What is the time complexity of it?

Question

I can calculate the time complexity of the following loop:

for (int i = 1; i < n; i++)

The time complexity of it is $O(n)$.

but I can't calculate the time complexity of the next two loops.

I can't find any hints.

for (int i = 0; i < n; i+= 2)

for (int j = 1; j < n; j*= 2)

Answer 1

Its hard to tell what the "complexity" of a loop without knowing what each iteration performs, but what we can do is count how many times each loop iterates:

for (int i = 1; i < n; i++)

You correctly stated this loop iterates $n$ times.

for (int i = 0; i < n; i+= 2)

The above loop iterates until $i \geq n$, or in other words, $k$ times, when $i+2k = n \rightarrow 0+2k = n$ (since you initiate $i=0$ and increment it by $2$). Solve the trivial equation, and you get $k=\frac{n}{2}$, which is $O(n)$ iterations.

for (int j = 1; j < n; j*= 2)

Same as the second time, we must solve $2^k = n$, since you multiply $j=1$ by $2$ each iteration. Solve and get $k=\log(n)$

Answer 2

for (int i = 0; i < n; i += 2)

has the time compexity O(n) and

for (int i = 0; i < n; i *= 2)

has the time compexity O(ln n).