Collection-aware optimum sequencing of operations and closed-form solutions for the distribution of a divisible load on arbitrary processor trees

The problem of optimally distributing a divisible load to the nodes of an arbitrary processor tree is tackled in this paper. The rigorous mathematical foundation presented allows the derivation of the sequence of operations that is necessary to obtain the minimum processing time, along with closed-form expressions that yield the solution in time O(NP), where P is the number of tree nodes and N their maximum degree. The main contributions of this work are: (1) both load distribution and result collection overheads are considered, thus providing better resource utilization, and (2) arbitrary processor trees are examined in contrast with previous approaches that examined either complete homogeneous trees, or single level trees. Additionally, approximate algorithms for solving the problem of specifying the optimum subset of active processors for a given load, are presented and evaluated.