Abstract
We introduce the class Structured NC (or SNC for short), consisting of all problems that can be solved by means of “structured” NC-circuits. A structured circuit is a circuit in which the gates are organized in a number of layers, each with the same number of gates and the same connections between gates in a layer. We will distinguish several subclasses of SNC depending on the depth of the circuits and the fan-in and fan-out of the gates. Relations among these subclasses and between these subclasses and NC and AC will be derived and we will show how to transform circuits from one class into equivalent circuits in other classes. Furthermore for some problems we present structured NC-circuits having fan-in as well as fan-out≤2; for Parallel Prefix Computation and Addition of n-bit numbers these circuits are of depth O(log n) with only O(n) gates in a layer, which is optimal for structured circuits. For Multiplication we will consider three methods and derive structured NC-circuits for them.
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References
Aho, A.V., J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.
Buss, S.R., The boolean formala value problem is in ALOGTIME, Proc 19th Annual ACM Symp. on Theory of Computing, 1987, pp.123–131.
Chandra, A.K., D.C. Kozen, L.J. Stockmeyer, Alternation, JACM 28 (1981), pp.114–133.
Cook, S.A., Towards a complexity theory of synchronous parallel computation, Enseign. Math. 27 (1981), pp.99–124.
Cook, S.A., A taxonomy of problems with fast parallel algorithms, Inform. and Control 64(1985), pp.2–22.
Furst, M., J.B. Saxe and M. Sipser, Parity, circuits and the polynomial time hierarchy, Math. Systems Theory 17(1984), pp.13–28.
Karp, R.M. and V.L. Ramachandran, A survey of parallel algorithms for shared-memory machines, in: J. van Leeuwen (Ed.), Handbook of Theoretical Computer Science, North-Holland Publ. Comp., Amsterdam (to appear).
Ladner, R.E. and M.J. Fischer, Parallel prefix computation, JACM 27 (1980), pp.831–838.
Mehlhorn, K. and F.P. Preparata, Area-time optimal VLSI integer multiplier with minimum computation time, IC 5(1983), pp.137–156.
Ofman, Y., On the algorithmic complexity of discrete functions, Sov. Phys. Dokl. 7 (1963), pp.589–591.
Ruzzo, W.L., Tree-size bounded alternation, J. Comp. Syst. Sci. 21 (1980), pp.218–235.
Ruzzo, W.L., On uniform circuit complexity, J. Comp. Syst. Sci 22 (1981), pp.365–383.
Savage, J.E., The Complexity of Computing, Wiley, New York, 1976.
Scholten, B. and J. van Leeuwen, Structured NC, Techn. Rep. RUU-CS-89-6, Dept. of Computer Science, University of Utrecht, Utrecht, 1989.
Schönhage, A. and V. Strassen, Schnelle Multiplikation grosser Zahlen, Computing 7(1971), pp.281–292.
Wallace, C.S., A suggestion for a fast multiplier, IEEE Trans. Comp. EC-13(1964), pp.14–17.
Wegener, I., The Complexity of Boolean Functions, Wiley-Teubner, Stuttgart, 1987.
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© 1989 Springer-Verlag Berlin Heidelberg
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Scholten, B., van Leeuwen, J. (1989). Structured NC. In: Dehne, F., Sack, J.R., Santoro, N. (eds) Algorithms and Data Structures. WADS 1989. Lecture Notes in Computer Science, vol 382. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51542-9_40
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DOI: https://doi.org/10.1007/3-540-51542-9_40
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