Randomized generation of acyclic orientations upon anonymous distributed systems
Introduction
In anonymous distributed systems, the processors do not have identity numbers. In this paper, we discuss two new randomized distributed algorithms for the generation of acyclic orientations upon anonymous distributed systems where no attribute information at all about the network is available, such as the network topology or the number of processors in it [32]. It is worth noticing that acyclic orientations are directly associated to graph colorings [30], notably with a great number of applications in the literature. The importance of coping with the lack of distinct names assignment in distributed systems is well known in other problems where symmetry breaking is essential, such as in leader election[1], edge election and topology recognition[32]. Additional applications and theoretical aspects concerning anonymous distributed algorithms can be found in [19], [20], [22], [25], [26], [31], [17], [28].
Given a target anonymous distributed system represented by an undirected graph , where and , our objective is to find an orientation over such that each vertex in has no directed path to itself, i.e., to define a directed acyclic graph (DAG). In the distributed algorithms introduced here, all nodes (processors) toss coins or dice in order to decide, based solely on local information, how to construct acyclic orientations over . Although all distributed algorithms introduced in this paper are of asynchronous nature (i.e, may not run according to a global clock, in order to facilitate the understanding of our analysis) it has assumed the view of having all processors/nodes operating simultaneously and executing the same identical algorithm.
We study the correctness, the expected time complexity and speed of convergence of two new randomized distributed algorithms, named Alg-Neighbors and Alg-Edges. Both algorithms are based on the preliminary ideas presented by Calabrese and França [14] and Arantes Jr. and França [4], [5], [2], respectively. In those randomized algorithms, the execution time defines a random variable whose associated properties are better described by probabilistic statements. Thus, once the correctness of both algorithms is assured (acyclicity), we evaluate the expected time complexity of the new proposed algorithms. Handling with randomized algorithms (in opposition to the deterministic ones) is especially interesting if one is looking for a way of generating a large amount of solutions with different characteristics. One such case is the finding of initial acyclic orientations in the Scheduling by Edge Reversal–ser–graph dynamics [7], [9]. Another situation is the case of quickly finding parallel schedules for constraint satisfaction problems [27].
The Alg-Neighbors algorithm operates with biased or unbiased dice with faces. It has been proved that, in the special case of , that the unbiased version has expected performance time complexity [6]. However, this implies a less restrictive kind of anonymous system, as considered in [32], where the cardinality of the network must be known a priori by each node. In order to overcome this hurdle, we introduce here a biased version of the Alg-Neighbors algorithm that can achieve the same linear performance, assuming constant, by assigning suitable probabilities to die faces at each node. This is obtained by introducing a new biasing function, solely based on local information, that changes the expected time complexity from sub-exponential, i.e., to , when biased dice are used. The Alg-Edges algorithm operates using unbiased dice, and its analysis, based on Karp’s probabilistic recurrence relations[24], revealed a expectation, over arbitrarily connected graphs and using dice with faces.
Furthermore, the quality of acyclic orientations produced by both Alg-Neighbors and Alg-Edges algorithms and their computational performances are discussed and a clear tradeoff is defined: while Alg-Edges can produce, at random, one acyclic orientation quasi instantaneously over an arbitrarily large system, Alg-Neighbors is able to produce acyclic orientations associated to the coloring of graphs with a relatively small number of colors. Experimental results corroborating our findings are also presented.
In Section 2, we introduce basic concepts on distributed anonymous systems and their representation using graphs. Section 3 presents and discusses all randomized distributed algorithms: both unbiased and biased versions of the Calabrese/França algorithm (Section 3.1), the Alg-Neighbors and the Alg-Edges algorithms (Sections 3.2 Alg-Neighbors, 3.3 Alg-Edges). In all cases, we study their correctness, expected time and speed of convergence. The tradeoffs between the convergence speed and quality of both Alg-Neighbors and Alg-Edges algorithms are discussed in the light of experimental data in Section 4. Finally, some conclusions and suggestions for future work are discussed in Section 5.
Section snippets
Anonymous systems
A distributed system may be described as a set of processing nodes connected by communication links (or channels), such that the communication between each pair of nodes is performed through the exchange of messages. It is usual to assume that the connections are bidirectional, which means that if two nodes are connected, they can simultaneously send and receive messages to/from each other. This is a generic definition and may be applied to a number of situations. A distributed system is said
Randomized and distributed generation of acyclic orientations
As discussed in [11], acyclic orientations can be easily obtained by deterministic distributed algorithms in the case of distributed systems composed of processing units, each having a distinct name. In this work, however, we will only consider anonymous distributed systems. Initially, unless stated otherwise, the only information assumed to be known by each node is the number of its associated neighbors (adjacent nodes). At every assortment, each probabilistic node plays a die and compares
Speed vs. Quality: Tradeoffs
Despite a worst complexity performance of Alg-Neighbors, when compared to Alg-Edges, some interesting characteristics may be highlighted in the quality of the acyclic orientations generated by each algorithm. Fig. 4 presents a comparison between Alg-Edges and Alg-Neighbors algorithms by showing their average convergence time, when applied to randomly generated connected graphs (50 nodes), by varying , the number of faces of the dice. In fact, the superiority of Alg-Edges is very contrasting in
Conclusions
We introduced two randomized distributed algorithms for the generation of acyclic orientations upon anonymous distributed systems of arbitrary topology. Both algorithms were analyzed in terms of correctness, expected complexity and rate of convergence. Moreover, when compared to Alg-Edges, we have shown the benefits of both Alg-Neighbors and Alg-Color, despite their worst computational complexity. As future work, one possibility is to study the generation of orientations on multigraphs as a
Gladstone M. Arantes Jr. was born in Rio de Janeiro, Brazil. He received his B.Sc. in Computer Science from the Universidade Federal do Rio de Janeiro (UFRJ), in 1995, the M.Sc. in Systems Engineering and Computer Science from COPPE/UFRJ, in 1999, and his D.Sc., also from COPPE/UFRJ, in 2006. During his academic career, he has worked with distributed algorithms and parallel and distributed computing. Since 1998, he has been with the Brazilian Social and Economic Development Bank (BNDES),
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Gladstone M. Arantes Jr. was born in Rio de Janeiro, Brazil. He received his B.Sc. in Computer Science from the Universidade Federal do Rio de Janeiro (UFRJ), in 1995, the M.Sc. in Systems Engineering and Computer Science from COPPE/UFRJ, in 1999, and his D.Sc., also from COPPE/UFRJ, in 2006. During his academic career, he has worked with distributed algorithms and parallel and distributed computing. Since 1998, he has been with the Brazilian Social and Economic Development Bank (BNDES), working as system engineer, system architect and executive.
Felipe M.G. França was born in Rio de Janeiro, Brazil. He received his Electronics Engineer degree from the Universidade Federal do Rio de Janeiro (UFRJ), in 1981, the M.Sc. in Computer Science from COPPE/UFRJ, in 1987, and his PhD from the Department of Electrical and Electronics Engineering of the Imperial College London, in 1994. Since 1996, he has been with the Systems Engineering and Computer Science (graduate) Program, COPPE/UFRJ, as Associate Professor, and he has research and teaching interests in asynchronous circuits, computational intelligence, computer architecture, cryptography, distributed algorithms and other aspects of parallel and distributed computing.
Carlos A. Martinhon received his B.Sc. in Mathematics from the Universidade Federal de Goiás (Goiânia/Brazil), in 1987. He received his M.Sc. in Production Engineering (in the area of Operational Research) in 1991 and his D.Sc. in Systems Engineering and Computer Science (in the area of Optimization) from the Universidade Federal do Rio de Janeiro (Rio de Janeiro/Brazil) in 1998. He did his Postdoctoral research in the Laboratoire de Recherche en Informatique (LRI) of the Paris-Sud University, Orsay, France, between Sept/2005 and Feb/2007. Currently, Carlos A. Martinhon is an Associate Professor at the Department of Computer Science of the Universidade Federal Fluminense (UFF). His current research interests include combinatorial optimization, randomized and approximation algorithms, graph theory, computational complexity and mathematical programming.