String Matching Algorithms Based upon the Uniqueness Property

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String Matching Algorithms Based upon the
Uniqueness Property
C. W. Lu and R. C. T. Lee, 2007, String Matching
Algorithms Based upon the Uniqueness Property,
The 24th Workshop on Combinatorial Mathematics
and Computation Theory, pp.385-392.

• Advisor Prof. R. C. T. Lee
• Speaker C. W. Lu

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• String matching problem
• Given a text string T of length n and a pattern
  string P of length m.
• Find all occurrences of P in T.

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Rule 1 The Suffix to Prefix Rule

• Suppose we have longest suffix u of a window
  which is also a prefix of P, we can move P in
  such a way that the prefix u of P matches with
  the suffix u of the window.

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The Uniqueness Property of a String

• For any substring V of P, if V occurs in P only
  once, V is a unique substring.
• When V matches with some substring of T, we can
  move P such a way that the prefix of P matches
  with the suffix of V.

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Example
P c a t a g t a g c c t Suppose we use the
substring cc as the unique substring.
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Algorithm 1- The Longest Prefix with Unique
Suffix Matching Algorithm

• We further modified the uniqueness by noting that
  the substring does not have to be unique in the
  entire pattern P. In fact, a substring which is
  unique in a prefix of P suffices.
• Therefore, we only have to find the longest
  prefix which contains a unique suffix in P.

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Example
P CACTAGCCACTCTC The substring TC occurs twice
in P, but it is unique in the prefix CACTAGCCACTC.
Move P 11 steps.
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Example
P CACTAGCCACTCTC The substring G is also unique
in the prefix CACTAG.
Move P 6 steps.
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P CACTAGCCACTCTC
In the above example, using the unique substring
TC, we could move P 11 steps if TC matches with
TC in T using the unique substring G, we could
move P 6 steps if G matches with G in T.
Is the unique substring TC better than the unique
substring G?
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• We should notice that if the unique substring
  appears in T many times, our algorithm would be
  efficient.
• In general, the probability of TC in P matching
  with TC in T exactly is 1/16 (Suppose the size of
  alphabet is 4), and the probability of G in P
  matching with G in T exactly is 1/4.
• Thus, the size of the unique substring is also
  important.

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P CACTAGCCACTCTC

• If the substring TC in P exactly matches with TC
  in T once and moves P by 11 steps, the substring
  G in P may match G in T four times and moves P by
  6 steps for each time. So, we expect that the
  substring G would be better than the substring TC
  in general.

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• We now define a ratio to determine which
  substring is better.
• Let S be the alphabet.
• The larger s is, the better efficiency can be
  achieved in the searching phase.

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Preprocessing Phase
P CAGACGACCCCAACAGC S A, C, G, T, S 4.
Find the longest prefix with an unique suffix
which size is one.
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Preprocessing Phase

• We have found the unique substring with size 1,
  and we could use it to move P 3 steps.
• Next, we try to find an unique substring with
  size 2 such that we could use this substring to
  move P more than 34 steps.
• Thus, we only consider the substrings of
  p12p13p16.

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Searching Phase
If the unique substring mismatches, move P one
step.
Move 1 step.
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Searching Phase
If the unique substring GC matches with GC in T,
move P 16 steps.
Move 16 steps.
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• As we discuss above, the size of the unique
  substring is important.
• In the following, we will introduce another
  algorithm which uses an unique substring with
  size one.

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Algorithm 2- Longest Substring with Unique
Character Matching Algorithm

• In the window, let x be any character. In order
  to have any meaningful matching of P with T, we
  must find the same x in P located in the left
  side of x in T.

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• In preprocessing phase, we try to find the
  longest substring p in P such that x in p
  occurs only once. That is,
• and pj occurs in p only once.

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• If the unique character x matches with x in T, we
  can move P p steps.

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Example
In this example, we would find the longest
substring p4p5p10 with a unique character p10.
If the character p10 matches with T, we can move
P 7 steps.
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Searching Phase
If p10 mismatches, move P one step.
Move 1 step.
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Searching Phase
If p10 matches with T, move P 7 steps.
Move 7 steps.
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Algorithm 3- The Unique Pairwise Substring
Algorithm

• The substring pipi1pj-1pj is called an unique
  pairwise substring if it satisfies the condition
  that pipi1pj-1pj occurs in the prefix
  p1p2pj-1pj of P exactly once, and no
  pkpk1pkj-i exists in p1p2pj-1 such that
  pk pi and pkj-i pj.

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Example
The substring TCG is an unique pairwise substring
because no pkpk1pk2 exists in p1p2p12 such
that pk p11 T and pk2 p13 G.
The substring CAC is not an unique pairwise
substring because there exists a substring p2p3p4
in p1p2p9 such that p2 p8 C and p4 p10 C.
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• Suppose pipi1pj-1pj is an unique pairwise
  substring.
• If pi and pj match with T, we have two cases to
  move P.

Case 1 such that pj pk, where
0?k?j-i-1. We can move P j-k steps.
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Case 2 pj ? pk, where 0?k ?j-i-1.
We can move P j1 steps.
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Example
If we choose p11p12p13 as the unique pairwise
substring, we can move P 14 steps when p11 and
p13 match with T.
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• There would be many unique pairwise substrings in
  the pattern.
• We will select the one which is located at
  rightest in the pattern.

Example
The substrings p5p6, p7p8p9 and p11p12p13 are all
unique pairwise substrings. We would select
p11p12p13 because it will have the largest move.
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Example
If p11 or p13 mismatch, move P one step.
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Example
If p11 and p13 match with T, move P 14 steps.
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References

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Thanks for your attention.