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<p><it>Abstract—</it>The problem of placing resources in a <math><tmath>$k$</tmath></math>-ary <math><tmath>$n$</tmath></math>-cube <math><tmath>$(k\,{\char'076}\,2)$</tmath></math> is considered in this paper. For a given <math><tmath>$j \geq 1,$</tmath></math> resources are placed such that each nonresource node is adjacent to <math><tmath>$j$</tmath></math> resource nodes. We first prove that perfect <math><tmath>$j$</tmath></math>-adjacency placements are impossible in <math><tmath>$k$</tmath></math>-ary <math><tmath>$n$</tmath></math>-cubes if <math><tmath>$n\,{\char'074}\,j\,{\char'074}\,2n.$</tmath></math> Then, we show that a perfect <math><tmath>$j$</tmath></math>-adjacency placement is possible in <math><tmath>$k$</tmath></math>-ary <math><tmath>$n$</tmath></math>-cubes when one of the following two conditions is satisfied: 1) if and only if <math><tmath>$j$</tmath></math> equals <math><tmath>$2n$</tmath></math> and <math><tmath>$k$</tmath></math> is even, or 2) if <math><tmath>$1 \leq j \leq n$</tmath></math> and there exist integers <math><tmath>$q$</tmath></math> and <math><tmath>$r$</tmath></math> such that <math><tmath>$q$</tmath></math> divides <math><tmath>$k$</tmath></math> and <math><tmath>$q^r - 1 = 2n/j.$</tmath></math> In each case, we describe an algorithm to obtain perfect <math><tmath>$j$</tmath></math>-adjacency placements. We also show that these algorithms can be extended under certain conditions to place <math><tmath>$j$</tmath></math> distinct types of resources in a such way that each nonresource node is adjacent to a resource node of each type. For the cases when perfect <math><tmath>$j$</tmath></math>-adjacency placements are not possible, we consider approximate <math><tmath>$j$</tmath></math>-adjacency placements. We show that the number of copies of resources required in this case either approaches a theoretical lower bound on the number of copies required for any <math><tmath>$j$</tmath></math>-adjacency placement or is within a constant factor of the theoretical lower bound for large <math><tmath>$k.$</tmath></math></p><p><it>Index Terms—</it>Resource allocation, multiprocessors, hypercubes, mesh connected computers, interconnection network, fault-tolerance.</p>
Suresh Chalasani, Parameswaran Ramanathan, "Resource Placement with Multiple Adjacency Constraints in k-ary n-Cubes", IEEE Transactions on Parallel & Distributed Systems, vol. 6, no. , pp. 511-519, May 1995, doi:10.1109/71.382319
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