Resource Placement with Multiple Adjacency Constraints in k-ary n-Cubes

• Authors: Parameswaran Ramanathan;Suresh Chalasani
• Venue: IEEE Transactions on Parallel and Distributed Systems
• Year: 1995

The problem of placing resources in a $k$-ary $n$-cube $(k\,{\char'076}\,2)$ is considered in this paper. For a given $j \geq 1,$ resources are placed such that each nonresource node is adjacent to $j$ resource nodes. We first prove that perfect $j$-adjacency placements are impossible in $k$-ary $n$-cubes if $n\,{\char'074}\,j\,{\char'074}\,2n.$ Then, we show that a perfect $j$-adjacency placement is possible in $k$-ary $n$-cubes when one of the following two conditions is satisfied: 1) if and only if $j$ equals $2n$ and $k$ is even, or 2) if $1 \leq j \leq n$ and there exist integers $q$ and $r$ such that $q$ divides $k$ and $q^r - 1 = 2n/j.$ In each case, we describe an algorithm to obtain perfect $j$-adjacency placements. We also show that these algorithms can be extended under certain conditions to place $j$ 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 $j$-adjacency placements are not possible, we consider approximate $j$-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 $j$-adjacency placement or is within a constant factor of the theoretical lower bound for large $k.$Index Terms驴Resource allocation, multiprocessors, hypercubes, mesh connected computers, interconnection network, fault-tolerance.