Time Lower Bounds for CREW-PRAM Computation of Monotone Functions

Abstract

It is shown that the time to compute a monotone boolean function depending upon $n$ variables on a CREW-PRAM satisfies the lower bound $T = \Omega$(log $l$ + (log $n$)/$l$), where $l$ is the size of the largest prime implicant. It is also shown that the bound is existentially tight by constructing a family of monotone functions that can be computed in $T = O$(log $l$ + (log $n$)/$l$), even by an EREW-PRAM. The same results hold if $l$ is replaced by $L$, the size of the largest prime clause. An intermediate result of independent interest is that $S (n,l)$, the size of the largest minimal vertex cover minimized over all (reduced) hypergraphs of $n$ vertices and maximum hyperedge size $l$, satisfies the bounds $\Omega(n^{1/l}) \leq S (n,l) \leq O (ln^{1/l}).$