Exact String Search

1
Exact String Search

• Lecture 6 September 20, 2005
• Algorithms in Biosequence Analysis
• Nathan Edwards - Fall, 2005

2
K-M-P preprocessing

• Definition
• For pattern P, position i, let spi(P) be the
  length of the longest proper suffix of P1i
  that matches a prefix of P.
• Definition
• For pattern P, position i, let spi(P) be the
  length of the longest proper suffix of P1i
  that matches a prefix of P and P(i1) ? P(spi1).

3
K-M-P shift rule

• For any alignment of P with T
• If the first mismatch (from l to r) is at
  position (i1) of P and position k of T, then
  shift P to the right so that P1spi aligns
  with T(k-spi)(k-1)
• i.e. shift P (i1) - (spi1) i - spi to
  the right so that spi1 aligns with k
• If an occurrence of P is found, shift P by n -
  spn

4
Speeding up the naïve algorithm

• 0 1
• 1234567890123
• Txabxyabxyabxz
• P abxyabxz

5
Speeding up the naïve algorithm

• 0 1
• 1234567890123
• Txabxyabxyabxz
• P abxyabxz

6
Speeding up the naïve algorithm

• 0 1
• 1234567890123
• Txabxyabxyabxz
• P abxyabxz
• abxyabxz

7
Speeding up the naïve algorithm

• 0 1
• 1234567890123
• Txabxyabxyabxz
• P abxyabxy

8
Speeding up the naïve algorithm

• 0 1
• 1234567890123
• Txabxyabxyabxz
• P abxyabxy
• abxyabxz

9
Speeding up the naïve algorithm

• 0 1
• 1234567890123
• Txabxyabxyabxz
• P abxyabxz

10
Speeding up the naïve algorithm

• 0 1
• 1234567890123
• Txabxyabxyabxz
• P abxyabxz
• abxyabxz

11
K-M-P outline

• Show that no occurrence of P is missed
• Show linear search time, given the spi values
• Show how to compute the spi values from the Zi
  values
• Knuth-Morris-Pratt psuedo-code
• Using failure functions
• Extentions / strong vs weak shift rule

12
K-M-P outline

• Knuth-Morris-Pratt psuedo-code
• Using failure functions
• Extentions / strong vs weak shift rule
• K-M-P as a DFA
• Original K-M-P pattern preprocessing