Knuth-Morris-PrattString matching algorithm

Ivaylo Kenov

Telerik Corporationhttp:/telerikacademy.

com

Telerik Academy Student

Table of Contents

1. Background and idea

2. The “naive” approach

3. Basic definitions

4. Preprocessing

5. Search algorithm

6. Complexity

7. Additional information

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Background and ideaWhat is the problem?

Background and idea The problem of string matching. We have string text and pattern word.

Check if word occurs in text. If so, return the position where pattern occurs.

If not, return -1.

The “naive” approach

New to string searching

The naive approach (1) Very obvious solution – compare element by element.

O(m*n) complexity – not good!

Example:String Text

Pattern Word

The naive approach (2) Step 1: compare word[0] with text[0]

Step 2: compare word[1] with text[1]

Text

Word

Text

Word

The naive approach (3) Step 1: compare word[2] with text[2]

Mismatch found – shift word one index to the right and repeat!

Text

Word

Text

Word

The naive approach (4) A match will be found after three shifts to the right of the word!

Problem with the “naive” approach – two much comparisons over the same character!

TextWord

The “naive” approach

Live demo

Knuth-Morris-PrattWithout repeating!

Knuth-Morris-Pratt Linear time algorithm for string matching.

O(n) complexity. Backtracking never occurs. Already visited characters are not repeated!

Useful with binary data and small-alphabet strings.

Basic definitionsEasy theory!

Basic definitions (1) Prefix – a substring with which our string starts. Example: “abcdef” starts with

“abc”.

Suffix – a substring with which our string ends. Example: “abcdef” ends with

“def”.

Proper prefix and proper suffix – if the length of the substring is less than the length of the string.

Basic definitions (2) Border - if a substring is proper prefix and proper suffix at the same time. Example: “ab” is border of

“abcab”.

Width of border – length of the border.

The empty string “” is proper prefix, proper suffix and border at the same time of any string!

Basic definitions (3) How much the algorithm shifts the pattern?

The shift distance is determined by the widest border of the matching prefix of word.

Distance = length of the matching prefix – length of the widest border.

PreprocessingBuilding every border!

Preprocessing (1) If a, b are borders of text and length of a < length of b, then a is border of b.

A border r of x can be extended by a, if ra is border of xa.

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Preprocessing (2) We build an array table, which contains information about border widths.

When preprocessing a value, we already know the previous ones and use the extending of the borders for checking.

Border can be extended if tableb[i] = tablei.

If not next border to check is table[table[i]].

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Preprocessing (3)

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void FailFunction(string word) { int index = 0; int borderWidth = -1; failureTable[index] = borderWidth; while (index < word.Length) { while (borderWidth >= 0 && word[index] != word[borderWidth]) { borderWidth = failureTable[borderWidth]; } index++; borderWidth++; failureTable[index] = borderWidth; } }

Algorithm for building the table:

Preprocessing (4)

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Example for table: For pattern ”ababaa” the widths of

the borders in array b have the following values. For instance we have table[5] = 3, since the prefix “ababa” of length 5 has a border of width 3.

Note: zero element is always -1.

PreprocessingLive demo

Search algorithmFinding the word!

Search algorithm (1)

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static int KMPSearch(string text, string word, int position) { int index = 0; int borderWidth = 0; int currentPosition = 1;

while (index < text.Length) { while (borderWidth >= 0 && text[index] != word[borderWidth]) { borderWidth = failureTable[borderWidth]; }

index++; borderWidth++;

Continues…

The search algorithm is similar:

Search algorithm (2)

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Continues…if (borderWidth == word.Length) { if (position == currentPosition) { return (index - borderWidth); } else { currentPosition++; } borderWidth = failureTable[borderWidth]; } }

return -1; }

Algorithm continues:

Search algorithm (3)

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How it works:

Example:

Search algorithmLive demo

ComplexityLinear time algorithm!

Complexity

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The table building algorithm is O(m) where m is the length of the pattern.

The search algorithm is O(n) where n is the length of the text.

Overall complexity therefore is O(n).

Additional information Wikipedia: http://en.wikipedia.org/wiki/Knuth%E2%80%93Morris%E2%80%93Pratt_algorithm#Worked_example_of_the_table-building_algorithm

Knuth-Morris-Pratt explained: http://www.inf.fh-flensburg.de/lang/algorithmen/pattern/kmpen.htm

Examples and concept: http://wcipeg.com/wiki/Knuth%E2%80%93Morris%E2%80%93Pratt_algorithm

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