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<p><b>Abstract</b>—In their 1982 paper, Adams and Siegel proposed an Extra Stage Cube Interconnection Network that tolerates one switch failure with one extra stage. We extend their results and discover a class of Extra Stage Interconnection Networks that tolerate multiple switch failures with a minimal number of extra stages. Adopting the same fault model as Adams and Siegel, the faulty switches can be bypassed by a pair of demultiplexer/multiplexer combinations. It is easy to show that, to maintain point to point and broadcast connectivities, there must be at least <tmath>$f$</tmath> extra stages to tolerate <tmath>$f$</tmath> switch failures. We present the first known construction of an Extra Stage Interconnection Network that meets this lower-bound. This <tmath>$n$</tmath>-dimensional Multistage Interconnection Network has <tmath>$n+f$</tmath> stages and tolerates <tmath>$f$</tmath> switch failures. An n-bit label called mask is used for each stage that indicates the bit differences between the two inputs coming into a common switch. We designed the fault-tolerant construction such that it repeatedly uses the singleton basis of the <tmath>$n$</tmath>-dimensional vector space as the stage mask vectors. This construction is further generalized and we prove that an <tmath>$n$</tmath>-dimensional Multistage Interconnection Network is optimally fault-tolerant if and only if the mask vectors of every n consecutive stages span the <tmath>$n$</tmath>-dimensional vector space.</p>
Multistage Interconnection Networks (MIN), fault tolerance, extra-stage, switch faults, stage masks.

C. C. Fan and J. Bruck, "Tolerating Multiple Faults in Multistage Interconnection Networks with Minimal Extra Stages," in IEEE Transactions on Computers, vol. 49, no. , pp. 998-1004, 2000.
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