What are the shift rules for Boyer–Moore string search algorithm?

Question

I have been trying to understand shift rules in Boyer–Moore string search algorithm but haven\'t understood them. I read here on wikipedia but that is too complex !

Answer 1

In the Boyer-Moore algorithm, you start comparing pattern characters to text characters from the end of the pattern. If you find a mismatch, you have a configuration of the type

....xyzabc....      <-text
  ....uabc          <- pattern
      ^
    mismatch

Now the bad character shift means to shift the pattern so that the text character of the mismatch is aligned to the last occurrence of that character in the initial part of the pattern (pattern minus last pattern character), if there is such an occurrence, or one position before the pattern if the mismatched character doesn't appear in the initial part of the pattern at all.

That could be a shift to the left, if the situation is

     v
...xyzazc...
 ....uazc
 ..uazc

so that alone doesn't guarantee a progress.

The other shift, the good suffix shift, aligns the matched part of the text, m, with the rightmost occurrence of that character sequence in the pattern that is preceded by a different character (including none, if the matched suffix is also a prefix of the pattern) than the matched suffix m of the pattern - if there is such an occurrence.

So for example

           v
....abcdabceabcfabc...
    ...xabcfabcfabc
        ...xabcfabcfabc

would lead to a good suffix shift of four positions, since the matched part m = abcfabc occurs in the pattern four places left of its suffix-occurrence and is preceded by a different character there (x instead of f) than in the suffix position.

If there is no complete occurrence of the matched part in the pattern preceded by a different character than the suffix, the good suffix shift aligns a suffix of the matched part of the text with a prefix of the pattern, choosing maximal overlap, e.g.

      v
...robocab....
    abacab
        abacab

The good suffix shift always shifts the pattern to the right, so guarantees progress.

Then, on every mismatch the advances of the bad character shift and the good suffix shift are compared, and the greater is chosen. It is explained in greater depth by Christian Charras and Thierry Lecroq here, along with many other string searching algorithms.

---

For the example you mentioned in the comments,

SSIMPLE EXAMPLE
EXAMPLE
  ^

the matched suffix is MPLE, and the mismatched text character is I. So the bad character shift looks for the last occurrence of I in the initial part of the pattern. There is none, so that bad character shift would shift the pattern so that the mismatched I is one before the start of the pattern

SSIMPLE EXAMPLE
   EXAMPLE

and the good suffix shift looks for the rightmost occurrence of MPLE in the pattern not preceded by an A, or the longest suffix of MPLE that is a prefix of the pattern. There is no complete occurrence of the matched part in the pattern before the suffix, so the longest suffix of the matched part that is also a prefix of the pattern determines the good suffix shift. In this case, the two suffixes of the matched part that are prefixes of the pattern are the single-character string E, and the empty string. The longest is obviously the nonempty string, so the good suffix shift aligns the one-character suffix E in the matched part of the text with the one-character prefix of the pattern

SSIMPLE EXAMPLE
      EXAMPLE

The good suffix shift shifts the pattern farther right, so that is the chosen shift.

Then there is an immediate mismatch at the last pattern position, and then the bad character shift aligns the P in the text with the P in the pattern (and the good suffix shift need not be considered at all if the mismatch occurs at the last pattern character, since in that case, it would never produce a larger shift than the bad character shift).

Then we have the complete match.

In the variant with the pattern TXAMPLE, the good suffix shift finds that no non-empty suffix of the matched part is a prefix of the pattern (and there is no occurrence of the complete matched part in the pattern not preceded by A), so the good suffix shift aligns the empty suffix of the matched part of the text (the boundary between the E and the space) with the empty prefix of the pattern (the empty string preceding the T), resulting in

SSIMPLE EXAMPLE
       TXAMPLE

(then in the next step, the bad character shift aligns the two Ls, and the next mismatch in the step thereafter occurs at the initial T of the pattern).