Optimal Co-Scheduling to Minimize Makespan on Chip Multiprocessors

On-chip resource sharing among sibling cores causes resource contention on Chip Multiprocessors (CMP), considerably degrading program performance and system fairness. Job co-scheduling attempts to alleviate the problem by assigning jobs to cores intelligently. Despite many heuristics-based empirical explorations, studies on optimal co-scheduling and its inherent complexity start only recently, and all have concentrated on the minimization of total performance degradations. There is another important criterion for scheduling, makespan, which determines the finish time of a job set. Its importance for job co-scheduling on CMP is increasingly recognized, especially with the rise of CMP-based compute cloud, data centers, and server farms. However, optimal coscheduling for makespan minimization still remains unexplored. This work compares makespan minimization problem with previously studied cost minimization (or degradation minimization) problem, revealing these connections as well as significant differences. It uncovers the computational complexity of the makespan minimization problem, and proposes a series of algorithms to either compute or approximate the optimal schedules. It proves that the problem is NP-complete in a general setting, but for a special case (dual-core without job migrations), the problem is solvable in O(n · log n) time (n is the number of jobs). In addition, this paper presents a series of algorithms to compute or approximate the optimal schedules in the general setting. Experiments on both real and synthetic problems verify the optimality of the optimal co-scheduling algorithms, and demonstrate the reasonable accuracy and scalability of the approximation algorithms. The findings may advance the current understanding of optimal co-scheduling, facilitate the evaluation of real co-scheduling systems, and provide insights for the development of practical co-scheduling algorithms.