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# 27.8. Rabin-Karp String Search Algorithm [Draft]¶

## 27.8.1. Rabin-Karp String Search Algorithm [Draft]¶

The Rabin-Karp algorithm is based on what could be imagined as a “perfect hash function for strings”. We will assume that our strings are drawn from an alphabet with $C$ possible characters. Denote the characters in string $S$ by $s_0, s_1, \ldots s_{n-1}$. Suppose that we have a mapping $c \rightarrow \hat{c}$ that associates with each character $c$ an integer $\hat{c}$ in the range $0 \ldots c - 1$. Then a “perfect hash function for strings” is:

$\widehat{s_0} \times C^{n-1} + \widehat{s_1} \times C^{n-2} + \ldots + \widehat{s_{n-2}} \times C + \widehat{s_{n-1}} \times C^0$

Suppose that we call this a string’s “magic number”. In effect it associates each string with a unique number in the base $C$ number system. However, nothing is perfect – these magic numbers for strings get very big very quickly. Hence the following sub-algorithm of Rabin-Karp to compute a string’s magic number (which is itself known as Horner’s polynomial evaluation algorithm) takes this into account by using the $mod$ operator to avoid an overflow condition.

Slideshow for Horner’s Method algorithm for computing Rabin-Karp “magic number” for a string

To check your understanding of this “magic number” computation try the following exercise in using Horner’s Method to compute a string’s “magic number” in a simple case

Because Horner’s Method cannot truly compute a magic number that is unique for every string, the Rabin-Karp algorithm must allow for two different strings having the same magic number. In effect, such a situation represents a “false positive” in which Rabin-Karp thinks it has found a match only to be disappointed. Watch Rabin-Karp in action in the following slideshow.

Finally try this exercise in tracing one step of the Rabin-Karp algorithm using the modified Horner’s algorithm to compute the “magic number” of a string.