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CS221: Computational Complexity Prof. Salil Vadhan
Lecture 21: Circuit Depth and Simultaneous
Size and Depth Bounds
11/13 Scribe: Alex Healy
Contents
1 Uniform Circuits 1
2 Circuit Depth 1
3 Simultaneous Size and Depth Bounds 2
1 Uniform Circuits
Definition 1 A circuit family C 1 , C 2 ,.. ., where Cn has n inputs, is said to be (log-space) uniform if there is a logspace machine M , such that M (1n) = Cn.
Using the fact that L ā P, that a polynomial-sized circuit can be evaluated in polynomial time and that any polynomial time algorithm can be transformed into a polynomial-sized circuit in logspace, we have the following:
Proposition 2 The class of languages decidable by uniform polynomial-sized circuits is exactly P.
2 Circuit Depth
Recall that the depth of a circuit is the length of the longest path from any input to the output.
Theorem 3
- If f : { 0 , 1 }n^ ā { 0 , 1 } has a circuit of depth d, then f has a formula of size at most 2 d.
- If f : { 0 , 1 }n^ ā { 0 , 1 } has a formula of size s, then f has a circuit (in fact, a formula) of depth O(log s).
Proof: The proof of 1 was given last lecture. The idea for part 2 is to ābalanceā the formula to obtain a formula of depth O(log s). To achieve this, we note that every formula of size s has a sub-formula (i.e. gate) G of size sā²^ ā [s/ 3 , 2 s/3], where the size of a sub-formula G is defined to be the size of the circuit below the gate G; this can be seen as follows: start at the root node (i.e., the output gate). Recursively travel to the āheavier/largerā child. This reduces the size of the remaining circuit by at most half in each step, and so it is not possible to pass from a circuit of
size greater than 2s/3 to a circuit of size less than s/3 in a single step. In particular, we eventually encounter a sub-formula with size sā²^ ā [s/ 3 , 2 s/3]. Now define the circuit F 0 to be the circuit obtained by hardwiring the output of G to be 0 (and removing the sub-formula under G), and similarly define F 1 to be the circuit obtained by hardwiring the output of G to be 1. We may now write F = (G ā§ F 1 ) āØ (Ā¬G ā§ F 0 ) where F 0 , F 1 and G are all of size between s/3 and 2s/3. Now we recursively balance F 0 , F 1 and G using the same technique. If we let D(s) denote the maximum depth that is needed for a formula of size s, then the above construction shows that D(s) satisfies the following recurrence: D(s) ā¤ D(2s/3) + 2
and in particular, solving for D(s), we have that D(s) ā¤ 2 Ā· dlog 3 / 2 se.
Corollary 4 Most functions f : { 0 , 1 }n^ ā { 0 , 1 } require depth approximately n.
Proof: depth(f ) ā„ log(formula size) ā„ log(cktsize(f )) and for most f , cktsize(f ) & log(2n/n).
Why study circuit depth? Because it would seem that lower bounds on circuit depth (or equiva- lently, formula size) should be easier to prove than general circuit lower bounds. Nonetheless, lower bounds for circuits with depth restrictions are non-trivial and so we will place constraints on both size and depth in the next section.
3 Simultaneous Size and Depth Bounds
Circuit lower bounds, are very difficult to come by. In the 1970ās, Valiant put forward a challenge to find an explicit function which cannot be computed by circuits of size O(n) and depth O(log n). To date, this problem is open. In this section we will study similarly restricted classes of circuits.
Definition 5 NCk is the class of languages decided by uniform circuits of polynomial size and depth O(logk^ n). The class NC is defined by NC =
k NCk.
The following proposition follows from the theorem in the previous section:
Proposition 6 Non-uniform NC 1 = polynomial-sized formulae.
It is a fact, although we will not prove it here, that NC is also the class of languages with parallel algorithms that run in poly-logarithmic time on polynomially many processors, i.e. āefficiently par- allelizable problemsā. It is an open question whether NC = P, that is, whether every polynomial- time algorithm can be efficiently parallelized. If this is not the case, then P-complete problems are inherently sequential, as NC is closed under logspace reductions (see problem set 5).
Theorem 7 (Borodin)
1. NC 1 ā L
2. NL ā NC 2
