Question

Find a) overall run time T(n) and b) worst case run timeO(N) of this recursive function.

int mystery (int n)

{

if (n<= 1)

{return 1

}

else

{

return mystery(n-1) + mystery(n-2);

}

}

Answer 1

a)
recurrence relation
overall runtime is T(n)
T(n) = T(n-1) + T(n-2) + O(1)

b)
worst-case running time is O(2^n)

Explanation:
----------------
T(n)
= T(n-1) + T(n-2) + 1
<= 2T(n-1) + 1
= 2(2T(n-2) + 1) + 1
= 2(2(2T(n-3) + 1) + 1) + 1
= 2^3T(n-3) + 2^2 + 2^1 + 1
= 2^nT(1) + 2^n-1 + ... + 1
= 2^n + 2^n-1 + ... + 1
= 2^(n+1) - 1
so, T(n) = O(2^n)

hence, worst-case running time is O(2^n)