Chapter 3 String Matching

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Chapter 3String Matching

• 3.1 Basic Terminologies of Strings
• 3.2 The KMP Algorithm
• 3.3 The Boyer-Moore Algorithm
• 3.4 Suffix Trees and Suffix Arrays
• 3.5 Approximate String Matching

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3.1 Basic Terminologies of String

• Sa alphabet set
• Sithe ith character
• Si,jSiSi1Sj
• Sa substring of string
• perfixSS1,s
• suffixSSs-s1,s

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It take O(nm) time
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The KMP Algorithm

• Case1
• The first mismatch occurs at P4 and T4
• slide the window all the way to T4 and Match P1
  with T4

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The KMP Algorithm

• Case2
• The first mismatch occurs at P7 and T7
• We can not slide the window all the way to match
  P1 with T7
• slide the window to match P1 with T6.

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The KMP Algorithm

• Case3
• The first mismatch occurs at P8 and T8
• go back to P7 and find out that P6,7 equal to
  the prefix P1,2
• We can slide the window to align P1 with T6 and
  P3 with T8.

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The KMP Algorithm
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The KMP Algorithm
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The KMP Algorithm

• The KMP algorithm consists of two phases.
• First phase computes the prefix function for the
  pattern P .
• Second phase searches the pattern.

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The KMP Algorithm
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The KMP Algorithm
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The Boyer-Moore Algorithm

• This algorithm compares the pattern with the
  substring within a sliding window in the
  right-to-left order.
• Assume that the first mismatch occurs when
  comparing Tsj-1 whit Pj

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The Boyer-Moore Algorithm

• Bad character rule
• Align Tsj-1 with Pj , whrer j is the rightmost
  position of Tsj-1 in P.
• Only one character is used.

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The Boyer-Moore Algorithm

• Good Suffix Rule 1
• ?Align Tsj-1 with Pj-mj
• ?j is the largest position such that Pj1,m is a
  suffix of P1,j
• ? Pj-mjltgtPj

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The Boyer-Moore Algorithm

• Good Suffix Rule 2
• ?Align Tsm-j with P1
• ?j is the largest position such that P1,j is a
  suffix of Pj1,m

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The Boyer-Moore Algorithm
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The Boyer-Moore Algorithm

• Let us consider g1(7). Note that g1(7)9.This
  mean that P8,12CATCA must be equal to P5,9 and
  P7ltgtP4
• Consider g2(4).Note that g2(4)4.This means that
  P1,4 is a suffix of P5,12.That is,P1,4 must be
  equal to P9,12.
• G(j)m-maxg1(j),g2(j)

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The Boyer-Moore Algorithm
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The Boyer-Moore Algorithm
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The Boyer-Moore Algorithm
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The Boyer-Moore Algorithm
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The Boyer-Moore Algorithm
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The Boyer-Moore Algorithm

• n24 and m12
• The first mismatch occurs at PjP12
• ssmaxG(j),m-B(Tsj-1)
• 1maxg(12),12-B(T12)
• 1max1,2
• 3
• Then we shift the windows to position s3

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Suffix Trees
If S is ATCACATCATCA, its 12 suffixes are listed
in Table 3.1.
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Suffix Tree
Suffix_array
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For Example
Consider the suffix tree for
SATCACATCATCA. Suppose PTCAT. Since P1 is
T, we follow the branch of TCA. Then we
match P1,3 with TCA. Since P4 is T, we
follow the branch of TCA. We now can report
that P is at position 7 in S because P4 matches
the first symbol of TCA and leaf 7 is reached
along the branch of TCA.
Suffix_tree
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Suffix Array
For example, the non-decreasing lexical order
of suffices of SATCACATCATCA is S(12) , S(4)
, S(9) , S(1) , S(6) , S(11) , S(3) , S(8) ,
S(5) , S(10) , S(2) and S(7) .Table 3.2 shows the
suffix array A.
Suffix_tree
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The longest common substring(1)
The longest common substring of strings X and
Y is a common substring of X and Y which has the
longest length. For example, PAT is the
longest common substring of XAPAT and
YPATT. We can create a suffix tree for X
and Y for finding the longest Common substring.
The suffices for X and Y are descrebed in Table
3.3. Figure 3.25 shows the suffix string for
X and Y.
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The longest common substring(2)
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Approximate String Matching
Given a text string T of length n, a pattern
string P of length m and a maximal number of
errors allowed k. For instance, if
Tpttapa, Ppatt and k2, the substrings
T1,2, T1,4,and T5,6 are all up to 2 errors with
P.
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The suffix edit distance
This is called the suffix edit distance which
is the minimum number of substitutions,
insertions and deletions, which will transform
some suffix of S1 and S2. Consider S1p and
S2p. The suffix edit distance between S1 and
S2 is 0. Consider S1ptt and S2p. The
suffix edit distance between S1 and S2 is 1 as we
can replace the last character t by p.
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Algorithm
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Dynamic Programming
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For Example
Consider E(4, 3). The arrows traced are E(4,
3) to E(3, 2) to E(2, 1) to E(1, 1). We ignore
E(2, 1) to E(1, 1). Thus we have obtained an
occurrence of approximate matching, T1,3ptt.
Algorithm