The Twisted N-Cube with Application to Multiprocessing

It is shown that by exchanging any two independent edges in any shortest cycle of the n-cube (n>or=3), its diameter decreases by one unit. This leads to the definition of a new class of n-regular graphs, denoted TQ/sub n/, with 2/sup n/ vertices and diameter n-1, which has the (n-1)-cube as subgraph. Other properties of TQ/sub n/ such as connectivity and the lengths of the disjoints paths are also investigated. Moreover, it is shown that the complete binary tree on 2/sup n/-1 vertices, which is not a subgraph of the n-cube, is a subgraph of TQ/sub n/. How these results can be used to enhance hypercube multiprocessors is discussed. >

[1]  Stephen T. Hedetniemi,et al.  ON CLASSES OF GRAPHS DEFINED BY SPECIAL CUTSETS OF LINES. , 1969 .

[2]  J. Spencer,et al.  EVOLUTION OF THE n-CUBE , 1979 .

[3]  Hans-J. Bandelt,et al.  Infinite median graphs, (0, 2)-graphs, and hypercubes , 1983, J. Graph Theory.

[4]  S. Foldes,et al.  A characterization of hypercubes , 1977, Discret. Math..

[5]  Frank Harary,et al.  Graph Theory , 2016 .

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

[7]  D. A. Reed,et al.  Networks for parallel processors: measurements and prognostications , 1988, C3P.

[8]  M. H. Schultz,et al.  Topological properties of hypercubes , 1988, IEEE Trans. Computers.

[9]  Sergiu Hart,et al.  A note on the edges of the n-cube , 1976, Discret. Math..

[10]  Kai Hwang,et al.  Hypernet: A Communication-Efficient Architecture for Constructing Massively Parallel Computers , 1987, IEEE Transactions on Computers.

[11]  Sartaj Sahni,et al.  Binary Trees and Parallel Scheduling Algorithms , 1981, IEEE Transactions on Computers.

[12]  Abdol-Hossein Esfahanian,et al.  Generalized Measures of Fault Tolerance with Application to N-Cube Networks , 1989, IEEE Trans. Computers.

[13]  Yousef Saad,et al.  Multigrid Algorithms on the Hypercube Multiprocessor , 1986, IEEE Transactions on Computers.

[14]  William J. Dally,et al.  Performance Analysis of k-Ary n-Cube Interconnection Networks , 1987, IEEE Trans. Computers.

[15]  Dirk Grunwald,et al.  Benchmarking hypercube hardware and software , 1986 .

[16]  Moshe Krieger,et al.  Multiple Microprocessor Systems: What, Why, and When , 1983, Computer.

[17]  Jon Gregory Kuhl.  Fault diagnosis in computing networks , 1980 .

[18]  Henry Martyn Mulder.  N-cubes and Median Graphs , 1980, J. Graph Theory.

[19]  Kai Hwang,et al.  Computer architecture and parallel processing , 1984, McGraw-Hill Series in computer organization and architecture.

[20]  Lionel M. Ni,et al.  Multicast in Hypercube Multiprocessors , 1990, J. Parallel Distributed Comput..

[21]  James R. Armstrong,et al.  Fault Diagnosis in a Boolean n Cube Array of Microprocessors , 1981, IEEE Transactions on Computers.

[22]  Ilse C. F. Ipsen,et al.  How to Embed Trees in Hypercubes. , 1985 .

[23]  John C. Peterson,et al.  The Mark III Hypercube-Ensemble Concurrent Computer , 1985, International Conference on Parallel Processing.

[24]  John P. Hayes,et al.  Architecture of a Hypercube Supercomputer , 1986, ICPP.

[25]  L. Nebeský.  On cubes and dichotomic trees , 1974 .