TY - JOUR

T1 - Synthesizing optimal bias in randomized self-stabilization

AU - Volk, Matthias

AU - Bonakdarpour, Borzoo

AU - Katoen, Joost-Pieter

AU - Aflaki, Saba

PY - 2022

Y1 - 2022

N2 - Randomization is a key concept in distributed computing to tackle impossibility results. This also holds for self-stabilization in anonymous networks where coin flips are often used to break symmetry. Although the use of randomization in self-stabilizing algorithms is rather common, it is unclear what the optimal coin bias is so as to minimize the expected convergence time. This paper proposes a technique to automatically synthesize this optimal coin bias. Our algorithm is based on a parameter synthesis approach from the field of probabilistic model checking. It over- and under-approximates a given parameter region and iteratively refines the regions with minimal convergence time up to the desired accuracy. We describe the technique in detail and present a simple parallelization that gives an almost linear speed-up. We show the applicability of our technique to determine the optimal bias for the well-known Herman’s self-stabilizing token ring algorithm. Our synthesis obtains that for small rings, a fair coin is optimal, whereas for larger rings a biased coin is optimal where the bias grows with the ring size. We also analyze a variant of Herman’s algorithm that coincides with the original algorithm but deviates for biased coins. Finally, we show how using speed reducers in Herman’s protocol improve the expected convergence time.

AB - Randomization is a key concept in distributed computing to tackle impossibility results. This also holds for self-stabilization in anonymous networks where coin flips are often used to break symmetry. Although the use of randomization in self-stabilizing algorithms is rather common, it is unclear what the optimal coin bias is so as to minimize the expected convergence time. This paper proposes a technique to automatically synthesize this optimal coin bias. Our algorithm is based on a parameter synthesis approach from the field of probabilistic model checking. It over- and under-approximates a given parameter region and iteratively refines the regions with minimal convergence time up to the desired accuracy. We describe the technique in detail and present a simple parallelization that gives an almost linear speed-up. We show the applicability of our technique to determine the optimal bias for the well-known Herman’s self-stabilizing token ring algorithm. Our synthesis obtains that for small rings, a fair coin is optimal, whereas for larger rings a biased coin is optimal where the bias grows with the ring size. We also analyze a variant of Herman’s algorithm that coincides with the original algorithm but deviates for biased coins. Finally, we show how using speed reducers in Herman’s protocol improve the expected convergence time.

U2 - 10.1007/s00446-021-00408-4

DO - 10.1007/s00446-021-00408-4

M3 - Article

SN - 1432-0452

JO - Distributed computing

JF - Distributed computing

ER -