KMP Algorithm - DSA

KMP AlgorithmKnuth-Morris-Pratt

String Searching Problem

Input: A word string W and a text string S

Check if W exists as a substring of S, and if it does then return its location.

Output: The position in S at which W is found

Brute Force

S:

W:

A B C A B C A B A B A C

C A B A

Worst Case of Brute Force

S:

W:

A A A A A A A A A A A A

A A A C

A A C

Worst Case of Brute Force

S:

W:

A A A A A A A A A A A A

A A A C

A A C

If |S|=n, |W|=m then the algorithm runs in O(mn) time.

Better AlgorithmsBackward AlgorithmBoyer and Moore AlgorithmColussi AlgorithmCrochemore and Perrin AlgorithmGalil Gianardo AlgorithmGalil and Seiferas AlgorithmHorsepool Algorithm Knuth Morris and Pratt AlgorithmKMP Skip AlgorithmMax-Suffix Matching AlgorithmMorris and Pratt AlgorithmQuick Searching Algorithm

Raita AlgorithmReverse Factor AlgorithmReverse Colussi AlgorithmSelf Max-Suffix AlgorithmSimon Algorithm Skip Search AlgorithmSmith Algorithm Tuned Boyer and Moore AlgorithmTwo Way AlgorithmUniqueness Algorithm Wide Window AlgorithmZhu and Takaoka Algorithm

KMP

Linear Time Avoids comparisons with elements of S that

have already been involved in a comparison, i.e. backtracking in S never occurs

Time: O(m+n) Space: O(m+n)

KMP Differs from brute force by always keeping

track of the information that it gains from previous comparisons

A failure function or partial matching table (T) is computed which tells us how much of the last comparison can be reused if it fails

T[i]=the longest prefix of W that is also a proper suffix of W[0..i]

KMPT shows how much of the beginning of W matches up to the portion of S immediately preceding the failed comparison.

. . A B C A B C A B A .

A B C A B A

A B C A B A

No need to repeat these comparisonsResume comparing here

Sliding Window Approach

Nearly all exact string matching algorithms use the slide window approach

Whenever a mismatch is found, slide the window to the right

Suffix to Prefix RuleFor a window to have any chance to match a pattern, in some way, there must be a suffix of the window which is equal to a prefix of the pattern.

KMPT shows how much of the beginning of W matches up to the portion of S immediately preceding the failed comparison.

. . A B C A B C A B A .

A B C A B A

A B C A B A

No need to repeat these comparisonsResume comparing here

KMP example

mSWi

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

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

A B C D A D

0 1 2 3 4 5

KMP

Calculating the longest valid suffix during runtime will be very inefficient

Pre-processing can eliminate the problem, as the suffix also exists in W itself

KMP

The algorithm preprocesses the word W to produce the prefix function, which gives the number of steps the pattern can skip for every possible location of a mismatch

Components of KMP

Compute Prefix Function: For a given W, compute a table T of equal length where T[i] gives the length of the longest prefix of W that is also a proper suffix of W[0..i].

KMP Matcher Function: Actual searching.

Example of a prefix function

WT

A C A A C A C B

0


WT

A C A A C A C B

0 0


WT

A C A A C A C B

0 0 1


WT

A C A A C A C B

0 0 1 1


WT

A C A A C A C B

0 0 1 1 2


WT

A C A A C A C B

0 0 1 1 2 3


WT

A C A A C A C B

0 0 1 1 2 3 2


WT

A C A A C A C B

0 0 1 1 2 3 2 0

Example

SWT

A C B A C A A C A A C A C A A C A B

A C A A C A B

0 0 1 1 2 3 0

Matcher FunctionKMP(String S, String W):

set T to prefixFunc(W) //Compute the partial match table

set q to 0 //Candidate character of W initially 0

for every i in range 0 to n-1

while q>0 and W[q] is not equal to S[i]

set q to T[q-1] //Mismatch, backtrack if you can

if W[q] is equal to S[i]

increment q //Match, move to next character

if q is equal to m

print i-m+1 //Entire W has been found

set q to T[q-1] //Find others

Prefix FunctionprefixFunc(List W):

set T[0] to 0 //Set first element of table to 0

set k to 0 //Candidate character initially 0

for every q in range 1 to m-1

while k>0 and W[k] is not equal to W[q]

set k to T[k-1] //Mismatch, backtrack if possible

if W[k] is equal to W[q]

increment k //Match, move to next character

Set T[q] to k //Store result

return T

Runtime AnalysisAlthough the algorithm as implemented here contains a loop within a loop, it runs in linear time. This is because the backtracking statement, which essentially shifts the sliding window to the right, can only execute a maximum of n times in the entire run of the for loop. The remaining body of the for loop runs executes exactly n times itself, giving a runtime of O(n) for the matching function.Similar reasoning applies to the prefix function.

KMP Algorithm - DSA

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