KMP by Rajeev K Krishna

8/14/2019 KMP by Rajeev K Krishna

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Introduction Knuth-Morris-Pratt algorithm Final remarks

The Knuth-Morris-Pratt algorithm

Kshitij Bansal

Chennai Mathematical Institute

Undergraduate Student

August 25, 2008
http://find/http://goback/


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IntroductionThe ProblemNaive algorithm

Knuth-Morris-Pratt algorithm

Can we do better?Improved algorithmComputing f

Final remarks

ConclusionThink/read aboutReferencesThankyou


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The Problem

Given a piece of text, find if a smaller string occurs in it.


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The Problem

Given a piece of text, find if a smaller string occurs in it.

Let T[1..n] be an array which holds the text. Call the smallerstring to be searched pattern: p[1..m].


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Naive Algorithm

Naive-String-Matcher(T[1..n],P[1..m])


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Naive Algorithm


1 for i from 0 to n m

I d i K h M i P l i h Fi l k


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Naive Algorithm


1 for i from 0 to n m2 if P[1 . . . m] = T[i + 1 . . . i + m]

I t d ti K th M i P tt l ith Fi l k


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Naive Algorithm



3 print Pattern found starting at position , i

Introduction Knuth Morris Pratt algorithm Final remarks


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Naive Algorithm



3 print Pattern found starting at position , i

Time complexity: O(mn). Space complexity: O(1).



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Final remarks




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Can we do better?

Text: abaabaabac

Pattern: abaabac



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Can we do better?

Text: abaabaabac

Pattern: abaabac

The naive algorithm when trying to match the seventh character ofthe text with the pattern fails. It discards all information abouttext read from first seven characters and starts afresh. Can we

somehow use this information and improve?



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g

Improved algorithm

Improved-String-Matcher(T[1..n],P[1..m],f[1..m])

1 q = 0 (number of characters matched)



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Improved algorithm



2 for i from 1 to n:



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Improved algorithm



2 for i from 1 to n:3 if P[q+ 1] = T[i]:

4 q = q+ 1



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Improved algorithm



2 for i from 1 to n:3 if P[q+ 1] = T[i]:

4 q = q+ 1

5 else if q > 0

6 q = f[q]



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Improved algorithm



2for

ifrom

1to

n:3 if P[q+ 1] = T[i]:

4 q = q+ 1

5 else if q > 0

6 q = f[q]7 goto line 3



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Improved algorithm



2for

ifrom

1to

n:3 if P[q+ 1] = T[i]:

4 q = q+ 1

5 else if q > 0


Does this really help? What happens when text is abaabadaba . . .?



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Improved algorithm



2 for i from 1 to n:

3 if P[q+ 1] = T[i]:

4 q = q+ 1

5 else if q > 0


Does this really help? What happens when text is abaabadaba . . .?Note that q can increase by atmost 1 at each step.



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Computing f

f[i] denotes the longest proper suffix of P[1 . . . i] which is aprefix of P[1 . . . n]



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Computing f

f[i] denotes the longest proper suffix of P[1 . . . i] which is aprefix of P[1 . . . n]

Just match the pattern with itself.



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Final remarks




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Conclusion

KMP algorithm Time complexity: O(n + m)



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Conclusion

KMP algorithm Time complexity: O(n + m)

Space complexity: O(m)



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Think/read about

Number of distinct substrings in a string



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Think/read about

Number of distinct substrings in a string Multiple patterns and single text



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Think/read about

Number of distinct substrings in a string Multiple patterns and single text

Matching regular expresssions



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References

Thomas H. Cormen; Charles E. Leiserson, Ronald L. Rivest,Clifford Stein (2001). Section 32.4: The Knuth-Morris-Prattalgorithm, Introduction to Algorithms, Second edition, MIT

Press and McGraw-Hill, 923931. ISBN 978-0-262-03293-3. The original paper: Donald Knuth; James H. Morris, Jr,

Vaughan Pratt (1977). Fast pattern matching in strings.SIAM Journal on Computing 6 (2): 323350.doi:10.1137/0206024.

An explanation of the algorithm and sample C++ code byDavid Eppstein:http://www.ics.uci.edu/ eppstein/161/960227.html



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Thank you

Questions?



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The End

KMP by Rajeev K Krishna

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