Optimal online algorithms for MapReduce scheduling on two uniform machines

We study online scheduling on two uniform machines in the MapReduce system. Each job consists of two sets of tasks, namely the map tasks and reduce tasks. A job’s reduce tasks can only be processed after all its map tasks are finished. The map tasks are fractional, i.e., they can be arbitrarily split and processed on different machines in parallel. Our goal is to find a schedule that minimizes the makespan. We consider two variants of the problem, namely the cases involving preemptive reduce tasks and non-preemptive reduce tasks. We provide lower bounds for both variants. For preemptive reduce tasks, we present an optimal online algorithm with a competitive ratio of $$\frac{\sqrt{s^{2}+2s+5}+1-s}{2}$$s2+2s+5+1-s2, where $$s\ge 1$$s≥1 is the ratio between the speeds of the two machines. For non-preemptive reduce tasks, we show that the $${ LS}$$LS-like algorithm is optimal and its competitive ratio is $$\frac{2s+1}{s+1}$$2s+1s+1 if $$s<\frac{1+\sqrt{5}}{2}$$s<1+52 and $$\frac{s+1}{s}$$s+1s if $$s\ge \frac{1+\sqrt{5}}{2}$$s≥1+52.