A Quantitative Study of Pure Parallel Processes

In this paper, we study the interleaving -- or pure merge -- operator that most often characterizes parallelism in concurrency theory. This operator is a principal cause of the so-called combinatorial explosion that makes very hard - at least from the point of view of computational complexity - the analysis of process behaviours e.g. by model-checking. The originality of our approach is to study this combinatorial explosion phenomenon on average, relying on advanced analytic combinatorics techniques. We study various measures that contribute to a better understanding of the process behaviours represented as plane rooted trees: the number of runs (corresponding to the width of the trees), the expected total size of the trees as well as their overall shape. Two practical outcomes of our quantitative study are also presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random sampling of concurrent runs. These provide interesting responses to the combinatorial explosion problem

Two problems on the generation of linear extensions of posets

We describe two results on the generation of linear extensions of a poset. First, we prove that the linear extensions of every poset can be generated by insertion. Next, we describe a constant average time algorithm to generate the linear extensions of a series-parallel poset in lexicographic order .Apresentamos dois resultados sobre a geração de extensões lineares de um poset. Primeiro provamos que as extensões lineares de todo poset podem ser geradas por inserção. A seguir, descrevemos um algoritmo de tempo médio constante para gerar as extensões lineares de um poset série-paralelo em ordem lexicográfica

Identifying Infection Sources and Regions in Large Networks

Identifying the infection sources in a network, including the index cases that introduce a contagious disease into a population network, the servers that inject a computer virus into a computer network, or the individuals who started a rumor in a social network, plays a critical role in limiting the damage caused by the infection through timely quarantine of the sources. We consider the problem of estimating the infection sources and the infection regions (subsets of nodes infected by each source) in a network, based only on knowledge of which nodes are infected and their connections, and when the number of sources is unknown a priori. We derive estimators for the infection sources and their infection regions based on approximations of the infection sequences count. We prove that if there are at most two infection sources in a geometric tree, our estimator identifies the true source or sources with probability going to one as the number of infected nodes increases. When there are more than two infection sources, and when the maximum possible number of infection sources is known, we propose an algorithm with quadratic complexity to estimate the actual number and identities of the infection sources. Simulations on various kinds of networks, including tree networks, small-world networks and real world power grid networks, and tests on two real data sets are provided to verify the performance of our estimators

The Combinatorics of Non-determinism

A deep connection exists between the interleaving semantics of concurrent processes and increasingly labelled combinatorial structures. In this paper we further explore this connection by studying the rich combinatorics of partially increasing structures underlying the operator of non-deterministic choice. Following the symbolic method of analytic combinatorics, we study the size of the computation trees induced by typical non-deterministic processes, providing a precise quantitative measure of the so-called "combinatorial explosion" phenomenon. Alternatively, we can see non-deterministic choice as encoding a family of tree-like partial orders. Measuring the (rather large) size of this family on average offers a key witness to the expressiveness of the choice operator. As a practical outcome of our quantitative study, we describe an efficient algorithm for generating computation paths uniformly at random