When to add and when to multiply to find time complexity

Question

I am trying to learn how to determine time complexity. Sometimes, I see complexity as $\log n + \log n = 2\log n$ but sometimes I see complexity as $\log n\cdot\log n$ which is $(\log n)^2$.

Suppose I need to do two binary searches to find two different elements, in one algorithm. Is this $\log n + \log n$ time, or would it be $(\log n)^2$ time?

Answer 1

You add time complexities when you have something of the form: do operation $A$, then do operation $B$. This would be (time complexity of $A$) + (time complexity of $B$).

You multiply time complexities when you have something of the form: do operation $A$, $X$ times (eg. in a for loop). This would be $X \cdot $(time complexity of $A$).

If you perform two binary searches, each of which has time complexity $\log n$, the total runtime is $\log n + \log n$. You might see a runtime of $(\log n)^2$ if you had to perform $\log n$ binary searches.

Answer 2

Don't look for rules that you can follow. Try to understand what is actually going on and then write down the mathematics that expresses that.

You would, I hope, have no difficulty answering the following questions:

• If I have a line of $m$ bricks and a line of $n$ bricks, how many bricks do I have?
• If I want to build a wall that is $m$ bricks tall and $n$ bricks wide, how many bricks do I need?

Use the same sort of reasoning to figure out what is going on in complexity analyses.