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<p><it>Abstract</it>—Hypercube network is an attractive structure for parallel processing due to its symmetry and regularity. In this paper, we use the concept of <it>free dimensions</it> to achieve fault tolerance in hypercubes without requiring additional spare processing nodes; such additional redundancy requires modification of hypercube structure. A free dimension is defined to be a dimension across which both end nodes are not faulty.</p><p>Given an <it>n</it>-dimensional hypercube, <it>Q</it><sub><it>n</it></sub>, and a set of <it>f</it>≤<it>n</it> faulty nodes, we present an efficient algorithm to find free dimensions, and show that at least <it>n</it>−<it>f</it>+ 1 free dimensions exist. Free dimensions can be used to partition <it>Q</it><sub><it>n</it></sub> into subcubes such that each subcube contains at most one fault. Such a partitioning helps in achieving fault tolerance via emulation, embedding, reconfiguration. It also helps in designing efficient routing and broadcasting algorithms in faulty hypercubes.</p>
Hypercubes, fault tolerance, embedding, reconfiguration, routing and broadcasting.

P. Yang, C. Raghavendra and S. Tien, "Free Dimensions-An Effective Approach to Achieving Fault Tolerance in Hypercubes," in IEEE Transactions on Computers, vol. 44, no. , pp. 1152-1157, 1995.
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