sabin m. thomas - string matching algorithms

8/8/2019 Sabin M. Thomas - String Matching Algorithms

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String MatchingAlgorithms

Sabin Thomas


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History of String Search

The brute force algorithm:

invented in the dawn of computer history

re-invented many times, stillcommon

Knuth & Pratt invented a better one in 1970

published 1976 as Knuth-Morris-Pratt

Boyer & Moore found a better one before 1976

Published 1977Karp & Rabin found a better one in 1980

Published 1987


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Brute-force

Worst O(m*n)

Best O(n)

algorithm brute-force:

input: an array of characters, T (the string to be analyzed) , length n

an array of characters, P (the pattern to be searched for), length m

for i := 0 to n-m do

for j := 0 to m-1do

compare T[j] with P[i+j]

ifnot equal, exit the inner loop


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Boyer-Moore

(Example 1)

t[0] t[1] t[2] t[3] t[4] t[5] t[6] t[7] t[8] t[9] t[10]

A B C D

p[0] p[1] p[2] p[3]

N

There is no E in the pattern : thus the pattern cant match ifanycharacters lie

under t[3]. So, move four boxes to the right.

A B C E F G A B C D E


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Boyer-Moore

(Example 1)

t[0] t[1] t[2] t[3] t[4] t[5] t[6] t[7] t[8] t[9] t[10]


A B C D

p[0] p[1] p[2] p[3]

N

Again, no match. But there is a B in the pattern. So move two boxes to the

right.


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Boyer-Moore

(Example 1)

t[0] t[1] t[2] t[3] t[4] t[5] t[6] t[7] t[8] t[9] t10]


A B C D

p[0] p[1] p[2] p[3]

YYYY


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Boyer-Moore

(Pseudocode)

Compares right to left

2 precomputed functions

Good suffix shift Bad character shift


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Boyer-Moore

(Performance)

Performance depends on length of pattern

O(n/m)

Longer patterns = better performance Smallest pattern = m = 1

O(n) linear search


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Knuth-Morris-Pratt

searches for occurrences of a "word" W

within a main "text string" S

Bypasses re-examination of previouslymatched characters.


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Knuth-Morris-Pratt

(Example 1)

t[0] t[1] t[2] t[3] t[4] t[5] t[6] t[7] t[8] t[9] t[10] t[11] t[12] t[13]

p[0] p[1] p[2] p[3] p[4] p[5] p[6]

Y

A B C A B C D A B A B C

NY Y

m = 0

A B C D A B D


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Knuth-Morris-Pratt

(Example 1)

t[0] t[1] t[2] t[3] t[4] t[5] t[6] t[7] t[8] t[9] t[10] t[11] t[12] t[13]

p[0] p[1] p[2] p[3] p[4] p[5] p[6]

Y


NY Y

m = 4

A B C D A B D

Y Y Y


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Knuth-Morris-Pratt

(Example 1)

t[0] t[1] t[2] t[3] t[4] t[5] t[6] t[7] t[8] t[9] t[10] t[11] t[12] t[13]

p[0] p[1] p[2] p[3] p[4] p[5] p[6]


N

m = 10

A B C D A B D


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Knuth-Morris-Pratt

(Example 1)

t[0] t[1] t[2] t[3] t[4] t[5] t[6] t[7] t[8] t[9] t[10] t[11] t[12] t[13]

p[0] p[1] p[2] ..

Y


Y

m = 11

A B C ..

Y


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Knuth-Morris-Pratt

PseudoCode

Search O(n)

algorithm kmp_search:

Input an array of characters, S (the text to be searched)

an array of characters, W (the word sought)

Output an integer (the zero-based position in S at which W is found)

define variables:

an integer, m 0 (the beginning of the current match in S)

an integer, i 0 (the position of the current character in W)

an array of integers, T (the table, computed elsewhere)

while m + i is less than the length of S, do:

ifW[i] = S[m + i],let i i + 1

ifi equals the length ofW,

return m

otherwise,

let m m + i - T[i],

ifi > 0,

let i T[i]


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Knuth-Morris-Pratt

PseudoCode

Partial Table Match O(k)

algorithm kmp_table:

input: an array of characters, W (the word to be analyzed)

an array of integers, T (the table to be filled)

define variables:

an integer, i 2 (the current position we are computing in T) an integer, j 0 (the zero-based index in W of the next character of the current candidate substring)

let T[0] -1, T[1] 0

while i is less than the length ofW, do:

(first case: the substring continues)

ifW[i - 1] = W[j], let T[i] j + 1, i i + 1, j j + 1

(second case: it doesn't, but we can fall back)

otherwise, ifj > 0, letj T[j]

(third case: we have run out of candidates. Note j = 0)

otherwise, let T[i] 0, i i + 1


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Karp-Rabin

Slower for Single pattern match

Fast for Multiple pattern match

Trick is Hash compare.


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Karp-Rabin

(Example 1)t[0] t[1] t[2] t[3] t[4] t[5] t[6] t[7] t[8] t[9] t[10]

A C D Bp[0] p[1] p[2] p[3]

A C D E A C A C C D E

Hash(ACDB) = 5

Hash(ACDE) = 10


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Karp-Rabin


A C D Bp[0] p[1] p[2] p[3]


Hash(ACDB) = 5

Hash(CDEA) = 6


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Karp-Rabin

(Caveats)

Good Hashing function with few collisions

Hashing result must be small number for

faster compare Rolling Hash

S.T1


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Slide 19

S.T1 Rolling Hash - Allows for faster recomputing of the hash. Instead of completely recomputing from scratch, make use of the fact that w

are computing the hash for just an extra letter. Do addition and subtraction to the original hash algorithm

Sabin Thomas, 4/11/2007


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Karp-Rabin

(Pseudocode)

algorithm RabinKarp:

Input an array of characters, S, length n

an array of characters sub, length m

hsub := hash(sub[1..m])

hs := hash(s[1..m])

fori from 1 to n-m+1

ifhs = hsub

ifs[i..i+m-1] = sub

return i

hs := hash(s[i+1..i+m])

return not found


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Karp-Rabin


A C D Bp[0] p[1] p[2] p[3]


Hash(ACDB) = 5

Hash(ACDE) = 10

F M D T

j[0] j[1] j[2] j[3]

Hash(FMDT) = 62

S.T2


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Slide 21

S.T2 Multiple Pattern SearchSabin Thomas, 4/11/2007


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Karp-Rabin


p[0] p[1] p[2] p[3]


Hash(CDEA) = 6

A C D B

Hash(ACDB) = 5

F M D T

j[0] j[1] j[2] j[3]

Hash(FMDT) = 62

S.T3


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Slide 22

S.T3 Hashes of the pattern have already been precomputed.Sabin Thomas, 4/11/2007


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Karp-Rabin

(Performance)

Single Pattern

BM O(n/m)

KMP O(n)

Karp-Rabin O(mn)

Multiple Pattern

BM, KMP O(n k)

Karp-Rabin O(n + k)


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(Applications)

BM - Text Editors search/replace

Karp-Rabin Plagiarism finder


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Questions?

sabin m. thomas - string matching algorithms

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