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Issue No.05 - May (1999 vol.48)

pp: 536-551

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/12.769436

ABSTRACT

<p><b>Abstract</b>—This paper introduces a new network topology, called <b>Multi-Mesh (MM)</b>, which uses multiple meshes as the basic building blocks interconnected in a suitable manner. The proposed network consists of <tmath>$n^4$</tmath> processors and is 4-regular with a diameter of <tmath>$2n$</tmath>. The network also contains a Hamiltonian cycle. Simple routing algorithms for point-to-point communication, one-to-all broadcast, and multicast have been described for this network. It is shown that a simple <tmath>$n^2\times n^2$</tmath> mesh can also be emulated on this network in <it>O</it>(1) time. Several application examples have been discussed for which this network is found to be more efficient with regard to computational time than the corresponding mesh with the same number of processors. As examples, <it>O</it><tmath>$(n)$</tmath> time algorithms for finding the sum, average, minimum, and maximum of <tmath>$n^4$</tmath> data values, located at <tmath>$n^4$</tmath> different processors have been discussed. Time-efficient implementations of algorithms for solving nontrivial problems, e.g., Lagrange's interpolation, matrix transposition, matrix multiplication, and Discrete Fourier Transform (DFT) computation have also been discussed. The time complexity of Lagrange's interpolation on this network is <it>O</it><tmath>$(n)$</tmath> for <tmath>$n^2$</tmath> data points compared to <it>O</it>(<tmath>$n^2$</tmath>) time on mesh of the same size. Matrix transpose requires <it>O</it><tmath>$(n^{0.5}$</tmath>) time for an <tmath>$n \times n$</tmath> matrix. The time for multiplying two <tmath>$n\times n$</tmath> matrices is <it>O</it><tmath>$(n^{0.6})$</tmath> with an AT-cost of <it>O</it><tmath>$(n^3)$</tmath>. DFT of <tmath>$n$</tmath> sample points can be computed in <it>O</it><tmath>$(n^{0.6})$</tmath> time on this network. Papers [<ref type="bib" rid="bibT05366">6</ref>], [<ref type="bib" rid="bibT05367">7</ref>] show that <tmath>$n^4$</tmath> data elements can be sorted on this network in <tmath>$O(n)$</tmath> time.</p>

INDEX TERMS

Mesh, multimesh, diameter, Hamiltonian cycle, point-to-point communication, one-to-all broadcast, multicast, fault-diameter, Lagrange's interpolation, matrix transpose, matrix multiplication, DFT.

CITATION

Debasish Das, Mallika De, Bhabani P. Sinha, "A New Network Topology with Multiple Meshes",

*IEEE Transactions on Computers*, vol.48, no. 5, pp. 536-551, May 1999, doi:10.1109/12.769436