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 consider ed in this paper. For a given j greater than or equal to 1, resources are placed such that each nonresource node is adjacent to j resource n odes. We first prove that perfect j-adjacency placements are impossibl e in k-ary n-cubes if n < j < 2n. Then, we show that a perfect j-adjac ency 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 ev en, or 2) if 1 less than or equal to j less than or equal to n and the re exist integers q and r such that q divides k and q(r) - 1 = 2n/j. I n each case, we describe an algorithm to obtain perfect j-adjacency pl acements. We also show that these algorithms can be extended under cer tain conditions to place j distinct types of resources in a such way t hat each nonresource node is adjacent to a resource node of each type. For the cases when perfect j-adjacency placements are not possible, w e consider approximate j-adjacency placements. We show that the number of copies of resources required in this case either approaches a theo retical lower bound on the number of copies required for any j-adjacen cy placement or is within a constant factor of the theoretical lower b ound for large k.