Python: Rabin-Karp algorithm hashing

Question

I am implementing Rabin-Karp algorithm for fun. I came across this pseudocode:

    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 t= 0
    6 for i = 1 to m
    / preprocessing
    /
    7 p = (dp + P [i]) mod q
    8 t = (dt + T [i]) mod q
    9 for s = 0 to n-m
    / matching
    /
    10     if p == t
    11         if P [1... m] == T [s + 1...s + m]
    12             print “Pattern occurs with shift” s
    13     if s < n-m
    14         t  = (d(t-T[s + 1]h) + T [s + m + 1]) mod q

I implemented in Python 2.7 like so:

def Rabin_Karp_Matcher(text, pattern, d, q):
    n = len(text)
    m = len(pattern)
    h = pow(d,m-1)%q
    p = 0
    t =0
    result = []
    for i in range(m): # preprocessing
        p = (d*p+ord(pattern[i]))%q
        t = (d*t+ord(text[i]))%q
    for s in range(n-m):
        if p == t: # check character by character
            match = True
            for i in range(m):
                if pattern[i] != text[s+i]:
                    match = False
                    break
            if match:
                result = result + [s]
        if s < n-m:
                t = (d*(t-ord(text[s+1])*h)+ord(text[s+m]))%q #index out of bounds here
    return result

where result is a list containing the index in text of pattern.

My code is failing to find the 26 in 3141592653589793 and I suspect it has something to do with my hashcode defined by line 14 of the pseudocode. Can anyone please help with this.

I passed in the following paramters:

P = "26" T = "3141592653589793" d = 257 q = 11

P and T must be strings/arrays of chars

Answer 1

Here is a working version of your code:

def Rabin_Karp_Matcher(text, pattern, d, q):
    n = len(text)
    m = len(pattern)
    h = pow(d,m-1)%q
    p = 0
    t = 0
    result = []
    for i in range(m): # preprocessing
        p = (d*p+ord(pattern[i]))%q
        t = (d*t+ord(text[i]))%q
    for s in range(n-m+1): # note the +1
        if p == t: # check character by character
            match = True
            for i in range(m):
                if pattern[i] != text[s+i]:
                    match = False
                    break
            if match:
                result = result + [s]
        if s < n-m:
            t = (t-h*ord(text[s]))%q # remove letter s
            t = (t*d+ord(text[s+m]))%q # add letter s+m
            t = (t+q)%q # make sure that t >= 0
    return result
print (Rabin_Karp_Matcher ("3141592653589793", "26", 257, 11))
print (Rabin_Karp_Matcher ("xxxxx", "xx", 40999999, 999999937))

The output is:

[6]
[0, 1, 2, 3]

On the first step, you check whether text[0..m] == pattern. On the second step, you want to check whether text[1..m+1] == pattern. Thus you remove text[0] from the hash (at the moment it is multiplied by your precomputed h): t = (t-h*ord(text[s]))%q. And then, add text[m] to it: t = (t*d+ord(text[s+m]))%q. On the next step, you remove text[1] and add text[m+1], and so on. The t = (t+q)%q line is there because a negative number modulo q yields remainder in the range (-q; 0], and we want it to be in the range [0; q).

Note that you want to check a total of n-m+1 substrings, not n-m, hence the correct loop is for s in range(n-m+1). Checked by the second example (finding "xx" in "xxxxx").

Also worth noting:

1. The line h = pow(d,m-1)%q may be too slow if m is large. It is better to take the result modulo q after each of the m-2 multiplications.

2. This algorithm is still O(nm) in the worst case. With text="a"*100000 and pattern="a"*50000, it will find 50001 positions where a substring of text matches the pattern, and it will check them all character-by-character. If you expect your code to work fast in such extreme cases, you should skip the character-by-character comparison and find a way to deal with false positives (i.e., hashes are equal but strings are not). Picking a large prime number q may help reduce the probability of a false positive to an acceptable level.