[Paper Review] Energy and Time Efficient Scheduling of Tasks with Dependencies on Asymmetric Multiprocessors

Ioannis Chatzigiannakis, Georgios Giannoulis+1 more|ArXiv.org|Apr 25, 2008

Scheduling and Optimization Algorithms10 references3 citations

TL;DR

This paper proposes energy- and time-efficient scheduling algorithms for task graphs with precedence constraints on asymmetric multiprocessors with two distinct speeds (fast and slow). It introduces the preemptive "Save-Energy" algorithm to reduce energy use without increasing makespan, and the non-preemptive "Remnants" algorithm combined with a refined randomized scheduling policy to achieve an expected makespan approximation factor of (3+o(1)), improving upon the prior best of 6 for two-speed systems.

ABSTRACT

In this work we study the problem of scheduling tasks with dependencies in multiprocessor architectures where processors have different speeds. We present the preemptive algorithm "Save-Energy" that given a schedule of tasks it post processes it to improve the energy efficiency without any deterioration of the makespan. In terms of time efficiency, we show that preemptive scheduling in an asymmetric system can achieve the same or better optimal makespan than in a symmetric system. Motivited by real multiprocessor systems, we investigate architectures that exhibit limited asymmetry: there are two essentially different speeds. Interestingly, this special case has not been studied in the field of parallel computing and scheduling theory; only the general case was studied where processors have $K$ essentially different speeds. We present the non-preemptive algorithm ``Remnants'' that achieves almost optimal makespan. We provide a refined analysis of a recent scheduling method. Based on this analysis, we specialize the scheduling policy and provide an algorithm of $(3 + o(1))$ expected approximation factor. Note that this improves the previous best factor (6 for two speeds). We believe that our work will convince researchers to revisit this well studied scheduling problem for these simple, yet realistic, asymmetric multiprocessor architectures.

Motivation & Objective

• Address the gap in scheduling theory for realistic asymmetric multiprocessor systems with only two distinct processor speeds, despite extensive study of the general K-speed case.
• Develop energy-efficient scheduling strategies that reduce power consumption without degrading makespan in asymmetric multiprocessor environments.
• Improve the approximation factor for makespan in non-preemptive scheduling of DAGs on two-speed systems beyond the prior best of 6.
• Provide a polynomial-time randomized algorithm that can be derandomized to yield a deterministic schedule with near-optimal makespan.
• Investigate the interplay between time efficiency and energy efficiency in modern heterogeneous computing platforms with limited asymmetry.

Proposed method

• Propose the preemptive "Save-Energy" algorithm to post-process an existing schedule, reallocating tasks to slower processors where possible to reduce energy use without affecting makespan.
• Introduce the non-preemptive "Remnants" algorithm to schedule chains of tasks by assigning longer chains to faster processors and shorter ones to slower ones, minimizing makespan.
• Refine a recent randomized scheduling policy from [5] by incorporating list-length and speed-assignment constraints to improve approximation guarantees.
• Apply Chernoff bounds to bound the tail probability of makespan deviation, ensuring high-probability performance close to the expected value.
• Derandomize the randomized schedule by repeating the process polynomially many times, guaranteeing a deterministic schedule with makespan at most (3+o(1)) times the optimal.
• Use linear programming relaxation to model task assignment and derive expected completion times, then optimize speed assignment based on expected load distribution.

Experimental results

Research questions

• RQ1 Can energy consumption be reduced in asymmetric multiprocessor scheduling without increasing the makespan?
• RQ2 Does preemptive scheduling on two-speed systems achieve a makespan that is as good as or better than symmetric systems?
• RQ3 What is the best achievable approximation factor for non-preemptive makespan minimization in DAG scheduling on two-speed processors?
• RQ4 Can a randomized scheduling policy be derandomized efficiently to yield a deterministic schedule with near-optimal makespan?
• RQ5 How does the performance of the proposed algorithms scale with list length and system size in realistic asymmetric architectures?

Key findings

• The "Save-Energy" algorithm reduces energy consumption in a given schedule without increasing the makespan, leveraging task reassignment to slower processors.
• Preemptive scheduling on asymmetric systems can achieve the same or better optimal makespan than on symmetric systems, demonstrating a performance advantage.
• The refined randomized scheduling policy achieves an expected makespan of at most (3+o(1)) times the optimal, improving upon the previous best factor of 6 for two-speed systems.
• For DAGs composed of long chains (length ≥ γ·n), the algorithm can produce a deterministic schedule with makespan at most (3+o(1))·T_opt in expected polynomial time.
• The probability that the actual makespan exceeds the expected value by a constant factor is bounded by O(1/n^{β²/2}), enabling efficient derandomization.
• The theoretical framework demonstrates that limited asymmetry (two speeds) is a viable and underexplored model for achieving both time and energy efficiency in modern multiprocessor systems.