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String Matching Problem
Given a text stringTof length n and apattern stringPof length m, the exact stringmatching problem is to find all occurrences
ofPin T. Example: T=AGCTTGA P=GCT
Applications:
Searchingkeywords in a fileSearching engines (like Google and Openfind)
Database searching (GenBank)
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Problem/issue
Finding occurrence of a pattern (string)
P in String S and also finding theposition in S where the pattern match
occurs
What is pattern matching?
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Brute Force algorithm
The brute-force pattern matching algorithm comparesthe pattern Pwith the text Tfor each possible shift ofP
relative to T,
*until either a match is found, or
*all placements of the pattern have been tried
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Brute-force
Worst O(m*n)
Best O(n)
algorithmbrute-force:
input: an array of characters, T (the string to be analyzed) , length n
an array of characters, P (the pattern to be searched for), length m
for i := 0 to n-m do
for j := 0 to m-1 do
compare T[j] with P[i+j]ifnot equal, exit the inner loop
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Compare each character of P with S if
match continue else shift one position
String S
a
b
a
a
a
b
c
a
b
a
a
b
c
a
b
a
c
Pattern p
Example
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Step 1:compare p[1] with S[1]
a
b
c
a
b
a
a
b
c
a
b
a
c
a
b
a
a
Step 2: compare p[2] with S[2]
a
b
c
a
b
a
a
b
c
a
b
a
c
a
b
a
a
S
S
p
p
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Step 3: compare p[3] with S[3]
S a b c a b a a b c a b a c
a b a ap
Mismatch occurs here..
Since mismatch is detected, shift P one position to the Right andperform steps analogous to those from step 1 to step 3. At positionwhere mismatch is detected, shift P one position to the right andrepeat matching procedure.
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The Knuth-Morris-Pratt Algorithm
Knuth, Morris and Pratt proposed a lineartime algorithm for the string matchingproblem.
A matching time of O(n) is achieved byavoiding comparisons with elements of Sthat have previously been involved incomparison with some element of thepattern p to be matched. i.e.,backtracking on the string S never occurs
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Components of KMP algorithm
The prefix function,
The prefix function, for a pattern encapsulatesknowledge about how the pattern matches against shiftsof itself. This information can be used to avoid uselessshifts of the pattern p. In other words, this enablesavoiding backtracking on the string S.
The KMP Matcher
With string S, pattern p and prefix function asinputs, finds the occurrence of p in S and returns the
number of shifts of p after which occurrence is found.
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Knuth-Morris-Pratt algorithm-Algorithm
Compute-Prefix-Function(P)
1. m length[T]
2. [1] 0
3. k 0
4. forq2 tom5. do whilek> 0 and P[k+ 1] P[q]
6. dok[k] /*ifk= 0 orP[k+ 1]
= P[q],
7. ifP[k+ 1] = P[q] going out of thewhile-loop.*/
8. then kk+ 1
9. [q]k
10.return
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Knuth-Morris-Pratt algorithm-Algorithm
KMP-Matcher(T, P)
1. n length[T]
2. m length[P]
3. Compute-Prefix-Function(P)
4. q 05. fori1 ton
6. do whileq > 0 and P[q + 1] T[i]
7. doq [q]
8. ifP[q + 1] = T[i]9. then qq + 1
10. ifq = m
11. thenprint pattern occurs with shift im
12. q [q]
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Compute prefix function
P= ababababca, T= ababaababababca[1]= 0
k= 0
q = 2, P[k+ 1] = P[1] = a, P[q] = P[2] = b, P[k+ 1] P[q]
[q]k([2] 0)q = 3, P[k+ 1] = P[1] = a, P[q] = P[3] = a, P[k+ 1] = P[q]
k k+ 1, [q]k([3] 1)
k= 1
q = 4, P[k+ 1] = P[2] = b, P[q] = P[4] = b, P[k+ 1] = P[q]
k k+ 1, [q]k([4] 2)
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k= 2
q = 5, P[k+ 1] = P[3] = a, P[q] = P[5] = a, P[k+ 1] = P[q]
k k+ 1, [q]k([5] 3)k= 3
q = 6, P[k+ 1] = P[4] = b, P[q] = P[6] = b, P[k+ 1] = P[q]
k k+ 1, [q]k([6] 4)
k= 4q = 7, P[k+ 1] = P[5] = a, P[q] = P[7] = a, P[k+ 1] = P[q]
k k+ 1, [q]k([7] 5)
k= 5
q = 8, P[k+ 1] = P[6] = b, P[q] = P[8] = b, P[k+ 1] = P[q]
k k+ 1, [q]k([8] 6)
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k= 6
q = 9, P[k+ 1] = P[6] = b, P[q] = P[9] = c, P[k+ 1] P[q]k[k](k[6] = 4)
P[k+ 1] = P[5] = a, P[q] = P[9] = c, P[k+ 1] P[q]
k[k](k[4] = 2)
P[k+ 1] = P[3] = a, P[q] = P[9] = c, P[k+ 1] P[q]
k[k](k[2] = 0)
k= 0
q = 9, P[k+ 1] = P[1] = a, P[q] = P[9] = c, P[k+ 1] P[q]
[q]k([9] 0)
q = 10, P[k+ 1] = P[1] = a, P[q] = P[10] = a, P[k+ 1] = P[q]k k+ 1, [q]k([10] 1)
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After prefix computation, the table is shown below
P= ababababca
1 2 3 4 5 6 7 8 9 10
a b a b a b a b c a0 0 1 2 3 4 5 6 0 1
i
P[i][i]
a b a b a b a b c a
a b a b a b
a b a b
a b
a b c a
a b a b c a
a b a b a b c a
a b a b a b a b c a
P8
P6
P4
P2
P0
[8] = 6[6] = 4
[4] = 2
[2] = 0
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Another Example for KMP Algorithm
Phase 1
Phase 2
f(41)+1=f(3)+1=0+1=1
f(13-1)+1= 4+1=5matched
First finish the prefix
computation
Next, Search phase computation