P problem NP problem, Study notes of Algorithms and Programming

Find whether P is present

in T or not

Naïve (Brute-Force) Algorithm:

• Check for pattern starting at each text position -- Recursive Formulation (naiveMatch_rec) -- Iterative Approach (naiveMatch_itr)

Algorithm naiveMatch_rec (T[ ], N, P[ ], M) if (N < M) then return 0; else if (M == -1) then return 1; else if (T[N] == P[M]) then return (naiveMatchRec (T, N-1, P, M-1)); else return (naiveMatchRec (T, N-1, P, M));

Rabin-Karp Algorithm:

Let  = {0,1,2,.. .,9}.

We can view a string of k consecutive characters as representing a length-k

decimal number.

Let p denote the decimal number for P[1..m]

Let ts denote the decimal value of the length-m substring T[s+1..s+m] of T[1..n]

for s = 0, 1,.. ., n-m.

ts = p if and only if

T[s+1..s+m] = P[1..m], and s is a valid shift.

p = P[m] + 10(P[m-1] +10(P[m-2]+... +10(P[2]+10(P[1]))

We can compute p in O(m) time.

Similarly we can compute t 0 from T[1..m] in O(m) time.

Rabin-Karp Algorithm:

p = P[m] + 10(P[m-1] +10(P[m-2]+... +10(P[2]+10(P[1]))

m =

Rabin-Karp Algorithm:

Computation of p and t 0 and the recurrence is done using modulus q.

In general, with a d-ary alphabet {0,1,…,d-1}, q is chosen such that d  q fits within a

computer word.

The recurrence equation can be rewritten as ts+1 = (d(ts –T[s+1]h)+ T[s+m+1]) mod q, where h = dm-1(mod q) is the value of the digit “1” in the high order position of an m- digit text window.

Note that ts  p mod q does not imply that ts = p.

However, if ts is not equivalent to p mod q ,

then ts  p, and the shift s is invalid.

We use ts  p mod q as a fast heuristic test to rule out the invalid shifts.

Further testing is done to eliminate spurious hits.

• an explicit test to check whether P[1..m] = T[s+1..s+m]

Rabin-Karp Algorithm:

ts+1 = (d(ts –T[s+1]h)+ T[s+m+1]) mod q h = dm-1(mod q) Example :

T = 31415; P = 26, n = 5, m = 2, q = 11

p = 26 mod 11 = 4 t0 = 31 mod 11 = 9 t1 = (10(9 - 3(10) mod 11 ) + 4) mod 11 = (10 (9- 8) + 4) mod 11 = 14 mod 11 = 3

Rabin-Karp Algorithm:

Figure: The same text string with values computed modulo 13 for each possible position of a length-5 window. Assuming the pattern P = 31415, we look for windows whose value modulo 13 is 7, since 31415 7 (mod 13). Two such windows are found, shown shaded in the figure. The first, beginning at text position 7, is indeed an occurrence of the pattern, while the second, beginning at text position 13, is a spurious hit.

ts+1 = (d(ts - T[s + 1]h) + T[s + m + 1]) mod q

Rabin-Karp Algorithm:

Procedure RABIN-KARP-MATCHER(T, P, d, q)

Input : Text T, pattern P, radix d ( which is typically = ), and the

prime q. Output : valid shifts s where P matches

1. n  length[T];
2. m  length[P];
3. h  dm-1^ mod q;
4. p  0;
5. t 0  0;
6. for i  1 to m
7. do p  (d p + P[i] mod q;
8. t 0  (d t 0 +T[i] mod q;
9. for s  0 to n-m
10. do if p = ts
11. then if P[1..m] = T[s+1..s+m]
12. then “pattern occurs with shift ‘s’
13. if s < n-m
14. then ts+1  (d(ts –T[s+1]h)+ T[s+m+1]) mod q;

The loop of line 9 takes 𝜽((n-m+1)m) time The loop 6-8 takes O(m) time The overall running time is O((n-m)m)