  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.
