On the Number of Acceptable Task Assignments in Distributed Computing Systems

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. >