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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. In 2003 ,Rytter proposed a constant space and linear time string matching algorithm

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## 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

• 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

### 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.

### 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

### 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

i+j+1

T:

mismatch

j

j

P:

u

v

j

P:

v

u

j

i

v’

T:

Why do we have to use short maximal suffix?

Suppose V’ is very long, then we move pattern which is incorrect.

j

i

v’

P:

u

v

j

j+1

T:

j

i

P:

u

v

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.

### 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 v[j + 1], we shift v with j + 1 characters and start next comparison at P[1] with T[i + 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)

Full Example

T=adadadaddadababadada

P=u.v = ababa .dada

T=adadadaddadababadada

P=u.v = ababa .dada

Shift 4 steps

T=adadadaddadababadada

P=u.v = ababa .dada

Shift 1 steps

T=adadadaddadababadada

P=u.v = ababa .dada

Shift Preiod(P) = 8 steps

Rule 1 again!

T=adadadaddadababadada

P=u.v = ababa .dada

Match!!

Shift Preiod(P) = 8 steps

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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