Probabilistic analysis of some distributed algorithms

Projet EURECAIn this paper, we analyze : a storage allocation algorithm which permits to maintain two stacks inside a shared (contiguous) memory area of a fixed size - the well-known banker algorithm which plays a fundamental role in parallel processing. The natural formulation of these problems is in terms of constrained random walks. Our results rely on diffusion techniques

Probabilistic model checking of complex biological pathways

Probabilistic model checking is a formal verification technique that has been successfully applied to the analysis of systems from a broad range of domains, including security and communication protocols, distributed algorithms and power management. In this paper we illustrate its applicability to a complex biological system: the FGF (Fibroblast Growth Factor) signalling pathway. We give a detailed description of how this case study can be modelled in the probabilistic model checker PRISM, discussing some of the issues that arise in doing so, and show how we can thus examine a rich selection of quantitative properties of this model. We present experimental results for the case study under several different scenarios and provide a detailed analysis, illustrating how this approach can be used to yield a better understanding of the dynamics of the pathway

Singletons for Simpletons: Revisiting Windowed Backoff with Chernoff Bounds

Backoff algorithms are used in many distributed systems where multiple devices contend for a shared resource. For the classic balls-into-bins problem, the number of singletons - those bins with a single ball - is important to the analysis of several backoff algorithms; however, existing analyses employ advanced probabilistic tools to obtain concentration bounds. Here, we show that standard Chernoff bounds can be used instead, and the simplicity of this approach is illustrated by re-analyzing some well-known backoff algorithms

RDA: A Coq Library to Reason about Randomised Distributed Algorithms in the Message Passing Model

Distributed algorithms have received considerable attention and were studied intensively in the past few decades. Under some hypotheses on the distributed system, there is no deterministic solution to certain classical problems. Randomised solutions are then needed to solve those problems. Probabilistic algorithms are generally simple to formulate. However, their analysis can become very complex, especially in the field of distributed computing. In this paper, we formally model in Coq a class of randomised distributed algorithms. We develop some tools to help proving impossibility results about classical problems and analysing this class of algorithms. As case studies, we examine the handshake and maximal matching problems. We show how to use our tools to formally prove properties about algorithms solving those problems

Probabilistic Tools for the Analysis of Randomized Optimization Heuristics

This chapter collects several probabilistic tools that proved to be useful in the analysis of randomized search heuristics. This includes classic material like Markov, Chebyshev and Chernoff inequalities, but also lesser known topics like stochastic domination and coupling or Chernoff bounds for geometrically distributed random variables and for negatively correlated random variables. Most of the results presented here have appeared previously, some, however, only in recent conference publications. While the focus is on collecting tools for the analysis of randomized search heuristics, many of these may be useful as well in the analysis of classic randomized algorithms or discrete random structures.Comment: 91 page

Probabilistic model checking of complex biological pathways

Probabilistic model checking is a formal verification technique that has been successfully applied to the analysis of systems from a broad range of domains, including security and communication protocols, distributed algorithms and power management. In this paper we illustrate its applicability to a complex biological system: the FGF (Fibroblast Growth Factor) signalling pathway. We give a detailed description of how this case study can be modelled in the probabilistic model checker PRISM, discussing some of the issues that arise in doing so, and show how we can thus examine a rich selection of quantitative properties of this model. We present experimental results for the case study under several different scenarios and provide a detailed analysis, illustrating how this approach can be used to yield a better understanding of the dynamics of the pathway

Probabilistic and parallel algorithms for centroidal Voronoi tessellations with application to meshless computing and numerical analysis on surfaces

Centroidal Voronoi tessellations (CVT) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. Such tessellations are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, optimal allocation of resources, and grid generation. A detailed review is given in chapter 1. In chapter 2, some probabilistic methods for determining centroidal Voronoi tessellations and their parallel implementation on distributed memory systems are presented. The results of computational experiments performed on a CRAY T3E-600 system are given for each algorithm. These demonstrate the superior sequential and parallel performance of a new algorithm we introduce. Then, new algorithms are presented in chapter 3 for the determination of point sets and associated support regions that can then be used in meshless computing methods. The algorithms are probabilistic in nature so that they are totally meshfree, i.e., they do not require, at any stage, the use of any coarse or fine boundary conforming or superimposed meshes. Computational examples are provided that show, for both uniform and non-uniform point distributions that the algorithms result in high-quality point sets and high-quality support regions. The extensions of centroidal Voronoi tessellations to general spaces and sets are also available. For example, tessellations of surfaces in a Euclidean space may be considered. In chapter 4, a precise definition of such constrained centroidal Voronoi tessellations (CCVT\u27s) is given and a number of their properties are derived, including their characterization as minimizers of a kind of energy. Deterministic and probabilistic algorithms for the construction of CCVT\u27s are presented and some analytical results for one of the algorithms are given. Some computational examples are provided which serve to illustrate the high quality of CCVT point sets. CCVT point sets are also applied to polynomial interpolation and numerical integration on the sphere. Finally, some conclusions are given in chapter 5

Leader Election in Anonymous Rings: Franklin Goes Probabilistic

We present a probabilistic leader election algorithm for anonymous, bidirectional, asynchronous rings. It is based on an algorithm from Franklin, augmented with random identity selection, hop counters to detect identity clashes, and round numbers modulo 2. As a result, the algorithm is finite-state, so that various model checking techniques can be employed to verify its correctness, that is, eventually a unique leader is elected with probability one. We also sketch a formal correctness proof of the algorithm for rings with arbitrary size