Bounds on Algorithm-Based Fault Tolerance in Multiple Processor Systems

An important consideration in the design of high- performance multiple processor systems should be in ensuring the correctness of results computed by such complex systems which are extremely prone to transient and intermittent failures. The detection and location of faults and errors concurrently with normal system operation can be achieved through the application of appropriate on-line checks on the results of the computations. This is the domain of algorithm-based fault tolerance, which deals with low-cost system-level fault-tolerance techniques to produce reliable computations in multiple processor systems, by tailoring the fault-tolerance techniques toward specific algorithms. This paper presents a graph-theoretic model for determining upper and lower bounds on the number of checks needed for achieving concurrent fault detection and location. The objective is to estimate ate the overhead in time and the number of processors required for such a scheme. Faults in processors, errors in the data, and checks on the data to detect and locate errors are represented as a tripartite graph. Bounds on the time and processor overhead are obtained by considering a series of subproblems. First, using some crude concepts for t-fault detection and t-fault location, bounds on the maximum size of the error patterns that can arise from such fault patterns are obtained. Using these results, bounds are derived on the number of checks required for error detection and location. Some numerical results are derived from a linear programming formulation.