Lower bounds for merging on the hypercube

We show non-trivial lower bounds for several prefix problems in the CRCW PRAM model. Our main result is an $\Omega(\alpha(n))$ lower bound for the chaining problem, matching the previously known upper bound. We give a reduction to show that the same lower bound applies to a parenthesis matching problem, again matching the previously known upper bound. We also give reductions to show that similar lower bounds hold for the prefix maxima and the range maxima problems