Question

3. (10 marks) a) (5 marks) For a given string S = p1p2...pn, the definition for next(i), 2 ≤ i ≤ n, is as follows: next(i) = the maximum j (0 < j < i − 1) such that p1p2...pj = pi−jpi−j+1...pi−1, 0 if no such j exists. (next(1) is defined as -1.) Compute the next function for string “abababc”.

3. (10 marks) a) (5 marks) For a given string S = P1P2...Pn, the definition for next(i), 2 si s n, is as follows: next(i) = the maximum j (0 < j < < – 1) such that P1P2...P; = Pi-jPi-j+1... Pi-1, O if no such j exists. (next(1) is defined as -1.) Compute the next function for string “abababc”.

Answer 1

ANSWER:

GIVEN THAT:

Given string S = p1p2...pn, the definition for next(i), 2 ≤ i ≤ n:

1. Given String is abababc and let thier indices be 1234567 , the string will be as below

9

2. Here S = p1p2...pn where p1=a , p2=b, p3=a, p4=b, p5=a, p6=b, p7=c

Given next(1) = -1

3. next(2) = 0 (as here 0<j<1 and we cant find any j)

4. next(3) = 0 (as here 0<j<2 and we will get i=3 and j=1 and equation is p1!=p2(a!=b) hence next(3) is 0)

5. next(4) = 1 (as here 0<j<3 we will get i=4 and j=1 and equation is p1=p3(a=a) and other combination is i=4 and j=2 p1p2!=p2p3(ab != ba) hence j=1)

6. next(5) = 2 (as here 0<j<4 we will get i=5 and j=1 and equation is p1!=p4(a=b) and other combination is i=5 and j=2 p1p2=p3p4(ab = ab) and other combination is i=5 and j=3 p1p2p3!=p2p3p4(aba = bab) hence j=2)

7. next(6) = 3 (as here 0<j<5 we will get i=6 and j=1 and equation is p1=p5(a=a) and other combination is i=6 and j=2 p1p2!=p4p5(ab = ba) and other combination is i=6 and j=3 p1p2p3=p3p4p5(aba = aba) and other combination is i=6 and j=4 p1p2p3p4!=p2p3p4p5(ababa=baba) hence j=3)

8. next(7) = 4 (as here 0<j<6 we will get i=7 and j=1 and equation is p1!=p6(a=b) and other combination is i=7 and j=2 p1p2=p5p6(ab = ab) and other combination is i=7 and j=3 p1p2p3!=p4p5p6(aba = bab) and other combination i=7 and j=4 p1p2p3p4=p3p4p5p6(abab=abab) and other combination i=7 and j=5 p1p2p3p4p5=p2p3p4p5p6(abab=babab) hence j=4)

So

9. next function of the string abababc is -1001234