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Issue No.05 - May (1995 vol.6)
pp: 511-519
ABSTRACT
<p><it>Abstract—</it>The problem of placing resources in a $<tmath>k</tmath>$-ary $<tmath>n</tmath>$-cube $<tmath>(k\,{\char'076}\,2)</tmath>$ is considered in this paper. For a given $<tmath>j \geq 1,</tmath>$ resources are placed such that each nonresource node is adjacent to $<tmath>j</tmath>$ resource nodes. We first prove that perfect $<tmath>j</tmath>$-adjacency placements are impossible in $<tmath>k</tmath>$-ary $<tmath>n</tmath>$-cubes if $<tmath>n\,{\char'074}\,j\,{\char'074}\,2n.</tmath>$ Then, we show that a perfect $<tmath>j</tmath>$-adjacency placement is possible in $<tmath>k</tmath>$-ary $<tmath>n</tmath>$-cubes when one of the following two conditions is satisfied: 1) if and only if $<tmath>j</tmath>$ equals $<tmath>2n</tmath>$ and $<tmath>k</tmath>$ is even, or 2) if $<tmath>1 \leq j \leq n</tmath>$ and there exist integers $<tmath>q</tmath>$ and $<tmath>r</tmath>$ such that $<tmath>q</tmath>$ divides $<tmath>k</tmath>$ and $<tmath>q^r - 1 = 2n/j.</tmath>$ In each case, we describe an algorithm to obtain perfect $<tmath>j</tmath>$-adjacency placements. We also show that these algorithms can be extended under certain conditions to place $<tmath>j</tmath>$ 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 $<tmath>j</tmath>$-adjacency placements are not possible, we consider approximate $<tmath>j</tmath>$-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 $<tmath>j</tmath>$-adjacency placement or is within a constant factor of the theoretical lower bound for large $<tmath>k.</tmath>$</p><p><it>Index Terms—</it>Resource allocation, multiprocessors, hypercubes, mesh connected computers, interconnection network, fault-tolerance.</p>
CITATION
Parameswaran Ramanathan, Suresh Chalasani, "Resource Placement with Multiple Adjacency Constraints in k-ary n-Cubes", IEEE Transactions on Parallel & Distributed Systems, vol.6, no. 5, pp. 511-519, May 1995, doi:10.1109/71.382319
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