The complexity of parallel prefix problems on small domains

We establish non-trivial lower bounds for several prefix problems in the CRCW PRAM model. The chaining problem is, given a binary input, for each 1 in the input, to find the index of the nearest 1 to its left. Our main result is that for an input ofnbits, solving the chaining problem usingO(n) processors requires inverse-Ackerman time. This matches the previously known upper bound. We also 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 problem