Embedding of rings and meshes onto faulty hypercubes using free dimensions

Journal Article · Sun May 01 04:00:00 EDT 1994 · IEEE Transactions on Computers (Institute of Electrical and Electronics Engineers); (United States)

DOI:https://doi.org/10.1109/12.280808· OSTI ID:6687158

Yang, P; Tien, Singban; Raghavendra, C S ^[1]

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1. Univ. of Southern California, Los Angeles, CA (United States)

Fault tolerance in hypercubes is achieved by exploiting inherent redundancy and executing tasks on faulty hypercubes. We consider tasks that require linear chain, ring, mesh, and torus structure, which are quite useful in parallel and pipeline computations. In this brief contribution, we assume the number of faults is on the order of the number of dimensions of the hypercube. Our techniques are based on a key concept called free dimension, which can be used to partition a cube into subcubes such that each subcube contains, at most, one faulty node. Subgraphs are embedded in each subcube and then merged to form the entire graph. Using this approach, we obtain the following results in an n-dimensional hypercube. A ring of length 2(sup n) - 2f or a chain of length 2(sup n) - 2f + 1 can be found, given any f less than or = n - 2 faulty nodes. The length of chains or rings obtained is optimal (longest) in the worst case of fault distribution. A mesh of size at least (2(sup m(1)) - 2f + 1) * 2(sup m(2)) * ... * 2(sup m(d)), or a torus of size at least (2(sup m(1)) - 2f) * 2(sup m(2)) * ... * 2(sup m(d)), can be found when f less than or = m(sub 1) - 2, where n = m(sub 1) + m(sub 2) + ... + m(sub d) and m(sub 1) is the largest direction of mesh. A mesh or torus of size at least (2(sup m(1)) - k) * 2(sup m(2)) *... * 2(sup m(d)) can be found with dilation 2 when 1 less thanEQ f less than or = n, where k = min (f, (2(sup m(1))/2(sup n - f + 1))). 14 refs.

• OSTI ID: 6687158
• Journal Information: IEEE Transactions on Computers (Institute of Electrical and Electronics Engineers); (United States), Journal Name: IEEE Transactions on Computers (Institute of Electrical and Electronics Engineers); (United States) Vol. 43:5; ISSN 0018-9340; ISSN ITCOB4
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ALGORITHMS
COMPUTERS
DESIGN
DIGITAL COMPUTERS
FAULT TOLERANT COMPUTERS
HYPERCUBE COMPUTERS
MATHEMATICAL LOGIC
PARALLEL PROCESSING
PROGRAMMING