Randomized generation of acyclic orientations upon anonymous distributed systems

• Published in: Journal of parallel and distributed computing Vol. 69; no. 3; pp. 239 - 246
• Main Authors: Arantes, Gladstone M., França, Felipe M.G., Martinhon, Carlos A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.03.2009; Elsevier
• Subjects: Anonymous systems; Applied sciences; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Exact sciences and technology; Graph coloring; Information retrieval. Graph; Randomized distributed algorithms; Software; Symmetry breaking; Theoretical computing; Graph coloring; Randomized distributed algorithms; Anonymous systems; Symmetry breaking; Probabilistic approach; Anonymity; Distributed system; Topology; Randomized algorithm; Orientation; Convergence speed; Randomization; Distributed algorithm; Convergence rate; Graph colouring; Time complexity; Numerical convergence; Biased estimation
• ISSN: 0743-7315; 1096-0848
• DOI: 10.1016/j.jpdc.2008.11.009

This paper presents two new randomized distributed algorithms for the generation of acyclic orientations upon anonymous distributed systems of arbitrary topology. Both algorithms, called Alg-Neighbors and Alg-Edges, make use of biased and unbiased dice having f ≥ 2 faces, and are analyzed in terms of correctness, expected time complexity and rate of convergence. First, the Alg-Neighbors algorithm is presented as a generalization of the Calabrese/França algorithm for dice (or coins) with 2 faces. It is proved that a convenient biasing function applied to all dice changes the expected time complexity from sub-exponential, i.e., O ( f ( f f − 1 ) n − 1 ) (for unbiased dice with f faces), to O ( n ) , where n is the number of the system’s nodes. Next, it is shown that the Alg-Edges algorithm is able to produce acyclic orientations in O ( log f m ) steps with high probability, where m denotes the total number of edges. Finally, a speed of convergence versus quality of acyclic orientation generation (associated number of colors) tradeoff is identified between Alg-Neighbors and Alg-Edges algorithms. Computational experiments were carried out in order to provide a more accurate description of the behavior of both algorithms.