Two Way Algorithm

1
Two Way Algorithm
Two-way string-matching Journal of the ACM
38(3)651-675, 1991 Crochemore M., Perrin D.

• Advisor Prof. R. C. T. Lee
• Speaker C. C. Yen

2

• In 2003 ,Rytter proposed a constant space and
  linear time string matching algorithm
• To achieving the good constant space , this
  algorithm avoids the preprocessing function table
  of the KMP algorithm
• Before introducing this algorithm , we shall
  define some characteristic of the strings

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The Property of Maximal Suffix

• Consider a string P. Let P uv where v
  MaxSuf(P). The property of the maximal suffix of
  a string is If u is non-empty, no suffix of u
  will be equal to a prefix of v.
• Example
• Consider a pattern ababadada.
• Let P uv ababa.dada
• No suffix of u is equal to a prefix of v.

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Short Maximal Suffix

• If a maximal suffix of a string x satisfies
• , we say that
  this maximal suffix of x is a short maximal
  suffix of x.
• Example
• Consider a string x abcdda ,dda is a maximal
  suffix of x and
  .
• Hence we say that dda is a short maximal suffix
  of x

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Short Prefixes Lemma

• Let the decomposition of P uv, where v is the
  maximal suffix of P and v is also a short maximal
  suffix. Suppose that we start to match v with T
  at position i, a part of v is matched and a
  mismatch occurs at the j 1-th position on v.
  Then we can shift P safely by j 1 positions
  without missing any occurrence of P in T.

i
ij1
T
mismatch
j
j
P
v
u
j
P
v
u
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• Why do we have to use short maximal suffix?
• Suppose V is very long, then we move pattern
  which is incorrect.

j
i
v
T
j
i
v
P
v
u
j
j1
T
j
i
P
v
u
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• In the following , we will introduce the basic
  rule of the Two Way Matching algorithm with short
  maximal pattern strings
• The basic rules are given in the next slides.

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Basic rule of the Two-Way algorithm with short
maximal

• 1. Let the decomposition of P uv, where v is the
  maximal suffix of P and v is also a short maximal
  suffix.
• We then find where v appears in T from left to
  right. Assume the comparison starts at position
  i. When a mismatch occurs at vj 1, we shift v
  with j 1 characters and start next comparison
  at P1 with Ti j 1.
• When the part of v has be found in T, we scan the
  part of u from right to left. If a mismatch
  occurs when scanning u, we shift P with Period(P)
• 4. If we find both the parts of v and u in T, we
  report an occurrence of P in T. We then shift v
  with Period(P)

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• Full Example
• Tadadadaddadababadada
• Pu.v ababa .dada

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
T a d a d a d a d d a d a b a d a d a
P a b a b a d a d a
1 2 3 4 5 6 7 8 9
10
Tadadadaddadababadada Pu.v ababa .dada
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
T a d a d a d a d d a d a b a d a d a
1 2 3 4
P a b a b a d a d a
1 2 3 4 5 6 7 8 9
Shift 4 steps
P a b a b a d a d a
1 2 3 4 5 6 7 8 9
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Tadadadaddadababadada Pu.v ababa .dada
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
T a d a d a d a d d a d a b a d a d a
1 2 3 4
P a b a b a d a d a
1 2 3 4 5 6 7 8 9
Shift 1 steps
P a b a b a d a d a
1 2 3 4 5 6 7 8 9
12
Tadadadaddadababadada Pu.v ababa .dada
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
T a d a d a d a d d a d a d a b a b a d a d a
1 2 3 4
P a b a b a d a d a
1 2 3 4 5 6 7 8 9
Shift Preiod(P) 8 steps
P a b a b a d a d a
1 2 3 4 5 6 7 8 9
Rule 1 again!
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Tadadadaddadababadada Pu.v ababa .dada
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
T a d a d a d a d d a d a d a b a b a d a d a
1 2 3 4
P a b a b a d a d a
1 2 3 4 5 6 7 8 9
Match!!
P a b a b a d a d a
1 2 3 4 5 6 7 8 9
Shift Preiod(P) 8 steps
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• References
• BRESLAUER, D., 1996, Saving comparisons in the
  Crochemore-Perrin string matching algorithm,
  Theoretical Computer Science 158(1-2)177-192.
• CROCHEMORE, M., 1997. Off-line serial exact
  string searching, in Pattern Matching Algorithms,
  ed. A. Apostolico and Z. Galil, Chapter 1, pp
  1-53, Oxford University Press.
• CROCHEMORE M., PERRIN D., 1991, Two-way
  string-matching, Journal of the ACM
  38(3)651-675.
• CROCHEMORE, M., RYTTER, W., 1994, Text
  Algorithms, Oxford University Press.

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• Thanks for your attention