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A computational problem is said to be in the class $NC^K$, if the result $f(x)$ can be computed in time $O((\log |x|)^k )$ on a multi-processor computer, while the total number of operations remain polynomial in the input size $|x|$, i.e. number of bits in $x$. Show that counting the number of 1’s in a binary word $x \in \left\{0,1\right\}^{n}$ is an $NC^2$ problem. Hint: You may use a one-bit adder as a unit of counting.

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Your function is in fact in $\mathsf{NC}^1$.

It is well-known that threshold functions are in $\mathsf{NC}^1$ (see for example this question on cstheory). In particular, the function $\sum_i x_i = w$ is in $\mathsf{NC}^1$ for all $w$. By computing these functions for all $w$, you can easily compute your function in $\mathsf{NC}^1$.

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  • $\begingroup$ We have to use one-bit adder in counting how to do I am not getting but how to prove that it is NC^2 problem $\endgroup$ Oct 7 at 15:30
  • $\begingroup$ I'm sorry but I'm not willing to offer help of this kind. $\endgroup$ Oct 7 at 15:43

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