11.7 [FTI90, KN91] Consider the single-level load-balancing scheme which works as follows: a designated processor called manager generates many subtasks and gives them one-by-one to the requesting processors on demand. The manager traverses the search tree depth-first to a predetermined cutoff depth and distributes nodes at that depth as subtasks. Increasing the cutoff depth increases the number of subtasks, but makes them smaller. The processors request another subtask from the manager only after finishing the previous one. Hence, if a processor gets subtasks corresponding to large subtrees, it will send fewer requests to the manager. If the cutoff depth is large enough, this scheme results in good load balance among the processors. However, if the cutoff depth is too large, the subtasks given out to the processors become small and the processors send more frequent requests to the manager. In this case, the manager becomes a bottleneck. Hence, this scheme has a poor scalability.

Figure 11.20 illustrates the single-level work

Assume that the cost of communicating a piece of work between any two processors is negligible. Derive analytical expressions for the scalability of the single-level loadbalancing scheme.