A Lower Bound for Parallel String Matching

This paper presents an $\Omega (\log \log m)$ lower bound on the number of rounds necessary for finding occurrences of a pattern string $P[1..m]$ in a text string $T[1..2m]$ in parallel using m comparisons in each round. The bound is within a constant factor of the fastest algorithm for this problem [D. Breslauer and Z. Galil, SIAM J. Comput.,19 (1990), pp. 1051–1058] and also holds for an m-processor CRCW-PRAM in the case of a general alphabet. Consequently, the paper derives the parallel complexity of the string matching problem using p processors for general alphabets, which is • $\Theta ( \frac{m}{p} )$ if $p \leq \frac{m}{{\log \log m}}$, • $\Theta (\log \log m)$ if $\frac{m}{{\log \log m}} \leq p \leq m$, • $\Theta (\log \log _{2p/m} p)$ if $m \leq p \leq m^2 $, • $\Theta (1)$ if $p \geq m^2 $, or in short $\Theta \lceil \frac{m}{p} \rceil + \log \log _{\lceil 1 + p/m \rceil } 2p)$.