- Author:
- Dino Karabeg,
- Chairperson:
- Janos Komlos
The possibility of solving problems on a parallel computer in sublogarithmic time is discussed. Known lower-bound results are reviewed in order to show which properties are required of a parallel computer, and to point out to the bounds on the speed of sublogarithmic-time computation. The $\Omega$(log n/log log n) lower bound by Beame and Hastad for computing the parity function on the Abstract CRCW PRAM model is generalized by showing that the same bound holds even for computing the parity function with probability $1\over2$ + ${1}\over{\rm log {\it n}}$.
An integer summation algorithm is proposed that produces the prefix sums of n integers, each having O(log n) bits, in time O(log n/log log n) on a CRCW PRAM parallel computer. The algorithm has both an optimal running time and an optimal speedup (a different algorithm for the same problem has been proposed by Cole and Vishkin).
A deterministic CRCW PRAM algorithm for sorting is proposed, that sorts all except a negligible proportion of inputs in time O(log n/log log n) using n processors (the PRAM is assumed to have a vector instruction for comparisons). The algorithm is shown to have an optimal running time.
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