Circuit and decision tree complexity of some number theoretic problems
Information and Computation
Approximate Compaction and Padded-Sorting on Exclusive Write PRAMs
IPPS '96 Proceedings of the 10th International Parallel Processing Symposium
Timing for Associative Operations on the MASC Model
IPDPS '01 Proceedings of the 15th International Parallel & Distributed Processing Symposium
Gossiping and broadcasting versus computing functions in networks
Discrete Applied Mathematics - Special issue on international workshop on algorithms, combinatorics, and optimization in interconnection networks (IWACOIN '99)
Complexity of some arithmetic problems for binary polynomials
Computational Complexity
Circuit complexity of testing square-free numbers
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
On the average sensitivity of testing square-free numbers
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
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It was shown some years ago that the computation time for many important Boolean functions of $n$ arguments on concurrent-read exclusive-write parallel random-access machines (CREW PRAMs) of unlimited size is at least $arphi (n) \approx 0.72\log_2 n$. On the other hand, it is known that every Boolean function of $n$ arguments can be computed in $arphi (n)+1$ steps on a CREW PRAM with $n\cdot 2^{n-1}$ processors and memory cells. In the case of the OR of $n$ bits, $n$ processors and cells are sufficient. In this paper, it is shown that for many important functions, there are CREW PRAM algorithms that almost meet the lower bound in that they take $arphi (n) + o(\log n)$ steps but use only a small number of processors and memory cells (in most cases, $n$). In addition, the cells only have to store binary words of bounded length (in most cases, length 1). We call such algorithms ``feasible.'' The functions concerned include the following: the $\PARITY$ function and, more generally, all symmetric functions; a large class of Boolean formulas; some functions over non-Boolean domains $\{0,\ldots ,k-1\}$ for small $k$, in particular, parallel-prefix sums; addition of $n$-bit numbers; and sorting $n/l$ binary numbers of length $l$. Further, it is shown that Boolean circuits with fan-in 2, depth $d$, and size $s$ can be evaluated by CREW PRAMs with fewer than $s$ processors in $arphi(2^d)+o(d) \approx 0.72d+ o(d)$ steps. For the exclusive-read exclusive-write (EREW) PRAM model, a feasible algorithm is described that computes $\PARITY$ of $n$ bits in $ 0.86\log_2 n$ steps.