Resource placement with multiple adjacency constraints in k-ary n-cubes

The problem of placing resources in a k-ary n-cube (k>2) is considered in this paper. For a given j⩾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<j<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⩽j⩽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