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TR13-041 | 14th March 2013 13:56
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#### 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.