Update on the Asymptotic Optimality of LPT

Abstract

When independent tasks are to be scheduled onto identical processors, the typical goal is to minimize the makespan. A simple and efficient heuristic consists in scheduling first the task with the longest processing time (LPT heuristic), and to plan its execution as soon as possible. While the performance of LPT has already been largely studied, in particular its asymptotic performance, we revisit results and propose a novel analysis for the case of tasks generated through uniform integer compositions. Also, we perform extensive simulations to empirically assess the asymptotic performance of LPT. Results demonstrate that the absolute error rapidly tends to zero for several distributions of task costs, including ones studied by theoretical models, and realistic distributions coming from benchmarks.

Asymptotic Performance and Energy Consumption of SLACK

“…This algorithm has proven to have good theoretical and even better practical performance. In particular, its rate of convergence has been studied, and new results were recently established when the distribution of task costs is generated using uniform integer compositions [5].…”

“…As for scheduling algorithms that have low complexities (and therefore low energy consumption), Lpt has been a well known algorithm for decades and is known to provide good theoretical and practical performance while keeping a low time complexity in O(n log n) [14]. A more recent algorithm, Slack, also remains with an O(n log n) time complexity, while providing results that are sometimes better than Lpt [7,5].…”

“…Loulou [16] and Piersma and Romeijn [17] do not look at specific distributions, but instead they study Lpt under the assumption that the execution times are independent and identically distributed random variables. More recently, Benoit et al [5] studied the asymptotic optimality of Slack and Lpt under the assumption that the execution times are generated using a distribution called the uniform integer composition.…”

“…With these experiments, we can both investigate the relation we stated in Section 4, and investigate the unexplained "wave pattern" presented in [5].…”

Exploring Scheduling Algorithms for Parallel Task Graphs: A Modern Game Engine Case Study

“…4). We nd that there is value in bringing to light new applications and knowledge on existing algorithms, as have done Benoit et al [5] for the asymptotic performance of the longest processing time (LPT) heuristic for the case of tasks originating from uniform integer compositions.…”