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Issue No. 09 - September (1970 vol. 19)
ISSN: 0018-9340
pp: 859-860
K.N. Levitt , Stanford Research Institute
Applying some simple, easily understood principles, Spira, in extending some earlier work of Winograd, points the way to a powerful theory of computation complexity. Spira considers a (d, r) combinational network which is an interconnection of r-input, single-output modules, with each input-output line carrying a value from the set {0, 1, ? , d -1}. A finite function f: X1 ? X2 ? ? Xn?Y is to be computed, but it is assumed that before the inputs are inserted into the network, each input can be individually (and arbitrarily) transformed by a set of maps gj: Xj?Ij. It is also assumed that there is a 1-1 output map h: Y?Oc, so in practice the (d, r) network will have as input [g1(x), ?, gn(xn)] and as output h(f(x1, ?, xn)). The problem is to bound the number of levels required of the network. Given a f for a particular output mapping, it is not difficult to specify a lower bound on the number of levels required, by identifying for each output line the number of different values of input variables which yield a different output value. The minimum number of levels required for each output line is then evaluated by noting that an output at level z can depend on at most r' input lines whence the output line requiring the most levels provides the bound.
K.N. Levitt, "R70-38 The Time Required for Group Multiplication", IEEE Transactions on Computers, vol. 19, no. , pp. 859-860, September 1970, doi:10.1109/T-C.1970.223066
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