Text Searching

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CHAPTER 9

• Text Searching

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Algorithm 9.1.1 Simple Text Search
This algorithm searches for an occurrence of a
pattern p in a text t. It returns the smallest
index i such that ti..i m- 1 p, or -1 if no
such index exists.
Input Parameters p, t Output Parameters
None simple_text_search(p, t) m p.length n
t.length i 0 while (i m n) j
0 while (ti j pj) j j
1 if (j m) return i i i
1 return -1
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Algorithm 9.2.5 Rabin-Karp Search
This algorithm searches for an occurrence of a
pattern p in a text t. It returns the smallest
index i such that ti..i m- 1 p, or -1 if no
such index exists.
Input Parameters p, t Output Parameters
None rabin_karp_search(p, t) m p.length n
t.length q prime number larger than m r
2m-1 mod q // computation of initial
remainders f0 0 pfinger 0 for j 0 to
m-1 f0 2 f0 tj mod q pfinger
2 pfinger pj mod q ...
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Algorithm 9.2.5 continued
... i 0 while (i m n) if (fi
pfinger) if (ti..i m-1 p) // this
comparison takes //time
O(m) return i fi 1 2 (fi- r
ti) ti m mod q i i 1 return -1
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Algorithm 9.2.8 Monte Carlo Rabin-Karp Search
This algorithm searches for occurrences of a
pattern p in a text t. It prints out a list of
indexes such that with high probability ti..i
m- 1 p for every index i on the list.
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Input Parameters p, t Output Parameters
None mc_rabin_karp_search(p, t) m
p.length n t.length q randomly chosen prime
number less than mn2 r 2m-1 mod q //
computation of initial remainders f0
0 pfinger 0 for j 0 to m-1 f0 2
f0 tj mod q pfinger 2 pfinger pj
mod q i 0 while (i m n) if (fi
pfinger) prinln(Match at position
i) fi 1 2 (fi- r ti) ti m
mod q i i 1
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Algorithm 9.3.5 Knuth-Morris-Pratt Search
This algorithm searches for an occurrence of a
pattern p in a text t. It returns the smallest
index i such that ti..i m- 1 p, or -1 if no
such index exists.
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Input Parameters p, t Output Parameters
None knuth_morris_pratt_search(p, t) m
p.length n t.length knuth_morris_pratt_shift(p
, shift) // compute array shift of shifts i
0 j 0 while (i m n) while (ti
j pj) j j 1 if (j
m) return i i i shiftj - 1 j
max(j - shiftj - 1, 0) return -1
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Algorithm 9.3.8 Knuth-Morris-Pratt Shift Table
This algorithm computes the shift table for a
pattern p to be used in the Knuth-Morris-Pratt
search algorithm. The value of shiftk is the
smallest s gt 0 such that p0..k -s ps..k.

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Input Parameter p Output Parameter
shift knuth_morris_pratt_shift(p, shift) m
p.length shift-1 1 // if p0 ? ti we
shift by one position shift0 1 // p0..- 1
and p1..0 are both // the empty string i
1 j 0 while (i j lt m) if (pi j
pj) shifti j i j j
1 else if (j 0) shifti i
1 i i shiftj - 1 j max(j -
shiftj - 1, 0 )

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Algorithm 9.4.1 Boyer-Moore Simple Text Search
This algorithm searches for an occurrence of a
pattern p in a text t. It returns the smallest
index i such that ti..i m- 1 p, or -1 if no
such index exists.
Input Parameters p, t Output Parameters
None boyer_moore_simple_text_search(p, t)
m p.length n t.length i 0
while (i m n) j m - 1 // begin at
the right end while (ti j pj)
j j - 1 if (j lt 0)
return i i i 1 return
-1
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Algorithm 9.4.10 Boyer-Moore-Horspool Search
This algorithm searches for an occurrence of a
pattern p in a text t over alphabet S. It returns
the smallest index i such that ti..i m- 1 p,
or -1 if no such index exists.
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Input Parameters p, t Output Parameters
None boyer_moore_horspool_search(p, t)
m p.length n t.length // compute the
shift table for k 0 to S - 1 shiftk
m for k 0 to m - 2 shiftpk m - 1 -
k // search i 0 while (i m n) j
m - 1 while (ti j pj)
j j - 1 if (j lt 0)
return i i i shiftti m -
1 //shift by last letter return -1
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Algorithm 9.5.7 Edit-Distance
The algorithm returns the edit distance between
two words s and t.
Input Parameters s, t Output Parameters
None edit_distance(s, t) m s.length n
t.length for i -1 to m - 1 disti, -1 i
1 // initialization of column -1 for j 0 to
n - 1 dist-1, j j 1 // initialization
of row -1 for i 0 to m - 1 for j 0 to n
- 1 if (si tj) disti, j
min(disti - 1, j - 1, disti - 1, j
1, disti, j - 1 1) else
disti, j 1 min(disti - 1, j - 1,
disti - 1, j, disti, j -
1) return distm - 1, n - 1
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Algorithm 9.5.10 Best Approximate Match
The algorithm returns the smallest edit distance
between a pattern p and a subword of a text t.
Input Parameters p, t Output Parameters
None best_approximate_match(p, t) m
p.length n t.length for i -1 to m - 1
adisti, -1 i 1 // initialization of
column -1 for j 0 to n - 1 adist-1, j
0 // initialization of row -1 for i 0 to m
- 1 for j 0 to n - 1 if (si
tj) adisti, j min(adisti - 1, j -
1, adist i - 1, j 1, adisti, j - 1
1) else adist i, j 1
min(adisti - 1, j - 1, adist i -
1, j, adisti, j - 1) return adist m - 1, n -
1
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Algorithm 9.5.15 Dont-Care-Search
This algorithm searches for an occurrence of a
pattern p with dont-care symbols in a text t
over alphabet S. It returns the smallest index i
such that ti j pj or pj ? for all j
with 0 j lt p, or -1 if no such index exists.
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Input Parameters p, t Output Parameters
None don t_care_search(p, t) m p.length
k 0 start 0 for i 0 to m ci 0
// compute the subpatterns of p, and store them
in sub for i 0 to m if (pi ?)
if (start ! i) // found the end of
a dont-care free subpattern
subk.pattern pstart..i - 1
subk.start start k k 1
start i 1 ...
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... if (start ! i) // end of the last
dont-care free subpattern subk.pattern
pstart..i - 1 subk.start start
k k 1 P sub0.pattern, . . . ,
subk - 1.pattern aho_corasick(P, t) for
each match of subj.pattern in t at position i
ci - subj.start ci - subj.start
1 if (ci - subj.start k)
return i - subj.start return - 1
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Algorithm 9.6.5 Epsilon
This algorithm takes as input a pattern tree t.
Each node contains a field value that is either
, , or a letter from S. For each node, the
algorithm computes a field eps that is true if
and only if the pattern corresponding to the
subtree rooted in that node matches the empty
word.
Input Parameter t Output Parameters
None epsilon(t) if (t.value ) t.eps
epsilon(t.left) epsilon(t.right) else if
(t.value ) t.eps epsilon(t.left)
epsilon(t.right) else if (t.value )
t.eps true epsilon(t.left) // assume only
child is a left child else // leaf with
letter in S t.eps false
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Algorithm 9.6.7 Initialize Candidates
This algorithm takes as input a pattern tree t.
Each node contains a field value that is either
, , or a letter from S and a Boolean field
eps. Each leaf also contains a Boolean field cand
(initially false) that is set to true if the leaf
belongs to the initial set of candidates.
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Input Parameter t Output Parameters
None start(t) if (t.value )
start(t.left) if (t.left.eps)
start(t.right) else if (t.value )
start(t.left) start(t.right)
else if (t.value ) start(t.left)
else // leaf with letter in S t.cand
true
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Algorithm 9.6.10 Match Letter
This algorithm takes as input a pattern tree t
and a letter a. It computes for each node of the
tree a Boolean field matched that is true if the
letter a successfully concludes a matching of the
pattern corresponding to that node. Furthermore,
the cand fields in the leaves are reset to false.
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Input Parameters t, a Output Parameters
None match_letter(t, a) if (t.value )
match_letter(t.left, a) t.matched
match_letter(t.right, a) else if (t.value
) t.matched match_letter(t.left,
a) match_letter(t.right, a) else if
(t.value ) t.matched
match_letter(t.left, a) else // leaf
with letter in S t.matched t.cand (a
t.value) t.cand false return
t.matched
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Algorithm 9.6.10 New Candidates
This algorithm takes as input a pattern tree t
that is the result of a run of match_letter, and
a Boolean value mark. It computes the new set of
candidates by setting the Boolean field cand of
the leaves.
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Input Parameters t, mark Output Parameters
None next(t, mark) if (t.value )
next(t.left, mark) if (t.left.matched)
next(t.right, true) // candidates following a
match else if (t.left.eps) mark)
next(t.right, true) else
next(t.right, false) else if (t.value )
next(t.left, mark) next(t.right,
mark) else if (t.value ) if
(t.matched) next(t.left, true) //
candidates following a match else
next(t.left, mark) else // leaf
with letter in S t.cand mark
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Algorithm 9.6.15 Match
This algorithm takes as input a word w and a
pattern tree t and returns true if a prefix of w
matches the pattern described by t.
Input Parameter w, t Output Parameters
None match(w, t) n w.length epsilon(t)
start(t) i 0 while (i lt n)
match_letter(t, wi) if (t.matched)
return true next(t, false) i
i 1 return false
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Algorithm 9.6.16 Find
This algorithm takes as input a text s and a
pattern tree t and returns true if there is a
match for the pattern described by t in s.
Input Parameter s, t Output Parameters
None find(s,t) n s.length epsilon(t)
start(t) i 0 while (i lt n)
match_letter(t, si) if (t.matched)
return true next(t, true) i
i 1 return false