Simple Algorithm complexity

Question

I have an algorithm and I need help finding the complexity of it (tightest possible upper bound)

for(int i = 0; i < n/2; i++)
    for(int j = 0; j < n/4; j++)
        for(int k = 0; k < n; k++)
            x++;

My analysis is that if n would not be divided in each for loop, it would be O(n^3). This complexity still holds true, since each "for loop" will reduce each operation to a O(log n) complexity since it divides n every time the loop executes, making it smaller and smaller (smaller than O(n) at least).

I would say that the answer is between O(log n) and O(n^3). Could you help me getting the tightest possible bound?

`n` is never modified => complexity `Θ(n * (n/4) * (n/2)) = Θ(n³)` — fabian, May 12 '15 at 11:34

score 3 · Accepted Answer · answered May 12 '15 at 11:36

3

start with inner loop :

for(int k = 0; k < n; k++)
    x++;

is obviously O(n).

now one layer above that :

for(int j = 0; j < n/4; j++)

is O(n) because it takes n/4 for j to reach the n and we know that O(n/4) = O(n)

and like this for outer loop is O(n).so the complexity is O(n^3) because you have three nested loop each with O(n) and non of them have effect on each other.

answered May 12 '15 at 11:36

Lrrr

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Ok! So the tightest bound possible for this algorithm is indeed O(n^3) ? – idelara May 12 '15 at 12:10

score 3 · Answer 2 · answered May 12 '15 at 12:32

3

Assume each step takes time C. For k-loop, time taken is Cn. For j-loop, time taken to complete iteration is (Cn)n/4=C(n^2)/4 For i-loop, time taken to complete iteration is (C*(n^2)/4)n/2=C(n^3)/8

So Total time taken=(C/8)*(n^3)

As C/8 is a constant it can be ignored when considering Big-O Notation. Thus, Time Complexity=O(n^3).

answered May 12 '15 at 12:32

Lucifer

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my answer in words is urs – pola sai ram May 12 '15 at 12:36

pola sai ram · Answer 3 · 2015-05-12T12:12:35.017

2

for(int i = 0; i < n/2; i++)    --> n/2
    for(int j = 0; j < n/4; j++)  --> n/4
        for(int k = 0; k < n; k++)  --> n
            x++;

Hence total complexity is O((n^3)/8) which is O(n^3)

edited May 12 '15 at 12:12

answered May 12 '15 at 11:33

pola sai ram

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How do you get from O(n^3/8) to O(n^3)? – idelara May 12 '15 at 12:11
8 is a constant , and can be ignored – pola sai ram May 12 '15 at 12:12

Simple Algorithm complexity

3 Answers3