Philippe Duchon

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.

On the enumeration and generation of generalized Dyck words

Abstract We provide a new algebraic grammar for generalized Dyck languages as introduced by Labelle and Yeh (Discrete Math. 82 (1990) 1–6). The study of these languages leads to the particular sublanguages of words without proper factors belonging to the studied language. A random generation scheme is shown for generalized Dyck languages, which leads to some asymptotic results. In the two-letter case, for which the words correspond to ‘rational slope Dyck paths’, more exact and asymptotic enumerative results are obtained, including the asymptotic average area to integer or 3 2 slope Dyck paths.

Towards small world emergence

We investigate the problem of optimizing the routing performance of a virtual network by adding extra random links. Our asynchronous and distributed algorithm ensures, by adding a single extra link per node, that the resulting network is a navigable small world, i.e., in which greedy routing, using the distance in the original network, computes paths of polylogarithmic length between any pair of nodes with probability 1-<i>O</i>(1/<i>n</i>). Previously known small world augmentation processes require the global knowledge of the network and centralized computations, which is unrealistic for large decentralized networks. Our algorithm, based on a careful multi-layer sampling of the nodes and the construction of a light overlay network, bypasses these limitations. For bounded growth graphs, i.e., graphs where, for any node <i>u</i> and any radius <i>r</i> the number of nodes within distance 2<i>r</i> from <i>u</i> is at most a constant times the number of nodes within distance <i>r</i>, our augmentation process proceeds with high probability in <i>O</i>(log <i>n</i> log <i>D</i>) communication rounds, with <i>O</i>(log <i>n</i> log <i>D</i>) messages of size <i>O</i>(log <i>n</i>) bits sent per node and requiring only <i>O</i>(log <i>n</i> log <i>D</i>) bit space in each node, where <i>n</i> is the number of nodes, and <i>D</i> the diameter. In particular, with the only knowledge of original distances, greedy routing computes, between any pair of nodes in the augmented network, a path of length at most <i>O</i>(log² <i>n</i> log² <i>D</i>) with probability 1 - <i>O</i>(1/<i>n</i>), and of expected length <i>O</i>(log <i>n</i> log² <i>D</i>). Hence, we provide a distributed scheme to augment any bounded growth graph into a small world with high probability in polylogarithmic time while requiring polylogarithmic memory. We consider that the existence of such a lightweight process might be a first step towards the definition of a more general construction process that would validate Kleinbergs model as a plausible explanation for the small world phenomenon in large real interaction networks.

Q-grammars and wall polyominoes

We use a variant of theq-grammar method to write functional equations for the generating functions of a subclass of vertically convex polyominoes and directed walks, according to specified parameters, which include the area. The form of these equations, and some simple singularity computations, are used to prove that the area of wall polyominoes of perimeter 2n has the Airy distribution as a limit law.

Could any graph be turned into a small-world?

In the last decade, effective measurements of real interaction networks have revealed specific unexpected properties. Among these, most of these networks present a very small diameter and a high clustering. Furthermore, very short paths can be effciently found between any pair of nodes without global knowledge of the network (i.e., in a decentralized manner) which is known as the small-world phenomenon [1]. Several models have been proposed to explain this phenomenon [2,3]. However, Kleinberg showed in [4] that these models lack the essential navigability property: in spite of a polylogarithmic diameter, decentralized routing requires the visit of a polynomial number of nodes in these models.

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.

New Strahler numbers for rooted plane trees

In this paper, wepresent an extension of Strahler numbers to rooted plane trees. Several asymptotic properties are proved; others are conjectured. We also describe several applications of this extension.

Optimal Randomized Self-stabilizing Mutual Exclusion on Synchronous Rings

We propose several self-stabilizing protocols for unidirectional, anonymous, and uniform synchronous rings of arbitrary size, where processors communicate by exchanging messages. When the size of the ring n is unknown, we better the service time by a factor of n (performing the best possible complexity for the stabilization time and the memory consumption). When the memory size is known, we present a protocol that is optimal in memory (constant and independant of n), stabilization time, and service time (both are in Θ(n)).

Distributed Approximation Algorithm for Resource Clustering

In this paper, we consider the clustering of resources on large scale platforms. More precisely, we target parallel applications consisting of independant tasks, where each task is to be processed on a different cluster. In this context, each cluster should be large enough so as to hold and process a task, and the maximal distance between two hosts belonging to the same cluster should be small in order to minimize latencies of intra-cluster communications. This corresponds to maximum bin covering with an extra distance constraint. We describe a distributed approximation algorithm that computes resource clustering with coordinates in i¾? in O(log2n) steps and O(nlogn) messages, where nis the overall number of hosts. We prove that this algorithm provides an approximation ratio of

Approximation algorithms for energy minimization in Cloud service allocation under reliability constraints

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