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Resource Placement with Multiple Adjacency Constraints in k-ary n-Cubes
May 1995 (vol. 6 no. 5)
pp. 511-519

Abstract—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—Resource allocation, multiprocessors, hypercubes, mesh connected computers, interconnection network, fault-tolerance.

[1] L. N. Bhuyan and D. P. Agrawal,“Generalized hypercube and hyperbus structures for a computer network,”IEEE Trans. Comput., vol. C-33, pp. 323–333, Apr. 1984.
[2] H.-L. Chen and N.-F. Tzeng,“Fault-tolerant resource placement in hypercube computers,”inProc. Int. Conf. Parallel Processing, Aug. 1991, pp. 517–524.
[3] G.-M. Chiu and C. S. Raghavendra,“Resource allocation in hypercube systems,”inProc. Distrib. Memory Comput. Conf., Apr. 1990, pp. 894–902.
[4] F.T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes.San Mateo, Calif.: Morgan Kaufmann, 1992.
[5] M. Livingston and Q. Stout,“Perfect dominating sets,”Congressus Numerantium, vol. 79, pp. 187–203, 1990.
[6] M. Livingston and Q.F. Stout,“Distributing resources in hypercube computers,” Proc. Third Conf. Hypercube Concurrent Computers and Applications, vol. I, pp. 222-231,Pasadena, Calif., Jan. 1988.
[7] A. L. N. Reddy, P. Banerjee, and S. G. Abraham,“I/O embedding in hypercubes,”inProc. Int. Conf. Parallel Processing, Aug. 1988, pp. 331–338.
[8] Y. Saad and M. H. Schultz,“Topological properties of hypercubes,”IEEE Trans. Comput.,vol. 37, pp. 867–871, July 1988.
[9] C.L. Seitz et al., "The Architecture and Programming of the Ametak Series 2010," Proc. Third Conf. Hypercube Concurrent Computers and Applications, pp. 33-37, Jan. 1988.

Parameswaran Ramanathan, Suresh Chalasani, "Resource Placement with Multiple Adjacency Constraints in k-ary n-Cubes," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 5, pp. 511-519, May 1995, doi:10.1109/71.382319
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