Online Scheduling of Task Graphs on Heterogeneous Platforms

Modern computing platforms commonly include accelerators. We target the problem of scheduling applications modeled as task graphs on hybrid platforms made of two types of resources, such as CPUs and GPUs. We consider that task graphs are uncovered dynamically, and that the scheduler has information only on the available tasks, i.e., tasks whose predecessors have all been completed. Each task can be processed by either a CPU or a GPU, and the corresponding processing times are known. Our study extends a previous <inline-formula><tex-math notation="LaTeX">$4\sqrt{m/k}$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>4</mml:mn><mml:msqrt><mml:mrow><mml:mi>m</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:math><inline-graphic xlink:href="simon-ieq1-2942909.gif"/></alternatives></inline-formula>-competitive online algorithm by Amaris et al. <xref ref-type="bibr" rid="ref1">[1]</xref> , where <inline-formula><tex-math notation="LaTeX">$m$</tex-math><alternatives><mml:math><mml:mi>m</mml:mi></mml:math><inline-graphic xlink:href="simon-ieq2-2942909.gif"/></alternatives></inline-formula> is the number of CPUs and <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="simon-ieq3-2942909.gif"/></alternatives></inline-formula> the number of GPUs (<inline-formula><tex-math notation="LaTeX">$m\geq k$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="simon-ieq4-2942909.gif"/></alternatives></inline-formula>). We prove that no online algorithm can have a competitive ratio smaller than <inline-formula><tex-math notation="LaTeX">$\sqrt{m/k}$</tex-math><alternatives><mml:math><mml:msqrt><mml:mrow><mml:mi>m</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msqrt></mml:math><inline-graphic xlink:href="simon-ieq5-2942909.gif"/></alternatives></inline-formula>. We also study how adding flexibility on task processing, such as task migration or spoliation, or increasing the knowledge of the scheduler by providing it with information on the task graph, influences the lower bound. We provide a <inline-formula><tex-math notation="LaTeX">$(2\sqrt{m/k}+1)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:msqrt><mml:mrow><mml:mi>m</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msqrt><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="simon-ieq6-2942909.gif"/></alternatives></inline-formula>-competitive algorithm as well as a tunable combination of a system-oriented heuristic and a competitive algorithm; this combination performs well in practice and has a competitive ratio in <inline-formula><tex-math notation="LaTeX">$\Theta (\sqrt{m/k})$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>Θ</mml:mi><mml:mo>(</mml:mo><mml:msqrt><mml:mrow><mml:mi>m</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msqrt><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="simon-ieq7-2942909.gif"/></alternatives></inline-formula>. We also adapt all our results to the case of multiple types of processors. Finally, simulations on different sets of task graphs illustrate how the instance properties impact the performance of the studied algorithms and show that our proposed tunable algorithm performs the best among the online algorithms in almost all cases and has even performance close to an offline algorithm.