The size and depth of Boolean circuits




Jang, Jing-Tang Keith

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We study the relationship between size and depth for Boolean circuits. Over four decades, very few results were obtained for either special or general Boolean circuits. Spira showed in 1971 that any Boolean formula of size s can be simulated in depth O(log s). Spira's result means that an arbitrary Boolean expression can be replaced by an equivalent "balanced" expression, that can be evaluated very efficiently in parallel. For general Boolean circuits, the strongest known result is that Boolean circuits of size s can be simulated in depth O(s / log s). We obtain significant improvements over the general bounds for the size versus depth problem for special classes of Boolean circuits. We show that every layered Boolean circuit of size s can be simulated by a layered Boolean circuit of depth O(sqrt{s log s}). For planar circuits and synchronous circuits of size s, we obtain simulations of depth O(sqrt{s}). Improving any of the above results by polylog factors would immediately improve the bounds for general circuits. We generalize Spira's theorem and show that any Boolean circuit of size s with segregators of size f(s) can be simulated in depth O(f(s)log s). This improves and generalizes a simulation of polynomial-size Boolean circuits of constant treewidth k in depth O(k² log n) by Jansen and Sarma. Since the existence of small balanced separators in a directed acyclic graph implies that the graph also has small segregators, our results also apply to circuits with small separators. Our results imply that the class of languages computed by non-uniform families of polynomial size circuits that have constant size segregators equals non-uniform NC¹. As an application of our simulation of circuits in small depth, we show that the Boolean Circuit Value problem for circuits with constant size segregators (or separators) is in deterministic SPACE (log² n). Our results also imply that the Planar Circuit Value problem, which is known to be P-Complete, is in SPACE (sqrt{n} log n). We also show that the Layered Circuit Value and Synchronous Circuit Value problems, which are both P-complete, are in SPACE(sqrt{n}). Our study of circuits with small separators and segregators led us to obtain space efficient algorithms for computing balanced graph separators. We extend this approach to obtain space efficient approximation algorithms for the search and optimization versions of the SUBSET SUM problem, which is one of the most studied NP-complete problems. Finally we study the relationship between simultaneous time and space bounds on Turing machines and Boolean circuit depth. We observe a new connection between planar circuit size and simultaneous time and space products of input-oblivious Turing machines. We use this to prove quadratic lower bounds on the product of time and space for several explicit functions for input-oblivious Turing machines.



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