String searching problemsdavid/courses/advalgorithms/strings.pdf · Another e cient exact string...

String searching problems

String searching: The problem - given the following data:

a SOURCE (or TEXT ) string s = s1s2s3 . . . sn, and

a PATTERN string p = p1p2p3 . . . pm.

Ask: does p occur as a substring of s and, if so, where? Can seekfirst occurrence, or all occurrences, etc.

For example, s could be the string “abracadabra” and p the string“cad”.

Applications: Used in text editors, file operations eg grep, websearch engines etc.Question: The problem is a special case of more general ‘patternmatching’ problems - to what extent do string search algorithmsgeneralise, or pattern matching algorithms specialise?

String searching: Basic algorithm

A naıve algorithm:

Look for a match starting at beginning of source text s.

Compare characters of pattern p with those of s until either

1 find two characters that differ – this means that no match ispresent with the current starting point

2 reach end of p – this means we’ve found a match

3 reach end of s – this means that no match is present.

If (1) then slide the pattern along one character and start searchagain at the beginning of the pattern.

Repeat until reach end of p (success) or end of s (failure).

Naive algorithm - example and complexity

Naıve string searching algorithm

If the pattern has length m and source text has length n, in worstcase need

Length.of .pattern×Number .of .possible.start.points = m×(n−m+1)

character comparisons (i.e. equality tests).

For example:

Pattern aaaabText aaaaaaaaaaaaab

takes 5× (14− 5 + 1) = 50 comparisons

Since, in general, n is much larger than m, algorithm is O(m × n).

Efficient exact string matching

The Knuth-Morris-Pratt algorithm (KMP)

The naıve algorithm will always find a match if one exists but oftendoes much more work than is necessary.

0123456Pattern: abcabcdText: abcabcabcd....

0123456789

Match fails at position 6.

The naıve algorithm shifts the pattern along by one and startschecking again at start of pattern. It throws away knowledge ofthe source text gained by checking and knowing that we havematched so far. We know that the text matches the first 6characters of the pattern, so, from the pattern alone, we knowwhat the text is and that moving by 1 will not match.

Knuth-Morris-Pratt: Reference

————–

This is the basis for the Knuth-Morris-Pratt algorithm:Donald E. Knuth, James H. Morris and Vaughan R. Pratt, FastPattern Matching in Strings, SIAM Journal on Computing, 6(2):323–350.

————–

Knuth-Morris-Pratt: Development I

In fact, shifting the pattern by 2 cannot match, but by 3, it maypossibly match:

0123456Pattern: abcabcdText: abcabcabcd.....

0123456789

This works because, at the point of the match failure, the string‘abc’

is both a prefix of the pattern (abcabcd)

and a proper suffix of the pattern up to the mismatch (abc abcd).(by ‘proper’ we mean it is not the whole substring).

Knuth-Morris-Pratt: Development II

In general, for each k with 0 < k < pattern length, define fail(k)to be

Largest r < k, such that p0 . . . pr−1 matchespk−r . . . pk−1.

If we define fail(0) to be −1, we have (for the above example):

i 0 1 2 3 4 5 6pi a b c a b c d

fail(i) -1 0 0 0 1 2 3

If a match failure occurs at character k , we know that previousfail(k) characters already match.

Note that matched characters in the text are visited only oncebecause we restart the checking at the fail point in the text. Thealgorithm has complexity O(m + n). Why?

Comment

Note: In this example, a failure at say character index 5 – thesecond ‘c’ – means that although the first ‘ab’ aligns with thesecond ‘ab’, we know that this alignment too must fail as the nexttext character is not a ‘c’. This improvement is not usuallyincorporated into the algorithm.

Knuth-Morris-Pratt: Search program

This program finds the first occurrence of the pattern in thetext:

private int[] failure;private int matchPoint;public boolean *match*() {int j = 0;if (text.length() == 0) return false;for (int i = 0; i < text.length(); i++) {

while (j > 0 && pattern.charAt(j) != text.charAt(i)){ j = failure[j - 1]; }if (pattern.charAt(j) == text.charAt(i)) { j++; }if (j == pattern.length()){ matchPoint = i - pattern.length() + 1;return true; } }

return false; }

Knuth-Morris-Pratt: Pattern pre-processing program

This program computes the failure function using a boot-strappingprocess, where the pattern is matched against itself.

private void *computeFailure*() {int j = 0;for (int i = 1; i < pattern.length(); i++) {

while (j > 0 &&pattern.charAt(j)!= pattern.charAt(i))

{ j = failure[j - 1]; }if (pattern.charAt(j) == pattern.charAt(i)) {j++;

}failure[i] = j;

}}

Knuth-Morris-Pratt: Conclusions

KMP is fast exactly where the naıve algorithm is slow – thatis, when the pattern and text contain repeated patterns ofcharacters.

KMP is particularly good when the alphabet is small, forexample bit patterns.

There is another algorithm which outperforms them both . . .

Another efficient exact string search

The Boyer-Moore Algorithm

This algorithm (R.M. Boyer and J.S. Moore 1977) uses a changeof approach together with two techniques to improve the amountby which the pattern is shifted whenever a match fails.

Change of approach:

Try to match the pattern from right to left, rather than left to rightStill move the pattern across the source text from left to right.

Boyer-Moore: Example

For example, if the pattern is ‘wish’ and the source text is ‘dish offruit’.

dish of fruit|

wish

We would successfully match ’h’,’s’ and ’i’ before failing on ’d’ and‘w’.

At first sight doesn’t look like a great idea!

Boyer-Moore: Development I

Idea 1.

If a match fails, move the pattern along so that, if possible, amatch is made with the source text character we are looking at.

This is made clearer with an example. Consider the source text:here is a piece of text which we wish to search,and the pattern:wish:

here is a piece of text which we wish to search|

wish

The ‘h’ fails to match against ‘e’, and no ‘e’ appears in thepattern, so no match can contain the ‘e’.

So can move pattern along past the ‘e’, i.e. 4 places (the width ofthe pattern).

We are now at the position:

here is a piece of text which we wish to search|

wish

Again match fails, this time against the space character. Since nospace in the pattern can shift along 4 again

here is a piece of text which we wish to search|

wish

Match fails again, but this time against an ‘i’, so move patternalong so that ‘i’ in the source text matches rightmost ‘i’ in thepattern. (Why rightmost?)

We are now at position:

here is a piece of text which we wish to search|

wish

There is no ‘c’ in pattern, so shift 4. A couple more moves like thisgive us:

here is a piece of text which we wish to search|

wish

The ‘h’ matches, so try the next character down the pattern (‘s’).This fails, so move along to match the ‘w’ in the text against therightmost ‘w’ of the pattern

here is a piece of text which we wish to search|

wish

etc etc...

The complete search is shown below:

here is a piece of text which we wish to search| | | | | | || | | ||

wish | | | | | || | | ||wish | | | | || | | ||

wish | | | || | | ||wish | | || | | ||

wish | || | | ||wish || | | ||

wish | | ||wish | ||

wish ||wish|wish

The match start occurs at the 34th character, but we have onlyhad to make 15 comparisons. The previous methods would havemade at least 37 attempts at matching.

The B-M approach means that many text characters are notlooked at all. In general, the longer the pattern, the fewer thecomparisons.

Given the text character on which match fails, we need to knowhow far can shift pattern

If the character isn’t in the pattern, then can shift the widthof the pattern

If the character ch is in the pattern, then can shift so thatrightmost occurrence of ch matches the occurrence of ch inthe text.

Set up an array containing these values, for each character in thecharacter set being used.

Boyer-Moore: Development II

Idea 2: The second technique used by B-M is essentially anadaptation of the fail array used in KMP, adapted to the right toleft pattern search.

Suppose we have a pattern batsandcats:

.......dats.......|

batsandcats

Mismatch occurs on text character ‘d’. The first heuristic wouldslide the pattern along until next ‘d’ in pattern matched text, i.e. 1character:

.......dats.......|

batsandcats

But we know that characters to right of current position are ‘ats’.

If the string ‘ats’ does not occur again in the pattern, can slide thepattern past it, otherwise we slide pattern along so that ‘ats’ intext matches next occurrence of ‘ats’ in the pattern

.......dats.......|

batsandcats

Boyer-Moore: Definition

In general, define MatchJump[k] to be the amount to incrementthe text position to begin the next pattern scan after a mismatchat character k of the pattern.

If m is the length of the pattern p, then, for k < m, let r belargest index so that:

pr . . . pr+m−k−1 matches pk+1 . . . pm

and pr−1 6= pk .

Define MatchJump[k] = m − r + 1.

If we can’t match the whole suffix pk+1 . . . pm, look for a q suchthat suffix of length q matches, then take

MatchJump[k] = m − k + m − q.

The Boyer-Moore string searching algorithm

The Boyer-Moore algorithm uses both these approaches and movesthe pattern along as much as it can: computing both shifts andtaking the maximum.

This combination produces a dramatic improvement over the naıvealgorithm and KMP, particularly for long patterns and text with alarge alphabet.

Applications: The Boyer-Moore algorithm is the algorithm ofchoice for searching in some text editors (e.g. in the emacs editor).

Hashing techniques for exact and approximate stringsearching

A quite different approach to string searching uses hashingtechniques. See accompanying paper ‘Implementation of substringtest’ by M.C. Harrison (C.A.C.M. 14:12. 1971).

See also course unit website for additional material, goodalgorithm sites, papers and recommended books.