Directed vs. Undirected Monotone Contact Networks for Threshold Functions

we consider the problem of Computing threshold functions wing directed and undirected monotone contact networks. Our main results are the following. First, we show that there exist directed monotone contact networks that compute T,“, 2 5 k 5 n 1, of size O(k(n k + 2)log(n k + 2)). This bound is almost optimal for small thresholds, since there exists an R(knlog(n/(k 1))) lower bound. OUT networks are described explicitly; the previously best upper bound known, obtained from the undirected networks of Dubiner and Zwick, w e d non-constructive arguments and gave directed networks of size O(k3.”nlog n). Second, we show a lower bound of R(n1og log log n) on the size of undirected monotone contact networks computing TZ-l, improving the 2(n 1) lower bound of Markov. Combined with OUT upper bound result, this shows that directed monotone contact networks compute some threshold functions more easily than undirected networks.