Rabin Carp String Matching Algorithm

1. Rabin-Karp Substring search
algorithm
1
Prepared By:
Sabiya Fatima
sabiya1990fatima@gmail.com

2. Objectives
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 What is Substring search problem
 Definition of the Rabin-Karp algorithm
 How Rabin-Karp works
 An example to illustrate Rabin-Karp
 Complexity Analysis
 Real Life applications

3. What is Substring search Problem
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We assume that the text is an array T [1..N] of length n and that the pattern is an array P [1..M]
of length m, where m << n.
We also assume that the elements of P and T are characters in the finite alphabet S.
(e.g., S = {a,b} We want to find P = ‘aab’ in T = ‘abbaabaaaab’)

4.  A string search algorithm which compares a string's hash values, rather than the strings
themselves.
 For efficiency, the hash value of the next position in the text is easily computed from the
hash value of the current position.
Definition of the Rabin-Karp Algorithm
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5. How Rabin-Karp Works
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 Let characters in both arrays T and P be digits in radix-S notation. S = (0,1,...,9)
 Let p be the value of the characters in P
 Choose a prime number q such that fits within a computer word to speed
computations.
 Compute (p mod q)
 The value of p mod q is what we will be using to find all matches of the pattern P in T.

6. How Rabin-Karp Works(Contd.)
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 Compute (T[s+1, .., s+m] mod q) for s = 0 .. n-m
 Test against P only those sequences in T having the same (mod q) value
 (T[s+1, .., s+m] mod q) can be incrementally computed by subtracting the high-order digit,
shifting, adding the low-order bit, all in modulo q arithmetic.

7. Algorithm
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RABIN-KARP-MATCHER(T,P,d,q)
1. n = T.length
2. m= P.length
3. h = d^(m-1) mod q
4. p = 0
5. t0 = 0
6. for i = 1 to m // preprocessing
7. p = (dp + p[i]) mod q
8. t0 = (dt0 + p[i]) mod q
9. for s = 0 to n-m // matching
10. if p == ts
11. if P[1 . . . . M] == T[ s+1 . . . . s+m]
12. print “Pattern occurs with shift” s
13. if s<(n + m)
14. ts+1 = (d(ts – T[s+1]h)+T[s+m+1]) mod q

8. An Example to illustrate Rabin-Karp
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• Given T = 31415926535 and P = 26
• We choose q = 11
• P mod q = 26 mod 11 = 4
13 14 95 62 35 5
13 14 95 62 35 5
14 mod 11 = 3 not equal to 4
31 mod 11 = 9 not equal to 4
13 14 95 62 35 5
41 mod 11 = 8 not equal to 4

9. An Example to illustrate Rabin-Karp(contd.)
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13 14 95 62 35 5
15 mod 11 = 4 equal to 4 -> spurious hit
13 14 95 62 35 5
59 mod 11 = 4 equal to 4 -> spurious hit
13 14 95 62 35 5
92 mod 11 = 4 equal to 4 -> spurious hit
13 14 95 62 35 5
26 mod 11 = 4 equal to 4 -> an exact match!!
13 14 95 62 35 5
65 mod 11 = 10 not equal to 4

10. An Example to illustrate Rabin-Karp(contd.)
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13 14 95 62 35 5
53 mod 11 = 9 not equal to 4
13 14 95 62 35 5
35 mod 11 = 2 not equal to 4
As we can see, when a match is found, further testing is done to insure that a match has
indeed been found.

11. Complexity Analysis 11
RABIN-KARP-MATCHER(T,P,d,q)
1. n = T.length
2. m= P.length
3. h = d^(m-1) mod q O(1)
4. p = 0
5. t0 = 0
6. for i = 1 to m O(m)
7. p = (dp + p[i]) mod q
8. t0 = (dt0 + p[i]) mod q
9. for s = 0 to n-m O((n-m+1)m)
10. if p == ts
11. if P[1 . . . . M] == T[ s+1 . . . . s+m]
12. print “Pattern occurs with shift” s
13. if s<n + m
14. ts+1 = (d(ts – T[s+1]h)+T[s+m+1]) mod q

12. Complexity Analysis Result
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 The running time of the Rabin-Karp algorithm in the worst-case scenario is
O((n-m+1))m but it has a good average-case running time.
 If the expected number of valid shifts is small O(1) and the prime q is chosen to be
quite large, then the Rabin-Karp algorithm can be expected to run in time O(n+m) plus
the time to required to process spurious hits.

13. Real Time Applications
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 Bioinformatics
• Used in looking for similarities of two or more proteins; i.e. high sequence
similarity usually implies significant structural or functional similarity.
Example:
Hb A_human
GSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKL
G+ +VK+HGKKV A++++++AH+ D++ ++ +++LS+LH KL
Hb B_human
GNPKVKAHGKKVLGAFSDGLAH LDNLKGTF ATLSELH CDKL
+ similar amino acids

14. 14
 Good for plagiarism, because it can deal with multiple pattern matching!
 With a good hashing function it can be quite effective and it’s easy to implement!
Real Time Applications

15. References
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.
 Cormen, Thomas S., et al. Introduction to Algorithms. 3rd ed. Boston: MIT Press, 2
 Go2Net Website for String Matching Algorithms
 [www.go2net.com/internet/deep/1997/05/14/body.html]
 Yummy Yummy Animations Site for an animation of the Rabin-Karp algorithm at work
[www.mills.edu/ACAD_INFO/MCS/CS/S00MCS125/String.Matching.Algorithms/animations.html]
 National Institute of Standards and Technology Dictionary of Algorithms, Data Structures, and Problems
 [hissa.nist.gov/dads/HTML/rabinKarpAlgo.html]
 Multi-Pattern String Matching with Very Large Pattern Sets
 [https://www.dcc.uchile.cl/~gnavarro/workshop07/lsalmela.pdf]

16. Thank You
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