NDLI: Size--Depth Tradeoffs for Threshold Circuits

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Saks, Michael E. Impagliazzo, Russell Paturi, Ramamohan
Copyright Year	1997
Abstract	The following size--depth tradeoff for threshold circuits is obtained: any threshold circuit of depth d that computes the parity function on n variables must have at least $n^{1 + c\theta^{-d }}$ edges, where $c>0$ and $\theta \leq 3$ are constants independent of n and d. Previously known constructions show that up to the choice of c and $\theta$ this bound is best possible. In particular, the lower bound implies an affirmative answer to the conjecture of Paturi and Saks that a bounded-depth threshold circuit that computes parity requires a superlinear number of edges. This is the first superlinear lower bound for an explicit function that holds for any fixed depth and the first that applies to threshold circuits with unrestricted weights. The tradeoff is obtained as a consequence of a general restriction theorem for threshold circuits with a small number of edges: For any threshold circuit with n inputs, depth d, and at most $kn$ edges, there exists a partial assignmentto the inputs that fixes the output of the circuit to a constant while leaving $\lfloor n/(c_1k)^{c_2\theta^{d}} \rfloor$ variables unfixed, where $c_1,c_2 > 0$ and $ \theta \leq 3$ are constants independent of n, k, and d.A tradeoff between the number of gates and depth is also proved: any threshold circuit of depth d that computes the parity of n variables has at least $(n/2)^{1/2(d-1)}$ gates. This tradeoff, which is essentially the best possible, was proved previously (with a better constant in the exponent) for the case of threshold circuits with polynomially bounded weights in [K. Siu, V. Roychowdury, and T. Kailath, IEEE Trans. Inform. Theory, 40 (1994), pp. 455--466]; the result in the present paper holds for unrestricted weights.
Starting Page	693
Ending Page	707
Page Count	15
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539792282965
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3
Volume Number	26
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-28
Access Restriction	Subscribed
Subject Keyword	circuit complexity threshold circuits lower bounds Complexity classes
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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