Knuth-Morris-Pratt

1
Knuth-Morris-Pratt
Reference Chapter 19, Algorithms in C by R.
Sedgewick.Addison Wesley, 1990.
2
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
a
a
b
a
a
a
a
a
a
a
a
b
3
4
5
6
0
1
2
accept state
3
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
a
a
b
a
a
a
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
4
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
a
b
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
5
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
a
b
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
6
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
a
b
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
7
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
a
b
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
8
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
a
b
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
9
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
a
b
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
10
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
a
b
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
11
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
a
b
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
12
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
a
b
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
13
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
a
b
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
14
Knuth-Morris-Pratt

• KMP algorithm.
• Use knowledge of how search pattern repeats
  itself.
• Build FSA from pattern.
• Run FSA on text.

Search Pattern
Search Text
a
a
b
a
a
a
a
a
b
a
a
a
b
a
a
a
b
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
15
FSA Representation

• FSA used in KMP has special property.
• Upon character match, go forward one state.
• Only need to keep track of where to go upon
  character mismatch.
• go to state nextj if character mismatches in
  state j

0
1
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5
a
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Search Pattern
b
0
0
3
0
0
3
a
a
b
a
a
a
next
0
0
2
0
0
3
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
accept state
b
b
16
KMP Algorithm

• Two key differences from brute force.
• Text pointer i never backs up.
• Need to precompute next table.

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FSA Construction for KMP

• FSA construction for KMP.
• FSA builds itself!
• Example. Building FSA for aabaaabb.
• State 6. p0..5 aabaaa
• assume you know state for p1..5 abaaa X 2
• if next char is b (match) go forward 6 1 7
• if next char is a (mismatch) go to state for
  abaaaa X 'a' 2
• update X to state for p1..6 abaaab X 'b'
  3

b
a
b
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a
a
a
a
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3
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0
1
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b
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b
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FSA Construction for KMP

• FSA construction for KMP.
• FSA builds itself!
• Example. Building FSA for aabaaabb.

b
a
b
a
a
a
a
a
b
b
3
4
5
6
0
1
2
7
b
b
b
a
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FSA Construction for KMP

• FSA construction for KMP.
• FSA builds itself!
• Example. Building FSA for aabaaabb.
• State 7. p0..6 aabaaab
• assume you know state for p1..6 abaaab X 3
  
• if next char is b (match) go forward 7 1 8
• next char is a (mismatch) go to state for
  abaaaba X 'a' 4
• update X to state for p1..7 abaaabb X 'b'
  0

b
a
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a
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a
a
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b
3
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6
0
1
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7
b
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b
a
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FSA Construction for KMP

• FSA construction for KMP.
• FSA builds itself!
• Example. Building FSA for aabaaabb.

b
a
a
b
a
a
a
a
a
b
b
b
3
4
5
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0
1
2
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8
b
b
b
a
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FSA Construction for KMP

• FSA construction for KMP.
• FSA builds itself!
• Crucial insight.
• To compute transitions for state n of FSA,
  suffices to have
• FSA for states 0 to n-1
• state X that FSA ends up in with input p1..n-1
• To compute state X' that FSA ends up in with
  input p1..n, it suffices to have
• FSA for states 0 to n-1
• state X that FSA ends up in with input p1..n-1

22
FSA Construction for KMP
X
Search Pattern
pattern1..j
j
a
a
b
a
a
a
b
b
a
b
b
a
0
1
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FSA Construction for KMP
Search Pattern
X
pattern1..j
j
next
a
a
b
a
a
a
b
b
0
0
0
0
a
1
b
0
b
a
0
1
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FSA Construction for KMP
Search Pattern
X
pattern1..j
j
next
a
a
b
a
a
a
b
b
0
0
0
a
1
1
0
0
1
a
1
2
b
0
0
b
a
a
0
1
2
b
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FSA Construction for KMP
Search Pattern
X
pattern1..j
j
next
a
a
b
a
a
a
b
b
0
0
0
a
1
1
0
a
b
0
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0
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a
1
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2
b
0
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3
a
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a
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0
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b
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FSA Construction for KMP
Search Pattern
X
pattern1..j
j
next
a
a
b
a
a
a
b
b
0
0
0
a
1
1
0
a
b
0
2
2
0
1
2
3
a
b
a
1
3
0
a
1
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2
4
b
0
0
3
0
a
b
a
a
a
b
3
4
0
1
2
b
b
27
FSA Construction for KMP
Search Pattern
X
pattern1..j
j
next
a
a
b
a
a
a
b
b
0
0
0
a
1
1
0
a
b
0
2
2
0
1
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3
4
a
b
a
1
3
0
a
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2
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5
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2
4
0
b
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3
0
0
b
a
b
a
a
a
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3
4
5
0
1
2
b
b
28
FSA Construction for KMP
Search Pattern
X
pattern1..j
j
next
a
a
b
a
a
a
b
b
0
0
0
a
1
1
0
a
b
0
2
2
0
1
2
3
4
5
a
b
a
1
3
0
a
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2
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a
b
a
a
2
4
0
b
0
0
3
0
0
3
a
b
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3
b
a
b
a
a
a
a
a
b
3
4
5
6
0
1
2
b
b
b
29
FSA Construction for KMP
Search Pattern
X
pattern1..j
j
next
a
a
b
a
a
a
b
b
0
0
0
a
1
1
0
a
b
0
2
2
0
1
2
3
4
5
6
a
b
a
1
3
0
a
1
2
2
4
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6
2
a
b
a
a
2
4
0
b
0
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3
0
0
3
7
a
b
a
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3
a
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3
6
2
b
a
b
a
a
a
a
a
b
b
3
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0
1
2
7
b
b
b
a
30
FSA Construction for KMP
X
Search Pattern
pattern1..j
j
next
0
0
a
a
b
a
a
a
b
b
0
a
1
1
0
a
b
0
2
2
0
1
2
3
4
5
6
7
a
b
a
1
3
0
a
1
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2
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2
4
a
b
a
a
2
4
0
b
0
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3
7
8
a
b
a
a
a
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3
a
b
a
a
a
b
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2
a
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a
a
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0
7
4
31
FSA Construction for KMP

• Code for FSA construction in KMP algorithm.