This Article 
 Bibliographic References 
 Add to: 
On Embedding Between 2D Meshes of the Same Size
August 1997 (vol. 46 no. 8)
pp. 880-889

Abstract—Mesh is one of the most commonly used interconnection networks and, therefore, embedding between different meshes becomes a basic embedding problem. Not only does an efficient embedding between meshes allow one mesh-connected computing system to efficiently simulate another, but it also provides a useful tool for solving other embedding problems. In this paper, we study how to embed an s1×t1 mesh into an s2×t2 mesh, where siti(i = 1, 2), s1t1 = s2t2, such that the minimum dilation and congestion can be achieved. First, we present a lower bound on the dilations and congestions of such embeddings for different cases. Then, we propose an embedding with dilation $\lfloor s_1/s_2 \rfloor + 2$ and congestion $\lfloor s_1/s_2 \rfloor + 4$ for the case s1s2, both of which almost match the lower bound $\lceil s_1/s_2 \rceil.$ Finally, for the case s1 < s2, we present an embedding which has a dilation less than or equal to $2\sqrt {s_1}.$

[1] S.R. Kosaraju and M.J. Atallah, "Optimal Simulations Between Mesh-Connected Arrays of Processors," J. ACM, vol. 35, pp. 635-650, 1988.
[2] R. Aleliunas and A.L. Rosenberg, "On Embedding Rectangular Grids in Square Grids," IEEE Trans. Computers, vol. 31, no. 9, pp. 907-913, Sept. 1982.
[3] F. Lombardi, D. Sciuto, and R. Stefanelli, "An Algorithm for Functional Reconfiguration of Fixed-Size Arrays," IEEE Trans. Computer-Aided Design, vol. 10, pp. 1,114-1,118, 1988.
[4] J. Ellis, "Embedding Rectangular Grids into Square Grids," IEEE Trans. Computers, vol. 40, no. 1, pp. 46-52, Jan. 1991.
[5] M.Y. Chan, "Dilation-2 Embedding of Grids into Hypercubes," Proc. Int'l Conf. Parallel Processing, vol. 3, pp. 295-298, 1988.
[6] M.Y. Chan,“Embedding of grids into optimal hypercubes,” SIAM J. Computing, vol. 20, pp. 834-864, Oct. 1991.
[7] B. Cong and H. Sudborough, "Dilation-4 Embedding of 2D Meshes into Star Graphs," Proc. First Int'l Conf. Computer Comm. and Networks (IC3N), pp. 6-9, 1992.
[8] M.Y. Chan and F.Y.L. Chin, "On Embedding Rectangular Grids in Hypercubes," IEEE Trans. Computers, vol. 37, no. 10, pp. 1,285-1,288, Oct. 1988.
[9] H. Liu and S.S. Huang, "Dilation-6 Embedding of 3D Dimensional Grids into Optimal Hypercubes," Proc. Int'l Conf. Parallel Processing, vol. 3, pp. 250-254, 1991.
[10] C.-T. Ho and S.L. Johnsson, "On the Embedding of Arbitrary Meshes in Boolean Cubes with Expansion Two Dilation Two," Proc. Int'l Conf. Parallel Processing, pp. 188-191, 1987.
[11] Y. Ma and L. Tao, "Embeddings Among Toruses and Meshes," Proc. Int'l Conf. Parallel Processing, pp. 178-187, 1987.
[12] R.A. DeMillo, S.C. Eisenstat, and R.J. Lipton, "Preserving Average Proximity in Arrays," Comm. ACM, vol. 21, pp. 228-231, 1978.
[13] S.N. Bhatt, F.R.K. Chung, F.T. Leighton, and A.L. Rosenberg, "Efficient Embeddings of Trees in Hypercubes," SIAM J. Computing, vol. 21, no. 1, pp. 151-162, 1992.
[14] A.K. Gupta and S.E. Hambrusch, "Embedding Complete Binary Trees into Butterfly Networks," IEEE Trans. Computers, vol. 40, no. 7, pp. 853-863, July 1991.
[15] A.Y. Wu, "Embedding of Tree Networks into Hypercubes," J. Parallel and Distributed Computing, vol. 2, pp. 238-249, 1985.
[16] J.-W. Hong, K. Mehlhorn, and A.L. Rosenberg, "Cost Tradeoffs in Graph Embeddings," J. ACM, vol. 30, no. 4, pp. 709-728, 1983.
[17] F.T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes.San Mateo, Calif.: Morgan Kaufmann, 1992.
[18] K. Hwang and F.A. Briggs,Computer Architecture and Parallel Processing.New York: McGraw Hill, 1984.
[19] "Embedding Between 2D Meshes of the Same Size," Proc. Fifth IEEE Symp. Parallel and Distributed Processing, pp. 712-719,Dallas, Tex., Dec.1-4, 1993.

Index Terms:
Dilation, embedding, mesh, parallel processing, vertex partition.
Xiaojun Shen, Weifa Liang, Qing Hu, "On Embedding Between 2D Meshes of the Same Size," IEEE Transactions on Computers, vol. 46, no. 8, pp. 880-889, Aug. 1997, doi:10.1109/12.609277
Usage of this product signifies your acceptance of the Terms of Use.