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Parallel sorting with limited bandwidth

We study the problem of sorting on a parallel computer with limited communication bandwidth. By using the PRAM(m) model, where p processors communicate through a globally shared memory which can service m requests per unit time, we focus on the trade-off between the amount of local computation and the amount of interprocessor communication required for parallel sorting algorithms. Our main result is a lower bound of Ω( n log m m log n ) on the time required to sort n numbers on the exclusive-read and queued-read variants of the PRAM(m). We also show that Leighton's Columnsort can be used to give an asymptotically matching upper bound in the case where m grows as a fractionalpower of n. The bounds are of a surprising form in that they have little dependence on the parameter p. This implies that attempting to distribute the workload across more processors while holding the problem size and the size of the shared memory fixed will not improve the optimal running time of sorting in this model. We also show that both the lower and the upper bounds can be adapted to bridging models that address the issue of limited communication bandwidth: the LogP model and the bulk-synchronous parallel (BSP) model. The lower bounds provide further convincing evidence that efficient parallel algorithms for sorting rely strongly on high communication bandwidth.