What is the time complexity of this nested for loop?

Question

What would be the time complexity be of these lines of code?

Begin
  sum = 0
  For i = 1 to n do
     For j = i to n do
       sum = sum + 1
End

I want to say O(n) = 2n^2 + 1 but I'm unsure because of the i=j part.

For O() you get rid of all non-leading polynomial terms (+1) and multiplicative constants (2*), so you're talking about O(n^2). — user14215102
– user14215102, Commented Nov 10, 2020 at 20:15

TheManbunlorian · Accepted Answer · 2020-11-10 20:40:15Z

1

Your answer is correct!

A nested loop instinctively leads you to the right answer which is O(n^2). In doing Big-O analysis, it usually doesn't matter being specific to the point i.e. saying the time complexity is O(2n^2) or O(3n^2 + 1) -- saying O(n^2) is enough since that is the dominating task of the function.

The i = j condition simply makes it so that there are...

i=1: n operations
i=2: (n-1) operations
...
i=n: 1 operation

So, the sum of all operations you do is 1 + 2 + ... n = n(n+1)/2 which is O(n^2).

edited Nov 10, 2020 at 20:40

answered Nov 10, 2020 at 20:21

TheManbunlorian

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