On the time required to sum n semigroup elements on a parallel machine with simultaneous writes

Abstract

Suppose we have a completely-connected network of random-access machines which communicate by reading and writing data from their neighbours, with simultaneous reads and writes allowed. In the case of write conflicts, we allow any protocol which results in one of the competing values being written into the target register. We consider the semigroup summation problem, that is, the problem of summing n semigroup elements. If the semigroup is finite, we find that it can be solved in time O(log n/log log n) using only n processors, regardless of the details of the write-conflict resolution scheme used. In contrast, we show that any parallel machine for solving the summation problem for infinite cancellative semigroups must take time [log₃n], again, regardless of the details of the conflict resolution scheme. We give an example where it is possible to sum n “polynomial-sized” elements in less than [log₃n] time using only polynomially many processors. We are also able to show that such a machine must obey the [log₃n] lower-bound for elements which are only polynomially larger. Our upper-bounds are for a machine with a reasonable local instruction-set, whilst the lower-bounds are based on a communication argument, and thus hold no matter how much computational power is available to each processor. Similar results hold for a parallel machine whose processors communicate via a shared memory.