Algorithms for Scheduling Imprecise Computations with Timing Constraints

Here the problem of scheduling tasks, each of which is logically decomposed into a mandatory subtask and an optional subtask, is considered. The mandatory subtask must be executed to completion in order to produce an acceptable result. The optional subtask begins after the mandatory subtask is completed and refines the result in order to reduce the error in the result. The optional subtask can be left incomplete. The error in the result of a task is equal to the processing time of the unfinished portion of the optional subtask. Two preemptive algorithms for scheduling, on a uniprocessor system, n dependent tasks with rational ready times, deadlines, and processing times are described. An algorithm is optimal in the following sense: whenever feasible schedules that meet the ready time and deadline constraints of all tasks exist, it finds one that has the minimum total error of all tasks. One of the algorithms is optimal when the tasks have identical weights, and its time complexity is $O(n\log n)$. The other algorithm has time complexity $O(n^2 )$, but is optimal when tasks have different weights. A schedule is said to satisfy the $0/1$ constraint when every optional subtask is either completed or discarded. The problem of finding an optimal feasible schedule that satisfies the $0/1$ constraints and minimizes the total processing time of the discarded optional subtasks is NP-complete. Two algorithms for finding optimal schedules of dependent tasks on a uniprocessor system for the special case when all optional subtasks have identical processing times are presented.