String Matching-Algorithm Design and Analysis for Strings-Lecture Slides, Slides of Design and Analysis of Algorithms

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String Matching

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String Matching Problem

 Given a text string T of length n and a pattern

string P of length m , the exact string matching

problem is to find all occurrences of P in T.

 Example: T=“AGCTTGA” P=“GCT”

 Applications:

 Searching keywords in a file

 Searching engines (like Google and Openfind)

 Database searching (GenBank)

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A Brute-Force Algorithm

Time: O( mn ) where m =| P | and n =| T |.

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Two-phase Algorithms

 Phase 1：Generate an array to indicate the

moving direction.

 Phase 2：Make use of the array to move and

match the string

 KMP algorithm:

 Proposed by Knuth, Morris and Pratt in 1977.

 Boyer-Moore Algorithm:

 Proposed by Boyer-Moore in 1977.

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Second Case for KMP Algorithm

 The first symbol of P appears in P again.

 T

7

 P

7 in (a). We have to slide to T 6 , since P 6

= P

1

= T

6

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Third Case for KMP Algorithm

 The prefix of P appears in P again.

 T

8

 P

8 in (a). We have to slide to T 6 , since P 6,

= P

1,

= T

6,

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Definition of the Prefix Function

f ( j ) =k

f ( j )=largest k < j such that P

1, k

=P

j–k +1 ,j

f ( j ) = 0 if no such k

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Calculation of the Prefix Function

determine f ( 5 )

Because , we get ( 5 ) 0

If , then we check if ;

If , then we get ( 5 ) ( 4 ) 1 ;

( 4 ) 1 , thus

5 1

5 2 5 1

5 2

4 1

P P f

P P P P

P P f f

f P P

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Calculation of the Prefix Function

f ( 4 )  1 9 (^91 )^14

f ( 9 ) 4 because P P P

f

 

( 4 ) 1 because "A"

4 ( 4 1 ) 1 1

 

f P P P

f

"T"

( 10 ) 2 because "T" "C"

(^10) ( 10 1 ) 1 ( ( 10 1 )) 1 ( 4 ) 1 2

10 ( 10 1 ) 1 5

   ^ 

 

P P P P P

f P P P

f f f^ f

f

To determine f (10):

Pattern Matching 14

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) < j )

 Hence, there are no more

than 2 m iterations of the

while-loop

Algorithm failureFunction ( P )

F [ 0 ]  0

i  1

j  0

while i < m

if P [ i ]  P [ j ]

{we have matched j + 1 chars}

F [ i ]  j + 1

i  i  1

j  j  1

else if j > 0 then

{use failure function to shift P }

j  F [ j  1]

else

F [ i ]  0 { no match }

i  i  1

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An Example for KMP Algorithm

Phase 1

Phase 2

f (4–1)+1= f (3)+1=0+1=

f (12)+1= 4+1=

matched

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Time Complexity of KMP Algorithm

 Time complexity : O ( m + n ) (analysis omitted)

 O ( m ) for computing function f

 O ( n ) for searching P

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 A suffix Tree for S=“ATCACATCATCA”

Suffix Trees

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Properties of a Suffix Tree

 Each tree edge is labeled by a substring of S.

 Each internal node has at least 2 children.

 Each S

( i )

has its corresponding labeled path

from root to a leaf, for 1 i  n.

 There are n leaves.

 No edges branching out from the same

internal node can start with the same

character.