Optimal Communication Primitives on the Generalized Hypercube Network

Efficient interprocessor communication is crucial to increasing the performance of parallel computers. In this paper, a special framework is developed on thegeneralized hypercube, a network that is currently receiving considerable attention. Using this framework as the basic tool, a number of spanning subgraphs with special properties to fit various communication needs are constructed on the network. The importance of these spanning subgraphs is demonstrated with the development of optimal algorithms for four fundamental communication problems, namely, theone-to-allandall-to-all broadcastingand theone-to-allandall-to-all scattering. Broadcastingis the distribution of the same group of messages from a source processor to all other processors, andscatteringis the distribution of distinct groups of messages from a source processor to each other processor. We consider broadcasting and scattering from a single processor of the network (one-to-all broadcasting and scattering) and simultaneously from all processors of the network (all-to-all broadcasting and scattering). For the all-to-all broadcasting and scattering algorithms, a special technique is developed on the generalized hypercube so that messages originating at individual nodes are interleaved in such a manner that no two messages contend for the same edge at any given time. The communication problems are studied under thestore-and-forward, all-portcommunication model. Lower bounds are derived for the above problems under the stated assumptions, in terms of time and number of message transmissions, and optimal algorithms are designed.

[1]  Yousef Saad,et al.  Data communication in parallel architectures , 1989, Parallel Comput..

[2]  Selim G. Akl,et al.  Optimal communication algorithms on the star interconnection network , 1993, Proceedings of 1993 5th IEEE Symposium on Parallel and Distributed Processing.

[3]  Lars Gårding,et al.  Algebra for computer science , 1988, Universitext.

[4]  S. Lennart Johnsson,et al.  Communication Efficient Basic Linear Algebra Computations on Hypercube Architectures , 1987, J. Parallel Distributed Comput..

[5]  Kenneth A. Ross,et al.  Discrete mathematics (3. ed.) , 1992 .

[6]  S. Lakshmivarahan,et al.  Symmetry in Interconnection Networks Based on Cayley Graphs of Permutation Groups: A Survey , 1993, Parallel Comput..

[7]  Emmanouel A. Varvarigos,et al.  Communication algorithms for isotropic tasks in hypercubes and wraparound meshes , 1992, Parallel Comput..

[8]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[9]  S. Lennart Johnsson,et al.  Distributed Routing Algorithms for Broadcasting and Personalized Communication in Hypercubes , 1986, ICPP.

[10]  John N. Tsitsiklis,et al.  Optimal Communication Algorithms for Hypercubes , 1991, J. Parallel Distributed Comput..

[11]  Sheldon B. Akers,et al.  The Star Graph: An Attractive Alternative to the n-Cube , 1994, ICPP.

[12]  Pierre Fraigniaud,et al.  Complexity Analysis of Broadcasting in Hypercubes with Restricted Communication Capabilities , 1992, J. Parallel Distributed Comput..

[13]  Emmanouel A. Varvarigos,et al.  Partial Multinode Broadcast and Partial Exchange Algorithms for d-Dimensional Meshes , 1994, J. Parallel Distributed Comput..

[14]  Selim G. Akl,et al.  A Framework for Optimal Communication on the Multidimensional Torus Network , 1994 .

[15]  Sheldon B. Akers,et al.  A Group-Theoretic Model for Symmetric Interconnection Networks , 1989, IEEE Trans. Computers.

[16]  Selim G. Akl,et al.  Optimal Communication algorithms on Star Graphs Using Spanning Tree Constructions , 1995, J. Parallel Distributed Comput..

[17]  Fikret Erçal,et al.  Time-Efficient Maze Routing Algorithms on Reconfigurable Mesh Architectures , 1997, J. Parallel Distributed Comput..

[18]  Dharma P. Agrawal,et al.  Generalized Hypercube and Hyperbus Structures for a Computer Network , 1984, IEEE Transactions on Computers.

[19]  Sudhakar Yalamanchili,et al.  Adaptive routing protocols for hypercube interconnection networks , 1993, Computer.

[20]  Selim G. Akl,et al.  Edge-Disjoint Spanning Trees on the Star Network with Applications to Fault Tolerance , 1996, IEEE Trans. Computers.

[21]  Yousef Saad,et al.  Data Communication in Hypercubes , 1989, J. Parallel Distributed Comput..

[22]  Pierre Fraigniaud,et al.  Methods and problems of communication in usual networks , 1994, Discret. Appl. Math..

[23]  Quentin F. Stout,et al.  Intensive Hypercube Communication. Prearranged Communication in Link-Bound Machines , 1990, J. Parallel Distributed Comput..

[24]  S. Lennart Johnsson,et al.  Optimum Broadcasting and Personalized Communication in Hypercubes , 1989, IEEE Trans. Computers.