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2004 Goodrich, Tamassia Pattern Matching 1

Pattern Matching

1

a b a c a a b

234

a b a c a b

a b a c a b


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StringsA string is a sequence ofcharacters

Eam!"es of strings# $a%a !rogram &TM' document ()A sequence ordinar* tet

An a"!ha+et is the setof !ossi+"e characters fora fami"* of strings

Eam!"e of a"!ha+ets# AS-- .nicode /0, 1 /A, , G, T

'etP+e a string of sie m A su+string P[i .. j]ofPis the

su+sequence of Pconsisting ofthe characters ith rans+eteen i andj

A !re of Pis a su+string ofthe t*!e P[0.. i]

A su5 of Pis a su+string ofthe t*!e P[i ..m 1]

Gi%en strings T6tet7 and P6!attern7, the !atternmatching !ro+"em consists of

nding a su+string of Tequa"toP

A!!"ications# tet editors search engines +io"ogica" research


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8rute9:orce

Pattern MatchingThe +rute9force !atternmatching a"gorithmcom!ares the !attern Piththe tet Tfor each !ossi+"eshift of Pre"ati%e to T, unti"either a match is found, or a"" !"acements of the

!attern ha%e +een tried

8rute9force !atternmatching runs in time O(nm)

Eam!"e of orst case# T =aaa ah P =aaah ma* occur in images and

()A sequences un"ie"* in Eng"ish tet

AlgorithmBruteForceMatch(T, P)

Inputtext T of size nand patternP of size m

Outputstarting index of asubstring of Tequal toP or 1

if no such substring existsfor i 0 to n m

{ test shift iof the pattern }

j 0

whilej


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8o*er9Moore &euristicsThe 8o*er9Moore;s !attern matching a"gorithm is +ased onto heuristics

'ooing9g"ass heuristic#om!arePith a su+sequence of Tmo%ing +acards

haracter9


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2004 Goodrich, Tamassia Pattern Matching >

'ast9Cccurrence :unction

8o*er9Moore;s a"gorithm !re!rocesses the !attern Pandthe a"!ha+et to +ui"d the "ast9occurrence functionLma!!ing to integers, hereL(c)is dened as the "argest inde isuch thatP[i]=c or

1if no such inde eistsEam!"e#

={a, b, c, d} P=abacab

The "ast9occurrence function can +e re!resented +* anarra* indeed +* the numeric codes of the characters

The "ast9occurrence function can +e com!uted in timeO(m +s), here mis the sie of Pand sis the sie of

c a b c d

L(c) ! " # 1


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2004 Goodrich, Tamassia Pattern Matching ?

mj

i

j l

$ $ $ $ $ $ a $ $ $ $ $ $

$ $ $ $ b a

$ $ $ $ b a

j

ase 1# j 1+ l

The 8o*er9Moore

A"gorithmAlgorithmBoyerMooreMatch(T, P,

)

L lastOccurenceFunction(P, )i m 1j m 1repeat

if T[i]=P[j]if j =0

return i { match at i}else

i i 1j j 1

else

{ character%&ump }l L[T[i]]i i +m' min(j 1+ l)j m 1

until i >n 1return 1 { no match }

m(1 +l)

i

jl

$ $ $ $ $ $

a $ $ $ $ $ $

$ a $ $ b $

$ a $ $ b $

1

+

l

ase 2# 1+ lj


7/13 2004 Goodrich, Tamassia Pattern Matching @

Eam!"e

1

a b a c a a b a d c a b a c a b a a b b

234

>

?

@

AB1012

a b a c a b

a b a c a b

a b a c a b

a b a c a b

a b a c a b

a b a c a b

1113


8/13 2004 Goodrich, Tamassia Pattern Matching

Ana"*sis

8o*er9Moore;s a"gorithmruns in time O(nm +s)

Eam!"e of orst case# T =aaa a

P =baaa

The orst case ma* occurin images and ()Asequences +ut is un"ie"*in Eng"ish tet

8o*er9Moore;s a"gorithm issignicant"* faster thanthe +rute9force a"gorithmon Eng"ish tet

11

1

a a a a a a a a a

234>?

b a a a a a

b a a a a a

b a a a a a

b a a a a a

@AB1012

13141>1?1@1A

1B2021222324


9/13 2004 Goodrich, Tamassia Pattern Matching B

The DMP A"gorithm

Dnuth9Morris9Pratt;sa"gorithm com!ares the!attern to the tet in left-to-right, +ut shifts the

!attern more inte""igent"*than the +rute9forcea"gorithm

=hen a mismatch occurs,hat is the moste can

shift the !attern so as toa%oid redundantcom!arisonsF

Anser# the "argest !re ofP[0$$j]that is a su5 of P[1$$j]

x

j

$ $ a b a a b $ $ $ $ $

a b a a b a

a b a a b a

)o need tore!eat thesecom!arisons

esumecom!aring

here


10/13 2004 Goodrich, Tamassia Pattern Matching 10

DMP :ai"ure :unctionDnuth9Morris9Pratt;sa"gorithm !re!rocesses the!attern to nd matches of!rees of the !attern iththe !attern itse"f

The failure functionF(j)is dened as the sie ofthe "argest !re of P[0$$j]that is a"so a su5 of P[1$$j]

Dnuth9Morris9Pratt;s

a"gorithm modies the+rute9force a"gorithm sothat if a mismatch occursatP[j] T[i] e set j F(j 1)

j 0 1 # ! 5

P[j] a b a a b a

F(j) 0 0 1 1 3

x

j

$ $ a b a a b $ $ $ $ $

a b a a b a

F(j 1)

a b a a b a


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The DMP A"gorithmThe fai"ure function can +ere!resented +* an arra*and can +e com!uted inO(m)time

At each iteration of the

hi"e9"oo!, either iincreases +* one, or

the shift amount i jincreases +* at "east one6o+ser%e that F(j 1)*j7

&ence, there are no morethan n iterations of thehi"e9"oo!

Thus, DMP;s a"gorithm runsin o!tima" time O(m +n)

AlgorithmMPMatch(T, P)

F !ailureFunction(P)i 0j 0while i 0j F[j 1]

elsei i +1

return 1 { no match }


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om!uting the :ai"ure

:unctionThe fai"ure function can +ere!resented +* an arra*and can +e com!uted inO(m)time

The construction is simi"arto the DMP a"gorithm itse"f

At each iteration of thehi"e9"oo!, either iincreases +* one, or

the shift amount i jincreases +* at "east one6o+ser%e that F(j 1)*j7

&ence, there are no morethan m iterations of thehi"e9"oo!

Algorithm!ailureFunction(P)

F[0]0i 1j 0

while i 0 then{use failure function to shiftP}j F[j 1]

elseF[i]0 { no match }i i +1


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Eam!"e

1

a b a c a a b a c a b a c a b a a b b

@

A

1B1A1@1>

a b a c a b

1?14

13

2 3 4 > ?

B

a b a c a b

a b a c a b

a b a c a b

a b a c a b

10 11 12

c

j 0 1 # ! 5

P[j] a b a c a b

F(j) 0 0 1 0 1 2

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