This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Solving Fundamental Problems on Sparse-Meshes
December 2000 (vol. 11 no. 12)
pp. 1324-1332

Abstract—A sparse-mesh, which has PUs on the diagonal of a two-dimensional grid only, is a cost effective distributed memory machine. Variants of this machine have been considered before, but none are as simple and pure as a sparse-mesh. Various fundamental problems (routing, sorting, list ranking) are analyzed, proving that sparse-meshes have great potential. It is shown that on a two-dimensional $n \times n$ sparse-mesh, which has $n$ PUs, for $h = \omega(n^\epsilon \cdot \log n)$, h-relations can be routed in $(h + o(h)) / \epsilon$ steps. The results are extended for higher dimensional sparse-meshes. On a $d$-dimensional $n \times \cdots \times n$ sparse-mesh, with $h = \omega(n^\epsilon \cdot \log n)$, h-relations are routed in $(6 \cdot (d - 1) / \epsilon - 4) \cdot (h + o(h))$ steps.

[1] A. Aggarwal and J. S. Vitter, The Input/Output Complexity of Sorting and related Problems Comm. ACM, vol. 31, no. 9, pp. 1116-1127, 1988.
[2] M. Baumslag and F. Annexstein, “A Unified Framework for Off-Line Permutation Routing in Parallel Networks,” Math. Systems Theory, vol. 24, no. 4, pp. 233–251, 1991.
[3] B.S. Chlebus, A. Czumaj, L. Gasieniec, M. Kowaluk, and W. Plandowski, “Parallel Alternating-Direction Access Machine,” Proc. 21st Math. Foundations of Computer Science, pp. 267–278, 1996.
[4] B.S. Chlebus, A. Czumaj, and J.F. Sibeyn, “Routing on the PADAM: Degrees of Optimality,” Proc. Third Euro-Par Conf., pp. 272–279, 1997.
[5] R. Cole and J. Hopcroft, “On Edge Coloring Bipartite Graphs,” SIAM J. Computing, vol. 11, pp. 540–546, 1982.
[6] J. J'aJ'a, An Introduction to Parallel Algorithms.New York: Addison-Wesley, 1992.
[7] B. Juurlink, J.F. Sibeyn, and P.S. Rao, “Gossiping on Meshes and Tori,” IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 6, pp. 513–525, June 1998.
[8] M. Kaufmann and J.F. Sibeyn, “Randomized Multipacket Routing and Sorting on Meshes,” Algorithmica, vol. 17, pp. 224-244, 1997.
[9] M. Kaufmann,J. Sibeyn, and T. Suel,"Derandomizing Algorithms for Routing and Sorting on Meshes," Proc. Fifth Ann. ACM-SIAM Symp. Discrete Algorithms, pp. 669-679,Arlington, Va., 1994.
[10] M. Kunde, “Block Gossiping on Grids and Tori: Deterministic Sorting and Routing Match the Bisection Bound,” Proc. European Symp. Algorithms, pp. 272–283, 1993.
[11] T. Leighton, "Tight Bounds on the Complexity of Parallel Sorting," IEEE Trans. Computers, vol. 34, no. 4, pp. 344-354, Apr. 1985.
[12] V. Leppänen and M. Penttonen, “Work-Optimal Simulation of PRAM Models on Meshes,” Nordic J. Computing, vol. 2, pp. 51–69, 1995.
[13] G. Lev, N. Pippenger, and L.G. Valiant, “A Fast Parallel Algorithm for Routing in Permutation Networks,” IEEE Trans. Computers, vol. 30, no. 2, pp. 93–100, Feb. 1981.
[14] S. Rajasekaran, “$k$-$k$Routing,$k$-$k$Sorting, and Cut-through Routing on the Mesh,” J. Algorithms, vol. 19, no. 3, pp. 361–382, 1995.
[15] J.F. Sibeyn, “List Ranking on Interconnection Networks,” Proc. Second Euro-Par Conf., pp. 799–808, 1996.
[16] J.F. Sibeyn, “Better Trade-offs for Parallel List Ranking,” Proc. Ninth Symp. Parallel Algorithms and Architectures, pp. 221–230, 1997.
[17] J.F. Sibeyn, “Sample Sort on Meshes,” Proc. Third Euro-Par Conf., pp. 389–398, 1997.
[18] J.F. Sibeyn, “From Parallel to External List Ranking,” Technical Report MPI-I-97-1021, Max-Planck Institut für Informatik, Saarbrücken, Germany, 1997.
[19] J.F. Sibeyn, F. Guillaume, and T. Seidel, “Practical Parallel List Ranking,” Proc. Fourth Symp. Solving Irregularly Structured Problems in Parallel, pp. 25–36, 1997.
[20] T. Suel, “Permutation Routing and Sorting on Meshes with Row and Column Buses,” Parallel Processing Letters, vol. 5, pp. 63–80, 1995.
[21] C.D. Thompson, "Area-Time Complexity for VLSI," Proc. 11th Ann. Symp. Theory of Computing, pp. 81-88, 1979.
[22] L.G. Valiant and G.J. Brebner,"Universal Schemes for Parallel Communication," Proc. 13th Ann. ACM Symp. Theory of Computing, pp. 263-277, May 1981.

Index Terms:
Theory of parallel computation, algorithms, networks, meshes, sorting, routing, list-ranking.
Citation:
Jop F. Sibeyn, "Solving Fundamental Problems on Sparse-Meshes," IEEE Transactions on Parallel and Distributed Systems, vol. 11, no. 12, pp. 1324-1332, Dec. 2000, doi:10.1109/71.895796
Usage of this product signifies your acceptance of the Terms of Use.