Fast Searching in Packed Strings

Abstract

Given strings P and Q the (exact) string matching problem is to find all positions of substrings in Q matching P. The classical Knuth-Morris-Pratt algorithm [SIAM J. Comput., 1977] solves the string matching problem in linear time which is optimal if we can only read one character at the time. However, most strings are stored in a computer in a packed representation with several characters in a single word, giving us the opportunity to read multiple characters simultaneously. In this paper we study the worst-case complexity of string matching on strings given in packed representation. Let m ≤ n be the lengths P and Q, respectively, and let σ denote the size of the alphabet. On a standard unit-cost word-RAM with logarithmic word size we present an algorithm using time

$$O\left(\frac{n}{\log_\sigma n} + m + {\mathrm{occ}}\right).$$

Here occ is the number of occurrences of P in Q. For m = o(n) this improves the O(n) bound of the Knuth-Morris-Pratt algorithm. Furthermore, if m = O(n/log _σ n) our algorithm is optimal since any algorithm must spend at least $\Omega(\frac{(n+m)\log \sigma}{\log n} + {\mathrm{occ}}) = \Omega(\frac{n}{\log_\sigma n} + {\mathrm{occ}})$ time to read the input and report all occurrences. The result is obtained by a novel automaton construction based on the Knuth-Morris-Pratt algorithm combined with a new compact representation of subautomata allowing an optimal tabulation-based simulation.