty places. For a noncomputer science-minded reader, we shall use the metaphore of n cars parking on m places: each car ci chooses a place pi at random, and if pi is occupied, ci tries successively pi + 1, pi + 2, until it finds an empty place. Pittel [42] proves that when l/m goes to some positive limit β > 1, the size B1m,l1 of the largest block of consecutive cars satisfies 2(β - 1 - log β)B1m,l - 3 log log m + Ξm, where Ξm converges weakly to an extreme-value distribution. In this paper we examine at which level for n a phase transition occurs between B1m,l = o(m) and m - B1m,l = o(m). The intermediate case reveals an interesting behavior of sizes of blocks, related to the standard additive coalescent in the same way as the sizes of connected components of the random graph are related to the multiplicative coalescent.

Random walks, Gaussian processes and list structures

Abstract An asymptotic analysis of list structure properties leads to limiting Gaussian Markovian processes. Several costs functions are shown to have asymptotic normal distributions.

Asymptotics of the Moments of Extreme-Value Related Distribution Functions

The asymptotic cost of many algorithms and combinatorial structures is related to the extreme-value Gumbel distribution exp(-exp(-x)). The following list is not exhaustive: Trie, Digital Search Tree, Leader Election, Adaptive Sampling, Counting Algorithms, trees related to the Register Function, Composition of Integers, some structures represented by Markov chains (Column-Convex Polyominoes, Carlitz Compositions), Runs and number of distinct values of some multiplicity in sequences of geometrically distributed random variables. Sometimes we can start from an exact (discrete) probability distribution, sometimes from an asymptotic analysis of the discrete objects (e.g., urn models) before establishing the relationship with the Gumbel distribution function. Also some Markov chains are either exactly and directly given by the structure itself, or as a limiting Markov process. The main motivation of the paper is to compute the asymptotic distribution and the moments of the random variables in question. The moments are usually given by a dominant part and a small fluctuating part. We use Laplace and Mellin transforms and singularity analysis and aim for a unified treatment in all cases. Furthermore, our goal is a purely mechanical computation of dominant and fluctuating components, with the help of a computer algebra system. We provide each time the first three moments, but the treatment is (almost) completely automatic. We need some real analysis for the approximations and apart from that only easy complex analysis; simple poles and a few special functions.

Approximation of eigencharacteristics in nearly-completely decomposable stochastic systems

We analyse the accuracy with which the eigensystem of a nearly-completely decomposable stochastic matrix may be expressed as a function of the eigensystems of its principal submatrices and of the maximum degree of coupling between these submatrices.

Probabilistic analysis of some distributed algorithms

In this paper we analyze: (i) a storage allocation algorithm (D.E. Knuth, The Art of Computer Programming-Vol. I, Addison-Wesley, Reading, MA, 1969, Ex. 2.2.2.13) which permits one to maintain two stacks inside a shared (continuous) memory area of a fixed size, and (ii) the well-known banker algorithm which plays a fundamental role in parallel processing (J. Francon, Combinatoire et Parallelisme; Journees Informatique et Mathematique; Lumigny, October 15-17, 1987; A. N. Habermann, in Current Trends in Programming Methodology, Vol. 3, K. M. Chandy and R. T. Yeh, Eds., Prentice-Hall, Englewood Cliffs, NJ, 1987; J. Peterson and A. Silberschatz, Operating Systems Concepts, Addison-Wesley, Reading, MA, 1983). The natural formulation of the problems to be solved here is in terms of random walks. For (i) the random walk Ym(-) takes place in a triangle in a two-dimensional lattice space with two reflecting barriers along the axes (a deletion has no effect on an empty stack) and one absorbing barrier along the diagonal (the algorithm stops when the combined sizes of the stacks exhaust the available storage). For (ii) the random walk takes place in a rectangle with broken corner and has four reflecting barriers and one absorbing barrier (see Fig. 10). With the help of diffusion techniques, we obtain, asymptotically as m: ∞. the distributions of the hitting place (Zm) and hitting time (Tm) on the absorbing boundary. the joint distribution of Zm and Tm. the distribution P[Ym(n) ≦ ym, n < Tm].

Random Sampling from Boltzmann Principles

This extended abstract proposes a surprisingly simple framework for the random generation of combinatorial configurations based on Boltzmann models. Random generation of possibly complex structured objects is performed by placingan appropriate measure spread over the whole of a combinatorial class. The resultingalg orithms can be implemented easily within a computer algebra system, be analysed mathematically with great precision, and, when suitably tuned, tend to be efficient in practice, as they often operate in linear time.

Brownian motion and algorithm complexity

The Brownian motion is shown to be a useful tool in analysing some sorting and tree manipulation algorithms.