Time complexity of nested for-loop

Question

I need to calculate the time complexity of the following code   for  i   1  i  lt   n  i        for j   1  j  lt   i  j              Some code         Is it O n 2

User · Answer

Yes  the time complexity of this is O n 2

User · Answer

Indeed  it is O n 2   See also a very similar example with the same runtime here

User · Answer

First we ll consider loops where the number of iterations of the inner loop is independent of the value of the outer loop s index  For example    for  i   0  i  lt  N  i           for  j   0  j  lt  M  j               sequence of statements               The outer loop executes N times  Every time the outer loop executes  the inner loop executes M times  As a result  the statements in the inner loop execute a total of N   M times  Thus  the total complexity for the two loops is O N2

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Yes  nested loops are one way to quickly get a big O notation   Typically  but not always  one loop nested in another will cause O n      Think about it  the inner loop is executed i times  for each value of i  The outer loop is executed n times   thus you see a pattern of execution like this  1   2   3   4         n times   Therefore  we can bound the number of code executions by saying it obviously executes more than n times  lower bound   but in terms of n how many times are we executing the code   Well  mathematically we can say that it will execute no more than n   times  giving us a worst case scenario and therefore our Big-Oh bound of O n      For more information on how we can mathematically say this look at the Power Series   Big-Oh doesn t always measure exactly how much work is being done  but usually gives a reliable approximation of worst case scenario     4 yrs later Edit  Because this post seems to get a fair amount of traffic  I want to more fully explain how we bound the execution to O n    using the power series  From the website  1 2 3 4    n    n     n  2   n   2   n 2  How  then are we turning this into O n     What we re  basically  saying is that n      n   2   n 2  Is this true  Let s do some simple algebra    Multiply both sides by 2 to get  2n      n     n   Expand 2n   to get n     n      n     n  Subtract n   from both sides to get  n      n    It should be clear that n      n  not strictly greater than  because of the case where n 0 or 1   assuming that n is always an integer   Actual Big O complexity is slightly different than what I just said  but this is the gist of it  In actuality  Big O complexity asks if there is a constant we can apply to one function such that it s larger than the other  for sufficiently large input  See the wikipedia page

User · Answer

On  the  1st  iteration  of  the  outer  loop   i     1    the  inner  loop  will  iterate  1  times On  the  2nd  iteration  of  the  outer  loop   i     2    the  inner  loop  will  iterate  2  time On  the  3rd  iteration  of  the  outer  loop   i     3    the  inner  loop  will  iterate  3  times      On  the  FINAL  iteration  of  the  outer  loop   i     n    the  inner  loop  will iterate  n  times  So   the  total  number  of  times  the  statements  in  the  inner  loop  will  be  executed will  be  equal  to  the  sum  of  the  integers  from  1  to  n   which  is     n  n    2      n 2  2    O n 2   times

User · Answer

A quick way to explain this is to visualize it   if both i and j are from 0 to N  it s easy to see O N 2   O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O   in this case  it s   O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O   This comes out to be 1 2 of N 2  which is still O N 2

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