Pattern Matching

1
Pattern Matching
2
Outline

• Problem description
• Brute Force algorithm
• Knuth-Morris-Pratt algorithm
• Boyer-Moore algorithm

3
Problem description

• What is pattern matching?
• Pattern quchjfng
• Text
• kadogqlxwwenkpjkmolpevnmqcsjakoheorplkzb
• zfjliar Iacwzjlvkxromavcdquchjfnganqxharnhkw
• eweg tkrmwm fjduwhqvhwpfg pymyajsaywf aas
• bwxwcmrncmslgeqztyoygdbnxjqoctslbrvngoyxe

4
Problem description

• Given two strings P (pattern) and T (text), find
  the first occurrence of P in T.
• Input two strings P T
• Output The starting index of the occurrence of P
  in T or Null .
• Applications
• Text editors
• Search engines
• Biological research

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Strings 1/2

• A string is a sequence of data elements, such as
  numbers or characters.
• An alphabet S is a set of basic symbols for a
  category of strings
• Examples of alphabet
• Character A, BZ, a, b z
• Number 0, 1, 29
• DNA A, C, G, T

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Strings 2/2

• Let P be a string of size m
• A substring Pi .. j is a string containing
  characters of string P with ranks between i and
  j. P P0 .. m-1.
• A prefix of P is a substring of the type P0 ..
  i
• A suffix of P is a substring of the type Pi ..m
  - 1

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Brute Force Algorithm 1/2
T
P TTAGCT TTATTAGTTAGC
P
7
8
2
3
Pattern size m, Text size n Time complexity
O(m(n-m)) Example of worst cast P
AAAAG T AAAAAAAAAAAAAAAAG
4
5
6
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Brute Force Algorithm 2/2
Algorithm BruteForceMatch(P,T) Input pattern P
of size m and text T of size n Output starting
index of the first substring of T equal to P or
-1 if no such substring exists begin for i ? 0
to n - m do j ? 0 while j lt m and Pj Ti
j do j ? j 1 if j m then return
i /match at i/ return -1 /no match
anywhere/ end
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How to speed up the Pattern Matching.

• Shift P by more than one position when mismatch
  occurs
• Never miss a occurrence of P in T
• Preprocessing approach
• Preprocessing P
• Knuth-Morris-Pratt Algorithm
• Boyer-Moore Algorithm
• Preprocessing T
• Suffix trees

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The Knuth-Morris-Pratt algorithm

• Best known
• For linear time for exact matching
• Preprocessing pattern P
• Compares from Left to Right

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The KMP Ideas

• Shift more than one position
• Reduce comparison

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The KMP Algorithm - Motivation

• When a mismatch occurs, what is the most we can
  shift the pattern?
• Answer the largest prefix of P0..j-1 that is a
  suffix of P1..j-1

T
--
--
--
--
--
P
j
No need to repeat these comparisons
Resume comparing here
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KMP Failure Function

• Knuth-Morris-Pratts algorithm preprocesses the
  pattern to find matches of prefixes of the
  pattern with the pattern itself
• The failure function F(j) is defined as the size
  of the largest prefix of P0..j that is also a
  suffix of P1..j

P
P
0
0
1
0
1
2
0
0
1
1
2
3
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The KMP Algorithm
Algorithm KMPMatch(T, P) begin F ?
failureFunction(P) i ? 0 j ? 0 while i lt n
do if Ti Pj then if j m - 1
then return i - j / match / else i ?
i 1 j ? j 1 else if j gt 0 then j ?
Fj - 1 else i ? i 1 return -1 / no
match / end
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Example
T
T
C
C
C
A
A
P
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The KMP Algorithm

• The failure function can be represented by an
  array and can be computed in O(m) time
• At each iteration of the while-loop, either
• i increases by one, or
• the shift amount i - j increases by at least one
  (observe that F(j - 1) lt j)
• Hence, there are no more than 2n iterations of
  the while-loop
• Thus, KMPs algorithm runs in optimal time O(m
  n)

Algorithm KMPMatch(T, P) begin F ?
failureFunction(P) i ? 0 j ? 0 while i lt n
do if Ti Pj then if j m - 1
then return i - j / match / else i ?
i 1 j ? j 1 else if j gt 0 then j ?
Fj - 1 else i ? i 1 return -1 / no
match / end
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Computing the Failure Function

• The failure function can be represented by an
  array and can be computed in O(m) time
• The construction is similar to the KMP algorithm
  itself
• At each iteration of the while-loop, either
• i increases by one, or
• the shift amount i - j increases by at least one
  (observe that F(j - 1) lt j)
• Hence, there are no more than 2m iterations of
  the while-loop

Algorithm failureFunction(P) begin F0 ? 0 i ?
1 j ? 0 while i lt m do if Pi Pj then /
we have matched j 1 chars / Fi ? j
1 i ? i 1 j ? j 1 else if j gt 0
then / use failure function to shift P / j ?
Fj - 1 else Fi ? 0 / no match / i ? i
1 end
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Boyer-Moores Algorithm

• The Boyer-Moores pattern matching algorithm is
  based on two principles
• Right to Left Scan
• Bad character shift
• Reference
• http//www.cs.utexas.edu/users/moore/best-ideas/st
  ring-searching

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Right to left Scan Rule
T
T
T
T
T
A
P
14
4
15
5
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Bad Character Shift Rule 1/3

• Shift more than one position at a time.
• When a mismatch occurs at Ti c
• If P contains c shift P to align the last
  occurrence of c in P with Ti

T
--
--
--
--
--
P
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Bad Character Shift Rule 2/3

• When a mismatch occurs at Ti c
• If P contains c shift P to align the last
  occurrence of c in P with Ti
• Else if P doesnt contains c shift P to
  align P0 with Ti 1

T
--
--
--
--
--
P
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Bad Character Shift Rule 3/3

• When a mismatch occurs at Ti c
• If P contains c shift P to align the last
  occurrence of c in P with Ti
• Else if P doesnt contains c shift P to
  align P0 with Ti 1
• What if we have already crossed the last
  occurrence?

T
--
--
--
--
--
P
shift one char only
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Example of BM Algorithm
T
T
T
T
T
A
P
7
4
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Last-Occurrence Function

• Preprocess the pattern P and the alphabet S to
  build the last-occurrence function L mapping S to
  integers, where L(c) is defined as L(c)
• Example
• S a, b, c, d
• P abacab
• The last-occurrence function can be computed in
  time O(m s), where m is the size of P and s is
  the size of S

4
5
3
-1
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The Boyer-Moore Algorithm
Case 1 j lt l
Algorithm BoyerMooreMatch(T, P, S ) L ?
lastOccurenceFunction(P, S ) i ? m - 1 j ? m -
1 repeat if Ti Pj then if j 0
then return i / match at i / else i ?
i - 1 j ? j - 1 else / character-jump
/ until i gt n - 1 return -1 / no
match /
i
l ? LTi i ? i m min(j, 1 l) j ? m - 1
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Time complexity Analysis

• Boyer-Moores algorithm is significantly faster
  than the Brute-Force Algorithm on English text
• Pattern size m, Text size n ,Number of
  alphabet s ?Time complexity O(nms)
• Example of worst caseT aaaaaaaa..aP
  caaaaa

T
a
a
a
a
a
a
a
a
a
P
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Extended Boyer-Moore algorithm

• Extended Bad character shift rule
• then shift P to the right so that the closest c
  to the left of position i in P is below the
  mismatched c in T.
• The original Boyer-Moore algorithm use the
  simpler bad character rule

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Extended Boyer-Moore algorithm

• Extended Bad character shift rule

T
--
--
--
--
--
R(E)0
P
The position of E is 0,3,4 Find the top number lt j