An optimal randomized routing algorithm for the Mesh and a class of efficient Mesh-like routing networks
We present an optimal oblivious randomized algorithm for permutation routing on the MIMD version of Mesh. Our routing algorithm routes n2 elements on an n×n Mesh in 2n+O(log n) parallel communication steps with very high probability. Further, the maximum queue length at any node at any time is at the most O(log n) with the same probability. Since 2n is the distance bound for the Mesh, our algorithm is indeed optimal. Generalization of this result to k-dimensional (for any k) Meshes yields an algorithm that runs in time equal to the diameter of the Mesh. A lower bound result of [Schnorr and Shamir 86] states that sorting of n2 elements takes at least 3n steps on an n × n MIMD Mesh (for indexing schemes of practical interest). Thus our algorithm demonstrates that routing is easier than sorting on the MIMD Mesh.
We also identify a class of Mesh-like networks (we call circular meshes) which have n2 nodes but less than 2n diameter. These meshes have the property that they can be laid out on a chip with a physical diameter same as the model diameter. Our routing algorithm runs on these networks to route n2 elements in d+O(log n) steps (d being the diameter of the network) with a very high probability. These circular meshes also have the potential of being adopted to run existing sorting and routing algorithms (on the regular SIMD and MIMD Meshes) with a corresponding reduction in their run times.
And finally we present a (possibly) non optimal oblivious routing algorithm for the Mesh that requires only O(1) queue size for any node at any time. In particular we present a randomized oblivious constant queue size routing algorithm that runs in time O(n1+ε), for any ε>0.
KeywordsNormal Path Length Queue Size Single Instruction Multiple Data Unique Node Mesh Connect
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- •.Lang,H.-W. et al., “Systolic Sorting on a Mesh Connected Network,” IEEE Transactions on Computers, volume c-34, no.7, 1985.Google Scholar
- •.Krizanc,D., Private Communication, 1986.Google Scholar
- •.Kumar,M., and Hirschberg,D.S., “An Efficient Implementation of Batcher's Odd-Even Merge Algorithm and its Application in Parallel Sorting Schemes,” IEEE Transactions on Computers, volume c-32, 1983.Google Scholar
- •.Ma,Y., Sen,S., and Scherson,D., “The Distance Bound for Sorting on Mesh Connected Processor Arrays is Tight,” Proc. of the IEEE FOCS, 1986.Google Scholar
- •.Nassimi,D., and Sahni,S., “Bitonic Sort on a Mesh Connected Parallel Computer,” IEEE Transactions on Computers, volume c-27, no.1, 1979.Google Scholar
- •.Nassimi,D., and Sahni,S., “Data Broadcasting in SIMD Computers,” IEEE Transactions on Computers, volume c-30, no.2, 1981.Google Scholar
- •.Pippenger,N., “Parallel Communication with Limited Buffers,” Proc. IEEE Symposium on FOCS, 1984, pp. 127–136.Google Scholar
- •.Sado,K., and Igarishi,Y., “Some Parallel Sorts on a Mesh Connected Processor Array and their Time Efficiency,” to appear in Journal of Parallel and Distributed Computing, 1987.Google Scholar
- •.Schnorr,C.P., and Shamir,A., “An Optimal Sorting Algorithm for Mesh Connected Computers,” Proc. of the ACM STOC, 1986.Google Scholar
- •.Thompson,C.D., and Kung,H.T., “Sorting on a Mesh Connected Parallel Computer,” Communications of the ACM, volume 20, no.4, 1977.Google Scholar
- •.Thompson,C.D., “The VLSI Complexity of Sorting,” IEEE Transactions on Computers, volume c-32, no.12, 1983.Google Scholar
- •.Valiant,L.G., and Brebner,G.J., “Universal Schemes for Parallel Communication,” Proc. of the ACM STOC, 1981.Google Scholar