Critical Path Length of Large Acyclic Task Graphs

Abstract

We consider the execution of a large set of interdependent tasks, represented by an acyclic task graph, in which each task has arbitrary execution time and the precedence relations between tasks have arbitrary information or message transfer times associated with them. The acyclic task graph can model the execution of a parallel program, in which each task represents the execution of a sequential process. It can also be used to model PERT diagrams. The task graph is acyclic because we are interested in executions which terminate. Furthermore, communication times related to the transfer of control or of information between tasks is also represented by weights or times associated with the arcs of the graph. We are interested in determining the maximum speed-up which can be attained by a parallel program. In order to do so, we show that the length of the critical path of an arbitrary task graph, which is equivalent to the best execution time of the set of tasks with an unlimited number of processors, grows (almost surely) linearly with the total number of tasks it contains. The result is obtained using sub-additive process theory. It implies that the best speedup that can be attained is some constant independent of the size of the task graph. This confirms and generalizes special instances of the result which have appeared in the litterature.