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.

Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1

The recursive method formalized by Nijenhuis and Wilf [15] and systematized by Flajolet, Van Cutsem and Zimmermann [8], is extended here to floating-point arithmetic. The resulting ADZ method enables one to generate decomposable data structures -- both labelled or unlabelled -- uniformly at random, in expected O(n 1+ffl ) time and space, after a preprocessing phase of O(n 2+ffl ) time, which reduces to O(n 1+ffl ) for context-free grammars.

We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1-, 2-, and 3-connected components. For 3-connected graphs we apply a recent random generation algorithm by Schaeffer and a counting formula by Mullin and Schellenberg.

this paper, section (1.2) offers an elementary combinatorial proof of Dobinski's formula which seems simpler than other proofs in the literature (Rota [35], Berge [5], p. 44, Comtet [9], p. 211). This argument involves identities whose probabilistic interpretations are brought out later in the paper. 1.1 Notation

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This paper addresses the problem of selecting finite test sets and automating this selection. Among these methods, some are deterministic and some are statistical. The kind of statistical testing we consider has been inspired by the work of Thevenod-Fosse and Waeselynck. There, the choice of the distribution on the input domain is guided by the structure of the program or the form of its specification. In the present paper, we describe a new generic method for performing statistical testing according to any given graphical description of the behavior of the system under test. This method can be fully automated. Its main originality is that it exploits recent results and tools in combinatorics, precisely in the area of random generation of combinatorial structures. Uniform random generation routines are used for drawing paths from the set of execution paths or traces of the system under test. Then a constraint resolution step is performed, aiming to design a set of test data that activate the generated paths. This approach applies to a number of classical coverage criteria. Moreover, we show how linear programming techniques may help to improve the quality of test, i.e. the probabilities for the elements to be covered by the test process. The paper presents the method in its generality. Then, in the last section, experimental results on applying it to structural statistical software testing are reported. 1

This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using "coupling from the past" to remove initialization bias from Monte Carlo randomization.

We propose a new way of automating statistical structural testing, based on the combination of uniform generation of combinatorial structures, and of randomized constraint solving techniques. More precisely, we show how to draw test cases which balance the coverage of program structures according to structural testing criteria. The control flow graph is formalized as a combinatorial structure specification. This provides a way of uniformly drawing execution paths which have suitable properties. Once a path has been drawn, the predicate characterizing those inputs which lead to its execution is solved using a constraint solving library. The constraint solver is enriched with powerful heuristics in order to deal with resolution failures and random choice strategies.

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 placing an appropriate measure spread over the whole of a combinatorial class. The resulting algorithms 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.