Fast optimal diagnosis procedures for k-out-of-n:G systems

A k-out-of-n:G system consists of a set of components, where each component is either faulty or fault-free. The system is working if at least k components are fault-free. The problem of finding an optimal diagnosis procedure for a given k-out-of-n:G system has been considered in several research fields including medical diagnosis, redundant-system testing, and searching data-files. A polynomial-time algorithm for this problem was presented first by Salloum, and later by Salloum and Breuer, and independently by Ben-Dov. This paper implements the Salloum-Breuer-Ben-Dov algorithm, leading to an optimal diagnosis procedure that can determine the state of any given system in O(n·log(n)) time complexity and O(n) space complexity. The efficiency is achieved by using a generalized radix sorting procedure that uses a heap data structure. For some k-out-of-n:G systems, including those with equal testing costs for all components, the components along the leftmost and rightmost paths in the optimal diagnostic tree uniquely determine the other components in the tree. This property is used to devise a faster optimal diagnosis procedure than the one for the general k-out-of-n:G system. With regard to complexity, these procedures are the best solutions for the problem under consideration. This conjecture is supported by the fact that all these procedures require a sorting operation which has O(n·log(n)) as a lower bound on its time complexity