Dynamic Scheduling on Parallel Machines. Anja Feldmann, Jiri Sgall, Shang-Hua Teng; Theoretical Computer Science 1994.
Abstract
We study the problem of online job-scheduling on parallel machines with different network topologies. An online scheduling algorithm schedules a collection of parallel jobs with known resource requirements but unknown running times on a parallel machine.
We give an O(sqrt{loglog N})-competitive algorithm for online scheduling on a two-dimensional mesh of N processors and we prove a matching lower bound of Omega(sqrt{loglog N}) on the competitive ratio. Furthermore, we show tight constant bounds of 2 for PRAMs and hypercubes, and present a 2.5-competitive algorithm for lines. We also generalize our two-dimensional mesh result to higher dimensions. Surprisingly, our algorithms become less and less greedy as the geometric structure of the network topology becomes more complicated. The proof of our lower bound for the two-dimensional mesh actually shows that no greedy-like algorithm can perform well.

Postscript