Abstract not found

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.

A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.

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

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.

Abstract not found

A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type (e x 2 ), that is, Gaussian. We exhibit here a new class of \universal" phenomena that are of the exponential-cubic type (e ix 3 ), corresponding to nonstandard distributions that involve the Airy function. Such Airy phenomena are expected to be found in a number of applications, when conuences of critical points and singularities occur. About a dozen classes of planar maps are treated in this way, leading to the occurrence of a common Airy distribution that describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and ne optimization of random generation algorithms for multiply connected planar graphs.

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.

supported by the DFG (GRK 588/1) Abstract. We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in polynomial time in n. To generate labeled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block structure, and compute the exact number of labeled outerplanar graphs. This allows us to make the correct probabilistic choices in a recursive generation of uniformly distributed outerplanar graphs. Next we modify our formulas to count rooted unlabeled graphs, and finally show how to use these formulas in a Las Vegas algorithm to generate unlabeled outerplanar graphs uniformly at random in expected polynomial time. random structures, outerplanar graphs, efficient counting, uniform generation 1

We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number gn of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and gn is asymptotically g n −5/2 ρ −n, where g ≈ 0.00909941 and ρ −1 ≈ 7.50360 can be approximated. Using our enumerative results we investigate several statistical properties of random unlabeled outerplanar graphs on n vertices, for instance concerning connectedness, the chromatic number, and the number of edges. To obtain the results we combine classical cycle index enumeration with recent results from analytic combinatorics.