Defect tolerance: fundamental limits and examples

TL;DR

This paper explores the fundamental limits of physical redundancy for defect tolerance, proposing a bipartite graph model, analyzing its bounds, and demonstrating that simple modular redundancy is often suboptimal.

Contribution

It introduces a bipartite graph model for defect-tolerant systems, characterizes its fundamental limits, and shows the suboptimality of simple modular redundancy.

Findings

Bipartite graph model effectively captures defect tolerance design.

Fundamental limits are characterized under various asymptotic regimes.

Simple modular redundancy is generally suboptimal.

Abstract

This paper addresses the problem of adding redundancy to a collection of physical objects so that the overall system is more robust to failures. In contrast to its information counterpart, which can exploit parity to protect multiple information symbols from a single erasure, physical redundancy can only be realized through duplication and substitution of objects. We propose a bipartite graph model for designing defect-tolerant systems in which defective objects are replaced by judiciously connected redundant objects. The fundamental limits of this model are characterized under various asymptotic settings and both asymptotic and finite-size systems that approach these limits are constructed. Among other results, we show that simple modular redundancy is in general suboptimal. As we develop, this combinatorial problem of defect tolerant system design has a natural interpretation as one of graph coloring, and the analysis is significantly different from that traditionally used in information redundancy for error-control codes.