KMP Pattern Matching algorithm

Knuth-Morris-Pratt
Algorithm
Prepared by: Kamal Nayan

The problem of String Matching
Given a string ‘S’, the problem of string matching deals with finding
whether a pattern ‘p’ occurs in ‘S’ and if ‘p’ does occur then
returning position in ‘S’ where ‘p’ occurs.

…. a O(mn) approach
One of the most obvious approach towards the
string matching problem would be to compare the
first element of the pattern to be searched ‘p’, with
the first element of the string ‘S’ in which to locate
‘p’. If the first element of ‘p’ matches the first
element of ‘S’, compare the second element of ‘p’
with second element of ‘S’. If match found proceed
likewise until entire ‘p’ is found. If a mismatch is
found at any position, shift ‘p’ one position to the
right and repeat comparison beginning from first
element of ‘p’.

How does the O(mn) approach
work
Below is an illustration of how the previously
described O(mn) approach works.
String S aa bb cc aa bb aa aa bb cc aa bb aa cc
aa bb aa aaPattern p

Step 1:compare p[1] with S[1]
S
aa bb cc aa bb aa aa bb cc aa bb aa cc
p aa bb aa aa
Step 2: compare p[2] with S[2]
S aa bb cc aa bb aa aa bb cc aa bb aa cc
p aa bb aa aa

Step 3: compare p[3] with S[3]
S
aa bb aa aap
Mismatch occurs here..
Since mismatch is detected, shift ‘p’ one position to the left and
perform steps analogous to those from step 1 to step 3. At position
where mismatch is detected, shift ‘p’ one position to the right and
repeat matching procedure.
aa bb cc aa bb aa aa bb cc aa bb aa cc

S aa bb cc aa bb aa aa bb cc aa bb aa cc
aa bb aa aap
Finally, a match would be found after shifting ‘p’ three times to the right side.
Drawbacks of this approach: if ‘m’ is the length of pattern ‘p’ and ‘n’ the length
of string ‘S’, the matching time is of the order O(mn). This is a certainly a very
slow running algorithm.
What makes this approach so slow is the fact that elements of ‘S’ with which
comparisons had been performed earlier are involved again and again in
comparisons in some future iterations. For example: when mismatch is
detected for the first time in comparison of p[3] with S[3], pattern ‘p’ would be
moved one position to the right and matching procedure would resume from
here. Here the first comparison that would take place would be between p[0]=‘a’
and S[1]=‘b’. It should be noted here that S[1]=‘b’ had been previously involved
in a comparison in step 2. this is a repetitive use of S[1] in another comparison.
It is these repetitive comparisons that lead to the runtime of O(mn).

The Knuth-Morris-Pratt Algorithm
Knuth, Morris and Pratt proposed a linear time algorithm for the string
matching problem.
A matching time of O(n) is achieved by avoiding comparisons with
elements of ‘S’ that have previously been involved in comparison
with some element of the pattern ‘p’ to be matched. i.e.,
backtracking on the string ‘S’ never occurs

Components of KMP algorithm
The prefix function, Π
The prefix function,Π for a pattern encapsulates
knowledge about how the pattern matches against
shifts of itself. This information can be used to avoid
useless shifts of the pattern ‘p’. In other words, this
enables avoiding backtracking on the string ‘S’.
The KMP Matcher
With string ‘S’, pattern ‘p’ and prefix function ‘Π’ as
inputs, finds the occurrence of ‘p’ in ‘S’ and returns
the number of shifts of ‘p’ after which occurrence is
found.

The prefix function, Π
Following pseudocode computes the prefix fucnction, Π:
Compute-Prefix-Function (p)
1 m  length[p] //’p’ pattern to be matched
2 Π[1]  0
3 k  0
4 for q  2 to m
5 do while k > 0 and p[k+1] != p[q]
6 do k  Π[k]
7 If p[k+1] = p[q]
8 then k  k +1
9 Π[q]  k
10 return Π

Example: compute Π for the pattern ‘p’ below:
p aa bb aa bb aa cc aa
qq 11 22 33 44 55 66 77
pp aa bb aa bb aa cc aa
ΠΠ 00 00
Initially: m = length[p] = 7
Π[1] = 0
k = 0
Step 1: q = 2, k=0
Π[2] = 0
Step 2: q = 3, k = 0,
Π[3] = 1
Step 3: q = 4, k = 1
Π[4] = 2
qq 11 22 33 44 55 66 77
ΠΠ 00 00 11
qq 11 22 33 44 55 66 77
pp aa bb aa bb aa cc AA
ΠΠ 00 00 11 22

Step 4: q = 5, k =2
Π[5] = 3
Step 5: q = 6, k = 3
Π[6] = 0
Step 6: q = 7, k = 0
Π[7] = 1
After iterating 6 times, the prefix
function computation is complete:

qq 11 22 33 44 55 66 77
ΠΠ 00 00 11 22 33
qq 11 22 33 44 55 66 77
ΠΠ 00 00 11 22 33 00
qq 11 22 33 44 55 66 77
ΠΠ 00 00 11 22 33 00 11
qq 11 22 33 44 55 66 77
pp aa bb AA bb aa cc aa
ΠΠ 00 00 11 22 33 00 11

The KMP Matcher
The KMP Matcher, with pattern ‘p’, string ‘S’ and prefix function ‘Π’ as input, finds a match of p in S.
Following pseudocode computes the matching component of KMP algorithm:
KMP-Matcher(S,p)
1 n  length[S]
2 m  length[p]
3 Π  Compute-Prefix-Function(p)
4 q  0 //number of characters matched
5 for i  1 to n //scan S from left to right
6 do while q > 0 and p[q+1] != S[i]
7 do q  Π[q] //next character does not match
8 if p[q+1] = S[i]
9 then q  q + 1 //next character matches
10 if q = m //is all of p matched?
11 then print “Pattern occurs with shift” i – m
12 q  Π[ q] // look for the next match
Note: KMP finds every occurrence of a ‘p’ in ‘S’. That is why KMP does not terminate in step 12, rather it searches
remainder of ‘S’ for any more occurrences of ‘p’.

Illustration: given a String ‘S’ and pattern ‘p’ as follows:
S
bb aa cc bb aa bb aa bb aa bb aa cc aa cc aa
aa bb aa bb aa cc aap
Let us execute the KMP algorithm to find
whether ‘p’ occurs in ‘S’.
For ‘p’ the prefix function, Π was computed previously and is as follows:
qq 11 22 33 44 55 66 77
pp aa bb AA bb aa cc aa
ΠΠ 00 00 11 22 33 11 11

bb aa cc bb aa bb aa bb aa bb aa cc aa aa bb
aa bb aa bb aa cc aa
Initially: n = size of S = 15;
m = size of p = 7
Step 1: i = 1, q = 0
comparing p[1] with S[1]
S
p
P[1] does not match with S[1]. ‘p’ will be shifted one position to the right.
S
p
Step 2: i = 2, q = 0
comparing p[1] with S[2]
P[1] matches S[2]. Since there is a match, p is not shifted.

Step 3: i = 3, q = 1
Comparing p[2] with S[3]
S
aa bb aa bb aa cc aap
S
p
S
p
p[2] does not match with S[3]
Backtracking on p, comparing p[1] and S[3]
Step 4: i = 4, q = 0
comparing p[1] with S[4] p[1] does not match with S[4]
Step 5: i = 5, q = 0
comparing p[1] with S[5] p[1] matches with S[5]

Step 6: i = 6, q = 1Step 6: i = 6, q = 1
S
p
Comparing p[2] with S[6] p[2] matches with S[6]
S
p
Step 7: i = 7, q = 2Step 7: i = 7, q = 2
Step 8: i = 8, q = 3Step 8: i = 8, q = 3
S
p

Step 9: i = 9, q = 4Step 9: i = 9, q = 4
Step 10: i = 10, q = 5Step 10: i = 10, q = 5
Step 11: i = 11, q = 4Step 11: i = 11, q = 4
S
S
S
p
p
p
p[6] doesn’t match with S[10]
Backtracking on p, comparing p[4] with S[10] because after mismatch q = Π[5] = 3
p[5] matches with S[9]

Step 12: i = 12, q = 5Step 12: i = 12, q = 5
S
S
p
p
Step 13: i = 13, q = 6Step 13: i = 13, q = 6
Pattern ‘p’ has been found to completely occur in string ‘S’. The total number of shifts
that took place for the match to be found are: i – m = 13 – 7 = 6 shifts.

Running - time analysis
 Compute-Prefix-Function (Π)
1 m  length[p] //’p’ pattern to be matched
2 Π[1]  0
3 k  0
4 for q  2 to m
5 do while k > 0 and p[k+1] != p[q]
6 do k  Π[k]
7 If p[k+1] = p[q]
8 then k  k +1
9 Π[q]  k
10 return Π
In the above pseudocode for computing the prefix
function, the for loop from step 4 to step 10
runs ‘m’ times. Step 1 to step 3 take constant
time. Hence the running time of compute
prefix function is Θ(m).
 KMP Matcher
1 n  length[S]
2 m  length[p]
3 Π  Compute-Prefix-Function(p)
4 q  0
5 for i  1 to n
6 do while q > 0 and p[q+1] != S[i]
7 do q  Π[q]
8 if p[q+1] = S[i]
9 then q  q + 1
10 if q = m
11 then print “Pattern occurs with shift” i – m
12 q  Π[ q]
The for loop beginning in step 5 runs ‘n’ times, i.e., as
long as the length of the string ‘S’. Since step 1 to step
4 take constant time, the running time is dominated
by this for loop. Thus running time of matching
function is Θ(n).

KMP Pattern Matching algorithm

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