Transcript 28. KMP Algorithm -

KMP algorithm
KNUTH D.E., MORRIS (Jr) J.H., PRATT V.R.,, Fast pattern matching in strings,
SIAM Journal on Computing 6(1), 1977, pp.323-350.
Advisor: Prof. R. C. T. Lee
Reporter: C. W. Lu
1
KMP Table
• The KMP algorithm constructs a table in
preprocessing phase.
• In searching phase, the window shifting will be
determined easily by looking up the table.
• Example:
P = bcbabcbaebcbabcba
KMP Table
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0
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10 11 12 13 14 15 16
P
b
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b
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-1
0
-1
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-1
0
-1
1
4
-1
0
-1
1
-1
0
-1
1
2
• Once the KMP table is constructed, whenever
a mismatch occurs at location i, for the KMP
algorithm, we move the pattern i-KMPtable(i)
steps under the assumption that the location
starts with 0.
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i
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P
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-1
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T: … b c b a c c b a e b c b a b c b a …
P:
b c b a b c b a e b c b a b c b a
b c b a b c b a e b c b a b c b a
Mismatch occurs at location 4 of P.
Move P (4 - KMPtable[4]) = 4 - (-1) = 5 steps.
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P
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-1
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T: … b c b a b c b a b b c b a b c b a …
P:
b c b a b c b a e b c b a b c b a
b c b a b c b a e b c b a b c b a
Mismatch occurs in position 8 of P.
Move P (8 - KMPtable[8]) = 8 - 4 = 4 steps.
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The Definition of the KMP Table
• For location i, if j is the largest such that
P(0,j-1) is a suffix of P(0,i-1) and P(i) not
equal to P(j), then KMPtable(i)=j.
• Example
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P
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-1
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-1
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-1
3
0
∵P(0, 2) is the longest
prefix which is equal to a
suffix of P(0, 6), and
P(7)≠P(3).
∴KMPtable[7] = 3.
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Condition for KMPtable[i] = -1
Condition A: P(0) = P(i)
Condition B: P(0, j) is a suffix of P(0, i-1)
Condition C: P(j+1) = P(i)
KMPtable(i) = -1 :
A & (B  (j )( B & C ))
 ( A & B)  (j )( A & ( B & C ))
 ( A & B)  (j )( A & B & C )
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• There is no suffix of P(0, 3) which is equal to a prefix
of P(0, 3). ( B)
• P(0) = P(4). (A)
• KMPtable[4] = -1 because it satisfies the condition
( A & B) .
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P
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-1
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• There are two suffixes of P(0, 14) which are equal to
a prefix of P(0, 14):
bcbabc (P(0, 5)) and P(6) = P(15);
bc (P(0, 1)), and P(2) = P(15).
( (j )( B & C ) )
• P(0) = P(4). (A)
• KMPtable[15] = -1 because it satisfies the condition
(j )( A & B & C ) .
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Condition for KMPtable[i] = 0
• Condition A: P(0) = P(i)
• Condition B: P(0, j) is a suffix of P(0, i-1)
• Condition C: P(j+1) = P(i)
KMPtable(i) = 0 :
A & (B  (j )(B & C ))
 (A & B)  (j )(A & ( B & C ))
 (A & B)  (j )(A & B & C )
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• There are two suffixes of P(0, 13) which are equal to
a prefix of P(0, 13):
bcbab (P(0, 4)) and P(5) = P(14);
b (P(0)), and P(1) = P(14).
( (j )( B & C ) )
• P(0) = P(4). (  A )
• KMPtable[14] = 0 because it satisfies the condition
(j )(A & B & C ) .
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How to Construct the KMP Table
Efficiently?
• Note that the KMP algorithm is actually an
improvement of the MP algorithm. Therefore,
we may now take a look at the table used in
the MP algorithm.
• We call the table used in the MP algorithm the
prefix table.
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The Definition of the Prefix Table.
• For location i, let j be the largest j, if it exists,
such that P(0,j-1) is a suffix of P(0,i),
Prefix(i)=j.
• If, for P(0,i), there is no prefix equal to a suffix,
Prefix (i)=0.
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Example
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P
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Prefix 0
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• Note that, in the MP algorithm, we move the
pattern i-Prefix(i)+1 steps when a mismatch
occurs at location i.
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How can we construct the Prefix Table
Efficiently?
• To compute Prefix(i), we look at Prefix(i-1).
• In the following example, since Prefix(11)=4,
we know that there exists a prefix of length 4
which is equal to a suffix with length 4 of
P(0,11). Besides, P(4)=P(12). We may
conclude that Prefix(12)=Prefix(11)+1=4+1=5.
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Prefix 0
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P
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Another Case
• Consider the following example.
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Prefix
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?
• Prefix(9)=4. But P(4)≠P(10).
• Can we conclude that Prefix(10)=0?
• No, we cannot.
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• There exists a shorter prefix with length 2
which is equal to a suffix of P(0, 9), and
P(10)=P(2). We should conclude that
Prefix(10)=2+1=3.
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P
Prefix
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?
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• In other words, we may use the pointer idea
expressed below:
j
i-1
Y
X
• It may be necessary to examine P(0, j) to see
whether there exists a prefix of P(0, j) equal to
a suffix of P(0, j).
• Thus the Prefix function can be found
recursively.
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Construct the Prefix
Function f
f [0]=0
For ( i=1 ; i<m ; i++ ){
t = f (i-1);
While(t>=0){
if ( P(i) = P(t) ) {
f [i] = t + 1;
break;
}
else{
if ( t != 0)
t = f [t-1];
else{
f [i] = 0;
break;
}
}
}
}
/*recursive*/
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Example:
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P
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Prefix 0
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P b
Prefix 0
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0
t = f[i-1] = f[0] = 0;
∵P[1] = c ≠ P[t] = P[0] = b ∴f [1] = 0.
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Example:
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Prefix 0
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t = f[i-1] = f[4] = 1;
∵P[5] = c = P[t] = P[1] = c ∴f [5] = t +1 = 2.
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P b
Prefix 0
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t = f[i-1] = f[7] = 4;
∵P[8] = e ≠ P[t] = P[4] = b,
t != 0;
t = f[t-1] = f[3] = 0;
∵P[8] = e ≠ P[t] = P[0] = b, ∴f [8] = 0.
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Example:
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Prefix 0
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t = f[i-1] = f[14] = 6;
∵P[15] = b = P[t] = P[6] = b ∴f [15] = t +1 = 7.
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Prefix 0
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• The KMP Table can also be constructed
recursively.
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The KMPtable
KMPtable[0] = -1
For ( i=1 ; i<m ; i++ ) {
t=f (i-1)
While ( t > 0 ) {
if ( P(i) ≠ P(t) ) {
KMPtable[i]=t
break
}
else t = f ( t – 1 )
/*recursive*/
}
if ( KMPtable[i] = ψ)
if ( P(i) = P(0)) .
KMPtable[i] = -1
else
KMPtable[i] = 0
}
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Example:
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P
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Prefix 0
KMPtable -1
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Prefix 0
KMPtable -1
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t = f[i-1] = f[2] = 1;
∵P[3] = a ≠ P[t] = P[1] = c, ∴ KMPtable [3] = t = 1.
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Example:
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P
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Prefix 0
KMPtable -1
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t = f[i-1] = f[6] = 3;
∵P[7] = a = P[t] = P[3] = a;
t = f[t-1] = f[2] = 1;
∵P[7] = a ≠ P[t] = P[1] = c;
∴ KMPtable [3] = t = 1.
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Example:
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Prefix 0
KMPtable -1
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t = f[i-1] = f[12] = 4;
∵P[13] = b = P[t] = P[4] = b;
t = f[t-1] = f[3] = 0;
∵P[13] = b = P[0] = b;
∴ KMPtable [13] = -1.
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Example:
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f 0
KMPtable -1
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t = f[i-1] = f[15] = 7;
∵P[16] = a = P[t] = P[7] = a;
t = f[t-1] = f[6] = 3;
∵P[16] = a = P[t] = P[3] = a;
t = f[t-1] = f[2] = 1;
∵P[16] = a ≠ P[t] = P[1] = c;
∴ KMPtable [16] = t = 1.
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Time Complexity
Preprocessing phase in O(m) space and time complexity.
Searching phase in O(n+m) time complexity.
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References
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AHO, A.V., 1990, Algorithms for finding patterns in strings. in Handbook of Theoretical Computer Science, Volume
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References
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Thank You!
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