Question

For the following code, figure out the recurrences (base case and recursive step), calculate the running time and establish the order of growth. mystery(n): if n == 1 then println("A"); else mystery(n/2); println("B") mystery(n/2); end if

Answer 1

mystery(n):
if n == 1 then
   println("A");
else
   mystery(n/2);
   println("B");
   mystery(n/2);
end if

ans)

base case:
if n == 1 then
   println("A");

recursive case :
else
   mystery(n/2);
   println("B");
   mystery(n/2);

computing time complexity:
   Recurrence relation of the above recursive function : T(n) = 2T(n/2) + 1, T(1) = 1

T(n) = 2T(n/2) + 1 ---- 1
T(n/2) = 2T(n/2^2) + 1 ---- 2

substitue 2 in 1

T(n) = 2^2T(n/2^2) + 2^1 + 2^0 ---- 3
T(n/2^2) = 2 T(n/2^3) + 1 ----- 4

substitue 4 in 3

T(n) = 2^3 T(n/2^3) + 2^2 + 2^1 + 2^0
.
.
.
.
T(n) = 2^k T(n/2^k) + (2^0 + 2^1 + 2^2 + .... + 2^(k-1))

assume 2^k = n ==> k = log n

So T(n) = 2^k T(1) + (2^0 + 2^1 + 2^2 + .... + 2^(k-1))
= 2^k + (2^0 + 2^1 + 2^2 + .... + 2^(k-1))
= n + (2^k - 1)
= n + n - 1
= 2n
T(n)= O(n)