Randomized generation of acyclic orientations upon anonymous distributed systems

Abstract

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.