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On the number of acceptable task assignments in distributed computing systems

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2 Author(s)
Shin, K.G. ; Dept. of Electr. Eng. & Comput. Sci., Michigan Univ., Ann Arbor, MI, USA ; Ming-Syan Chen

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

Published in:

Computers, IEEE Transactions on  (Volume:39 ,  Issue: 1 )