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On the Number of Acceptable Task Assignments in Distributed Computing Systems
January 1990 (vol. 39 no. 1)
pp. 99-110

A distributed computing system and cooperating tasks can be represented by a processor graph G/sub p/=(V/sub p/, E/sub p/) and a task graph G/sub T/=(V/sub T/, E/sub T/), respectively. An edge between a pair of nodes in G/sub T/ represents the existence of direct communications between the two corresponding tasks. The maximal number of hops between two processors in G/sub p/ to which two adjacent tasks in G/sub T/ are assigned is called dilation of that assignment. Characterization and use of the number of acceptable assignments for given G/sub T/ and G/sub P/ are treated. Assignments with the dilation less than or equal to one are considered. This dilation constraint represents a special case in which two adjacent tasks in G/sub T/ must be assigned to either a single processor or two adjacent processors in G/sub p/. For the case where N(G/sub T/, G/sub P/) denotes the numbers of acceptable assignments under this constraint, N(G/sub T/, G/sub P/) are derived for arbitrary G/sub T/ and G/sub P/, and a recursive expression is formulated for N(G/sub T/, G/sub P/) when G/sub T/ is a tree. For some restricted cases, either closed-form or recursive-form expressions of N(G/sub T/, G/sub P/) are derived. The results on N(G/sub T/, G/sub P/) are extended to the completely general case, assignments with dilations greater than one, where two adjacent tasks in G/sub T/ can be assigned to any two processors in G/sub P/ which are not necessarily adjacent to each other.

[1] W. W. Chu, L. J. Holloway, M. T. Lan, and K. Efe, "Task allocation in distributed data processing,"IEEE Comput. Mag., vol. 13, pp. 57-69, Nov. 1980.
[2] W. W. Chu and L. M.-T. Lan, "Task allocation and precedence relations for distributed real-time systems,"IEEE Trans. Comput., vol. C-36, pp. 667-679, June 1987.
[3] C. L. Seitz, "The Cosmic Cube,"Commun. ACM, pp. 22-33, Jan. 1985.
[4] C. C. Shen and W. H. Tsai, "A graph matching approach to optimal task assignment in distributed computing systems using a minimax criterion,"IEEE Trans. Comput., vol. C-34, no. 3, pp. 197-203, Mar. 1985.
[5] F. Harary, "The topological cubical dimension of a graph," inLecture Notes First Japan Conf., Graph Theory Appl., June 3, 1986.
[6] R. C. Read and D. G. Corneil, "The graph isomorphism disease,"J. Graph Theory, pp. 339-363, 1977.
[7] M. R. Garey and D. S. Johnson,Computers and Intractability: A Guide to Theory of NP-Completeness. San Francisco, CA: Freeman, 1979.
[8] N. Nilsson,Principles of Artificial Intelligence. Palo Alto, CA: Tioga, 1980.
[9] S. H. Bokhari, "On the mapping problem,"IEEE Trans. Comput., vol. C-30, no. 3, pp. 207-214, Mar. 1981.
[10] K. Efe, "Heuristic models of task assignment scheduling in distributed systems,"IEEE Comput. Mag., vol. 15, pp. 50-56, June 1982.
[11] G. S. Rao, H. S. Stone, and T. C. Hu, "Assignment of tasks in a distributed processing system with limited memory,"IEEE Trans. Comput., vol. C-28, no. 4, pp. 291-299, Apr. 1979.
[12] F. Harary,Graph Theory. Reading, MA: Addison-Wesley, 1969.
[13] A. W. Marshall and I. Olkin,Inequalities: Theory of Majorization and Its Applications. New York: Academic, 1969.
[14] D. G. Corneil and D. G. Kirkpatrick, "A theoretical analysis of various heuristics for the graph isomorphism problem,"SIAM J. Comput., vol. 9, no. 2, pp. 281-297, May 1980.

Index Terms:
acceptable task assignments; distributed computing systems; cooperating tasks; processor graph; hops; dilation; distributed processing; graph theory.
Citation:
K.G. Shin, M.-S. Chen, "On the Number of Acceptable Task Assignments in Distributed Computing Systems," IEEE Transactions on Computers, vol. 39, no. 1, pp. 99-110, Jan. 1990, doi:10.1109/12.46284
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