Analysis of an Efficient Distributed Algorithm for Mutual Exclusion (Average-Case Analysis of Path Reversal)

TL;DR

This paper provides a comprehensive average-case analysis of a distributed mutual exclusion algorithm using path reversal in trees, revealing its expected cost and complexity bounds through combinatorial and probabilistic methods.

Contribution

It introduces a novel approach to compute the moments of path reversal cost, completing the average-case analysis of the algorithm and offering tools applicable to other tree-based distributed algorithms.

Findings

Expected path reversal cost is exactly H_{n-1}

Provides time and message complexity bounds

Offers probabilistic bounds on worst-case message complexity

Abstract

The algorithm analysed by Na\"{i}mi, Trehe and Arnold was the very first distributed algorithm to solve the mutual exclusion problem in complete networks by using a dynamic logical tree structure as its basic distributed data structure, viz. a path reversal transformation in rooted n-node trees; besides, it was also the first one to achieve a logarithmic average-case message complexity. The present paper proposes a direct and general approach to compute the moments of the cost of path reversal. It basically uses one-one correspondences between combinatorial structures and the associated probability generating functions: the expected cost of path reversal is thus proved to be exactly $H_{n-1}$. Moreover, time and message complexity of the algorithm as well as randomized bounds on its worst-case message complexity in arbitrary networks are also given. The average-case analysis of path reversal and the analysis of this distributed algorithm for mutual exclusion are thus fully completed in the paper. The general techniques used should also prove available and fruitful when adapted to the most efficient recent tree-based distributed algorithms for mutual exclusion which require powerful tools, particularly for average-case analyses.