It is known that the size of monotone arithmetic $(+,\ast)$ circuits can be exponentially decreased by allowing just one division "at the very end," at the output gate. A natural question is: can the size of $(+,\ast)$ circuits be substantially reduced if we allow divisions "at the very beginning," that ... more >>>

We investigate the number of pairwise comparisons sufficient to sort $n$ elements chosen from a linearly ordered set. This number is shown to be $\log_2(n!) + o(n)$ thus improving over the previously known upper bounds of the form $\log_2(n!) + \Theta(n)$. The new bound is achieved by the proposed group ... more >>>

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, ... more >>>