# Searching for Patterns | Set 3 (Rabin-Karp Algorithm)

Given a text txt[0..n-1] and a pattern pat[0..m-1], write a function search(char pat[], char txt[]) that prints all occurrences of pat[] in txt[]. You may assume that n > m.

Examples:
1) Input:

```  txt[] =  "THIS IS A TEST TEXT"
pat[] = "TEST"```

Output:

Pattern found at index 10

2) Input:

```  txt[] =  "AABAACAADAABAAABAA"
pat[] = "AABA"
```

Output:

```   Pattern found at index 0
Pattern found at index 9
Pattern found at index 13
```

The Naive String Matching algorithm slides the pattern one by one. After each slide, it one by one checks characters at the current shift and if all characters match then prints the match.
Like the Naive Algorithm, Rabin-Karp algorithm also slides the pattern one by one. But unlike the Naive algorithm, Rabin Karp algorithm matches the hash value of the pattern with the hash value of current substring of text, and if the hash values match then only it starts matching individual characters. So Rabin Karp algorithm needs to calculate hash values for following strings.

1) Pattern itself.
2) All the substrings of text of length m.

Since we need to efficiently calculate hash values for all the substrings of size m of text, we must have a hash function which has following property.
Hash at the next shift must be efficiently computable from the current hash value and next character in text or we can say hash(txt[s+1 .. s+m]) must be efficiently computable from hash(txt[s .. s+m-1]) and txt[s+m] i.e., hash(txt[s+1 .. s+m])= rehash(txt[s+m], hash(txt[s .. s+m-1]) and rehash must be O(1) operation.

The hash function suggested by Rabin and Karp calculates an integer value. The integer value for a string is numeric value of a string. For example, if all possible characters are from 1 to 10, the numeric value of “122” will be 122. The number of possible characters is higher than 10 (256 in general) and pattern length can be large. So the numeric values cannot be practically stored as an integer. Therefore, the numeric value is calculated using modular arithmetic to make sure that the hash values can be stored in an integer variable (can fit in memory words). To do rehashing, we need to take off the most significant digit and add the new least significant digit for in hash value. Rehashing is done using the following formula.

hash( txt[s+1 .. s+m] ) = d ( hash( txt[s .. s+m-1]) – txt[s]*h ) + txt[s + m] ) mod q

hash( txt[s .. s+m-1] ) : Hash value at shift s.
hash( txt[s+1 .. s+m] ) : Hash value at next shift (or shift s+1)
d: Number of characters in the alphabet
q: A prime number
h: d^(m-1)

## C/C++

```/* Following program is a C implementation of Rabin Karp
Algorithm given in the CLRS book */
#include<stdio.h>
#include<string.h>

// d is the number of characters in input alphabet
#define d 256

/* pat -> pattern
txt -> text
q -> A prime number
*/
void search(char pat[], char txt[], int q)
{
int M = strlen(pat);
int N = strlen(txt);
int i, j;
int p = 0; // hash value for pattern
int t = 0; // hash value for txt
int h = 1;

// The value of h would be "pow(d, M-1)%q"
for (i = 0; i < M-1; i++)
h = (h*d)%q;

// Calculate the hash value of pattern and first
// window of text
for (i = 0; i < M; i++)
{
p = (d*p + pat[i])%q;
t = (d*t + txt[i])%q;
}

// Slide the pattern over text one by one
for (i = 0; i <= N - M; i++)
{

// Check the hash values of current window of text
// and pattern. If the hash values match then only
// check for characters on by one
if ( p == t )
{
/* Check for characters one by one */
for (j = 0; j < M; j++)
{
if (txt[i+j] != pat[j])
break;
}

// if p == t and pat[0...M-1] = txt[i, i+1, ...i+M-1]
if (j == M)
printf("Pattern found at index %d \n", i);
}

// Calculate hash value for next window of text: Remove
if ( i < N-M )
{
t = (d*(t - txt[i]*h) + txt[i+M])%q;

// We might get negative value of t, converting it
// to positive
if (t < 0)
t = (t + q);
}
}
}

/* Driver program to test above function */
int main()
{
char txt[] = "GEEKS FOR GEEKS";
char pat[] = "GEEK";
int q = 101; // A prime number
search(pat, txt, q);
return 0;
}
```

## Python

```# Following program is the python implementation of
# Rabin Karp Algorithm given in CLRS book

# d is the number of characters in input alphabet
d = 256

# pat  -> pattern
# txt  -> text
# q    -> A prime number

def search(pat, txt, q):
M = len(pat)
N = len(txt)
i = 0
j = 0
p = 0    # hash value for pattern
t = 0    # hash value for txt
h = 1

# The value of h would be "pow(d, M-1)%q"
for i in xrange(M-1):
h = (h*d)%q

# Calculate the hash value of pattern and first window
# of text
for i in xrange(M):
p = (d*p + ord(pat[i]))%q
t = (d*t + ord(txt[i]))%q

# Slide the pattern over text one by one
for i in xrange(N-M+1):
# Check the hash values of current window of text and
# pattern if the hash values match then only check
# for characters on by one
if p==t:
# Check for characters one by one
for j in xrange(M):
if txt[i+j] != pat[j]:
break

j+=1
# if p == t and pat[0...M-1] = txt[i, i+1, ...i+M-1]
if j==M:
print "Pattern found at index " + str(i)

# Calculate hash value for next window of text: Remove
if i < N-M:
t = (d*(t-ord(txt[i])*h) + ord(txt[i+M]))%q

# We might get negative values of t, converting it to
# positive
if t < 0:
t = t+q

# Driver program to test the above function
txt = "GEEKS FOR GEEKS"
pat = "GEEK"
q = 101 # A prime number
search(pat,txt,q)

# This code is contributed by Bhavya Jain
```

Output:
```Pattern found at index 0
Pattern found at index 10```

The average and best case running time of the Rabin-Karp algorithm is O(n+m), but its worst-case time is O(nm). Worst case of Rabin-Karp algorithm occurs when all characters of pattern and text are same as the hash values of all the substrings of txt[] match with hash value of pat[]. For example pat[] = “AAA” and txt[] = “AAAAAAA”.

References:

http://net.pku.edu.cn/~course/cs101/2007/resource/Intro2Algorithm/book6/chap34.htm

http://www.cs.princeton.edu/courses/archive/fall04/cos226/lectures/string.4up.pdf

http://en.wikipedia.org/wiki/Rabin-Karp_string_search_algorithm

• Prateek Surana

Nice explanation
``` a.out: main.c search.c gcc -o a.out main.c main.h search.c clean: rm -vf a.out *~ ```

• MM

In the code, I don’t see the variable “h” being used anywhere. What’s the use of the below lines in the program?

// The value of h would be “pow(d, M-1)%q”

for (i = 0; i < M-1; i++)

h = (h*d)%q;

• RAHUL KUMAR

chk this line of code….
t = (d*(t – txt[i]*h) + txt[i+M])%q;

• renu

why are we choosing ‘q’ to be a prime number?and also in hashing we always chose prime number of buckets.why it is so?can someone please reply?

• RAHUL KUMAR

to calculate the mod value we need prime number .. go through ur basic u will come to know why only prime… good luck

• lizard

This part in the code is wrong…

``` ```
// Calculate the hash value of pattern and first window of text
for (i = 0; i < M; i++)
{
p = (d*p + pat[i])%q;
t = (d*t + txt[i])%q;
}
``` ```

This should be….

``` ```
// Calculate the hash value of pattern and first window of text
for (i = M-1; i >=0; i--)
{
p = (d*p + pat[i])%q;
t = (d*t + txt[i])%q;
}
``` ```

Please go through this for my say,
http://en.wikipedia.org/wiki/Rabin%E2%80%93Karp_algorithm#Hash_function_used

• kartik

It looks correct to me.

Consider the strings as strings of digits. When you want to calculate integer value of say “2314”, you start from from first digit.
p = 2
p = 2*10 + 3
p = 23*10 + 1
p = 231*10 + 4
p = 2314

• lizard

Thanks….got it..

• abhishek08aug

Intelligent 😀

• Vibhu Tiwari

This is the source code for pattern searching in much less effort with the time complexity of O(n).You can check it for various strings by passing the lengths of the two strings to be matched.The statement pattern match gets printed the number of times that substring occurs in the string.
#include
#include
void patternsearch(char *a,char *b,int n,int m)
{ int k,count=0,j=0,i=0,c=0;
while(i!=n)
{ if(j==m)
{j=0;
c=c+1;count=0;
i=c;}
k=a[i]-b[j];
if(k==0){
count++;}
if(count==m)
{printf(“Pattern Match found\n”);}
i=i+1;
j=j+1;
}
}
main()
{ char *a=”ABABABCABABABCABABABC”;
char *b=”ABABCA”;
patternsearch(a,b,21,6);
getch();
}

• Castle Age

How? if the a pattern not found, i starts from the C+1. The worst case is m * (n-m -1).

complexity is not n in the following case:
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAB.
AAAAAAAB

• Atri

We can save the string matching by using multiple hashing.For e.g. Use one hash function to calculate one hash value,and pass the first hash value as input to a second(different) hash value.If the values given by the second hash function are the same for two strings,we can safely assume that the two strings are same.This can be improved with introducing more hash functions.

• NitHish Divakar

This can only give probablistic matching no matter howmany hash function you use….ofcourse more hashing=> probablity is high…which is acceptable for most practical applications

• sparco

@sandeep

Can u explain the modular arithmetic in calculation of next range’s hash value using the value of h

``` ```

// The value of h would be "pow(d, M-1)%q"
for (i = 0; i < M-1; i++)
h = (h*d)%q;

t = (d*(t - txt[i]*h) + txt[i+M])%q;

``` ```
• Ishara
``` ```
Great Article! Simple & easy to understand. Thx.
``` ```
• Abhinav Kumar

In Rabin Karp Algo we can modify it with the initial and last character checking in the main string for the pattern at index i and then if both are same then we check the chars in between otherwise we increment i.
This reduces the computation involves in hashing.

• ricky

@geeksforgeek….Most of The program posted by you is not running when i pasted it on ideone online complier every program is saying so please try to take care of formatting & re-post it again ..hope u got my point

prog.c:3: error: stray ‘\302’ in program
prog.c:3: error: stray ‘\240’ in program
prog.c:6: error: stray ‘\302’ in program
prog.c:6: error: stray ‘\240’ in program
prog.c:9: error: stray ‘\302’ in program
prog.c:9: error: stray ‘\240’ in program
& so on

• shubh
• kp101090

Its Awesome !! Thanks for the Link .

• Anuj Jindal

http://courses.csail.mit.edu/6.006/spring11/rec/rec06.pdf

h(Si+1) = [(h(Si) (10^5
 ?rst digit of Si))  10 + next digit after Si
] mod m

SHOULD BE
h(Si+1) = [(h(Si) (10^4
 ?rst digit of Si))  10 + next digit after Si
] mod m

correct me if I am wrong..

• shek8034

Nice Explanation.. Thanks for the link

``` ```
/* Paste your code here (You may delete these lines if not writing code) */
``` ```
• hari6988

The integer value for a string is just multiplication of the ASCII values of all characters of the string

But, in the code, this is not done…can u explain how u calculate the integer hash value . what does the number d=256 stand for ?

• Sandeep

@hari6988: As mentioned in the post, the multiplication is done using modular arithmetic.

The following loop calculates hash values (or modular multiplication) for the pattern and for the first substring of size m of text.

``` ```
// Calculate the hash value of pattern and first window of text
for (i = 0; i < M; i++)
{
p = (d*p + pat[i])%q;
t = (d*t + txt[i])%q;
}
``` ```

And the following line of code calculates hash values (or modular multiplication) for other substrings of size m of text

``` ```
t = (d*(t - txt[i]*h) + txt[i+M])%q;
``` ```

d is the number of possible characters in pattern and text. If we consider the ASCII character set then its value is 256.

• martin

@sandeep..can you also please post the The Boyer-Moore string searching Algorithm..??

• martin

@sandeep ..also can you please post the comparison between all string searching algo till you have posted ..???

• sparco

Example)

String bacbabababacaab
Pattern ababaca

LPS Array
0123456
ababaca
0012301 lps[4]=3=>pattern[4]=String[9]
By using same criteria to form lps array , the last matched sub-pattern in pattern is known . So avoiding backtracking to the first element.

i=10 j=4
bacbabababa|caab
ababa|ca

Pattern found at i – len(pattern)

• sparco