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• W2169252125 abstract "Let k ges 4 be even and let n ges 2. Consider a faulty k-ary n-cube Q <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sup> <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</sub> in which the number of node faults fv and the number of link faults f <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>e</sub> are such that f <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>v</sub> + f <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>e</sub> les 2n-2. We prove that given any two healthy nodes s and e of Q <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sup> <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</sub> , there is a path from s to e of length at least k <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</sup> - 2 f <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>v</sub> -1 (respectively, k <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</sup> - 2 f <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>v</sub> -2) if the nodes s and e have different (respectively the same) parities (the parity of a node in Q <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sup> <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</sub> is the sum modulo 2 of the elements in the n-tuple over {0,1,...,k 1} representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan et al. (2007) and by Fu (2006). Furthermore, we extend known results, obtained by Kim and Park (2000), for the case when n=2." @default.
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• W2169252125 date "2008-08-01" @default.
• W2169252125 modified "2023-09-27" @default.
• W2169252125 title "Embedding Long Paths in k-Ary n-Cubes with Faulty Nodes and Links" @default.
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• W2169252125 doi "https://doi.org/10.1109/tpds.2007.70787" @default.
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