Optimal Self-Routing of Linear-Complement Permutations in Hypercubes

In this paper we describe an algorithm to route the class of linear-complement permutations on Hypercube SIMD computers. The class of linearcomplement permutations are extremely useful in devising storage schemes for parallel array access. The proposed algorithm is self-routing and minimal, that is, the path established by the algorithm between each pair of source and destination processors is via a minimal path using only the destination processor address. Furthermore, the algorithm requires only the optimal number of routing steps to realize any linearcomplement permutation. The best known previous routing algorithms for the Hypercubes are for the class of bit-permute-complement permutations, a subset of the class of linear-complement permutations. Those algorithms are either non-optimal or not self-routing. The algorithm presented is self-routing, optimal, and it routes a larger class of permutations. Also, this algorithm can route the class of linear-complement permutations in multi-dimensional meshes in optimal number of routing steps.

[1]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[2]  P. S. Aleksandrov,et al.  An introduction to the theory of groups , 1960 .

[3]  Jon S. Squire,et al.  Programming and design considerations of a highly parallel computer , 1963, AFIPS '63 (Spring).

[4]  Michael J. Flynn,et al.  Very high-speed computing systems , 1966 .

[5]  Kenneth E. Batcher,et al.  Sorting networks and their applications , 1968, AFIPS Spring Joint Computing Conference.

[6]  Harold S. Stone,et al.  Parallel Processing with the Perfect Shuffle , 1971, IEEE Transactions on Computers.

[7]  Michael J. Flynn,et al.  Some Computer Organizations and Their Effectiveness , 1972, IEEE Transactions on Computers.

[8]  Marshall C. Pease,et al.  The Indirect Binary n-Cube Microprocessor Array , 1977, IEEE Transactions on Computers.

[9]  M. Newman,et al.  Topics in Algebra , 1978 .

[10]  Sartaj Sahni,et al.  An optimal routing algorithm for mesh-connected Parallel computers , 1980, JACM.

[11]  Sartaj Sahni,et al.  Data broadcasting in SIMD computers , 1981, IEEE Transactions on Computers.

[12]  Leslie G. Valiant,et al.  Universal schemes for parallel communication , 1981, STOC '81.

[13]  Sartaj Sahni,et al.  A Self-Routing Benes Network and Parallel Permutation Algorithms , 1981, IEEE Transactions on Computers.

[14]  Sartaj Sahni,et al.  Optimal BPC Permutations on a Cube Connected SIMD Computer , 1982, IEEE Transactions on Computers.

[15]  Charles L. Seitz,et al.  The cosmic cube , 1985, CACM.

[16]  William Jalby,et al.  XOR-Schemes: A Flexible Data Organization in Parallel Memories , 1985, ICPP.

[17]  Cauligi S. Raghavendra,et al.  On Methods for Fast and Efficient Parallel Memory Access , 1990, ICPP.

[18]  Cauligi S. Raghavendra,et al.  On Self-Routing in Benes and Shuffle-Exchange Networks , 1991, IEEE Trans. Computers.