TR13-041 | 14th March 2013 13:56
On the complexity of parallel prefix circuits
Abstract:
It is shown that complexity of implementation of prefix sums of m variables (i.e. functions x_1 \cdot \ldots\cdot x_i, 1\le i \le m) by circuits of depth \lceil \log_2 m \rceil in the case m=2^n is exactly
3.5\cdot2^n - (8.5+3.5(n \bmod 2))2^{\lfloor n/2\rfloor} + n + 5.
As a consequence, for an arbitrary
m an upper bound
(3.5-o(1))m holds. In addition, an upper bound
\left(3\frac{3}{11}-o(1)\right)m for complexity of the minimal depth prefix circuit with respect to XOR operation is obtained. Some new bounds under different restrictions on the circuit depth are also established.
