# On efficient embeddings of grids into grids in PARIX

## Abstract

A hardware independent method of programming a massively parallel machine (MPP) can best be supported by a well-designed run-time environment. An important problem in this design is the ability of efficiently simulating networks different from the hardware topology. We will describe the mapping kernel of the virtual processors library for the commercial run-time system PARIX^{3}. This kernel contains description classes for several topologies (so-called *virtual topologies*) and implementations of respective embeddings which map given instances of virtual topologies onto others or onto the hardware. Using these functions, PARIX is able to establish concrete virtual topologies with corresponding communication channels. The implemented functions were selected with respect to the well-known criteria for graph embeddings: *equal load* and *small dilation*. Additionally, we focus on *fast distributed computation* and *universal applicability*. As an example, we will show new methods for efficiently embedding an arbitrary 2-dimensional grid as a guest graph into any 2-dimensional grid as a host graph.

## Key words

parallel run-time system PARIX virtual processors embedding grids## Preview

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## References

- 1.R. Aleliunas, A. Rosenberg:
*On Embedding Rectangular Grids in Square Grids*, IEEE Transactions on Computers, Vol. C-31, No. 9, September 1982.Google Scholar - 2.F.S. Annexstein:
*Parallel Implementations of Graph Embeddings*, Parallel Architectures and their efficient use, Lecture Notes in Computer Science, Vol. 678, 1992.Google Scholar - 3.M. Baumslag, M.C. Heydemann, J. Opatrny, D. Sotteau:
*Embeddings of shuffle-like graphs in hypercubes*, Parallel Architectures and Languages Europe (PARLE'91), Springer LNCS 505, pp. 179–190, 1991.Google Scholar - 4.J.E. Boillat, P.G. Kropf:
*A fast distributed Mapping Algorithm*, Proc. of CONPAR '90, Springer LNCS 457, 1990.Google Scholar - 5.M.Y. Chan:
*Embedding of Grids into Optimal Hypercubes*, SIAM J. Computing, Vol. 20, No. 5, pp. 834–864, 1991.Google Scholar - 6.M.Y. Chan, F.Y.L. Chin:
*Parallelized simulation of grids by hypercubes*, Technical Report, University of Hong Kong, October 1990.Google Scholar - 7.R. Diekmann, R. Lüling, J. Simon:
*Problem Independent Distributed Simulated Annealing and its Applications*, Applied Simulated Annealing, Lecture Notes in Economics and Mathematical Systems, Springer LNEMS 396, 1993.Google Scholar - 8.R. Heckmann, R. Klasing, B. Monien, W. Unger:
*Optimal Embedding of Complete Binary Trees into Lines and Grids*, Proc. 17th Int. Workshop on Graph-Theoretic Concepts in Computer Science (WG91), Springer LNCS 570, pp. 25–35, 1991.Google Scholar - 9.C.T. Ho, S.L. Johnsson:
*Embedding Meshes in Boolean Cubes by Graph Decomposition*, Journal of Parallel an Distributed Computing, 8, pp. 325–339, 1990.Google Scholar - 10.S.-H. Huang, H.L. Liu, R. Verma:
*A New Combinatorial Approach to Optimal Embeddings of Rectangles*, Intern. Parallel Processing Symposium, 1994.Google Scholar - 11.S.R. Kosaraj, M.J. Atallah:
*Optimal Simulations Between Mesh-Connected Arrays of Processors*, ACM Symposium on Theory of Computing, pp. 264–272, 1986.Google Scholar - 12.E. Ma, D.G. Shea:
*The Embedding Kernel on the IBM Victor Multiprocessor for Program Mapping and Network Reconfiguration*, Proceedings of the Second IEEE Symposium on Parallel and Distributed Processing, 1990.Google Scholar - 13.Z. Miller, I.H. Sudborough:
*Compressing Grids into Small Hypercubes*, Networks, Vol. 24, pp. 327–358, 1994.Google Scholar - 14.B. Monien, I.H. Sudborough:
*Embedding one Interconnection Network in Another*, Computing Suppl. 7, pp. 257–282, 1990.Google Scholar - 15.J. Philbin:
*Virtual Topologies: A New Concurrency Abstraction for High-Level Parallel Languages*, DIMACS Workshop on Interconnection Networks, 1994.Google Scholar - 16.M. Röttger, U.-P. Schroeder, J. Simon:
*Implementation of a Parallel and Distributed Mapping Kernel for PARIX*, Intern. Conference and Exhibition on Highperformance Computing and Networking, (HPCN Europe'95), pp. 781–786, 1995.Google Scholar - 17.M. Röttger, U.-P. Schroeder, W. Unger:
*Embedding 3-dimensional Grids into optimal Hypercubes*, Proc. of the 1st Canada-France Conference on Parallel Computing (CFCP '94), Springer LNCS 805, pp. 81–94, 1994.Google Scholar - 18.F.C. Sang, I.H. Sudborough:
*Embedding Large Meshes into Small Ones*, Proc. of the IEEE Symposium on Circuits and Systems, Vol. 1, pp. 323–326, 1990.Google Scholar