How do I find the Big O Notation for this for-loop line of code
for (int j = 0; pow(j,2) < n; j++) ?
Does anyone know?
I have read a little on Big O Notation and it’s a very confusing topic to understand. I know that usually for-loop like this one → for (int n = 0; n < 20; ++n), have a Big O notation of O(1), as input increases by 13 so does its output, with linear complexity. Is that the same situation as above?
A loop like this:
for (int i = 0; i < n; ++i) {
doSomething(i);
}
iterates n times, so if doSomething has O(1) running time, then the loop as a whole has O(n) running time.
Similarly, a loop like this:
for (int j = 0; pow(j, 2) < n; j++) {
doSomething(j);
}
iterates ⌈√n⌉ times, so if doSomething has O(1) running time, then the loop as a whole has O(√n) running time.
By the way, note that although pow(j, 2) is O(1) running time — so it doesn't affect your loop's asymptotic complexity — it's nonetheless pretty slow, because it involves logarithms and exponentiation. For most purposes, I'd recommend this instead:
for (int j = 0; j * j < n; j++) {
doSomething(j);
}
or perhaps this:
for (int j = 0; 1.0 * j * j < n; j++) {
doSomething(j);
}
Related
If we assume the statements inside of a for loop is O(1).
for (i = 0; i < N; i++) {
sequence of statements
}
Then the above time complexity should be o(n)
another example is:
for (i = 0; i < N; i++) {
for (j = i+1; j < N; j++) {
sequence of statements
}
}
the above time complexity should be o(n^2)
It seems like one for loop symbolize n times
However, I've seen some discussion said that it's not simply like this
Sometimes there are 2 for loops but it doesn't mean the time complexity must be o(n^2)
Can somebody give me an example of two or more for loops with only O(n) time complexity (with explanation will be much better)???
It occurs when either the outer or inner loop is bounded by a fixed limit (constant value) on iterations. It happens a lot in many problems, like bit manipulation. An example like this:
for (int i = 0; i < n; i++) {
int x = ...; //some constant time operations to determine int x
for (int j = 0; x != 0; j++) {
x >>>= 1;
}
}
The inner loop is limited by the number of bits for int. For Java, it is 32, so the inner loop is strictly limited by 32 iterations and is constant time. The complexity is linear time or O(n).
Loops can have fixed time if the bounds are fixed. So for these nested loops, where k is a constant:
// k is a constant
for (int i=0; i<N; i++) {
for (int j=0; j<k; j++) {
}
}
// or this
for (int i=0; i<k; i++) {
for (int j=0; j<N; j++) {
}
}
Are both O(kN) ~= O(N)
The code in the post has a time complexity of:
Given this is a second order polynomial, (N^2 - N)/2, you correctly assessed the asymptotic time complexity of your example as O(N^2).
I have some trouble finding the time complexity of the code below. I figured that the if statement will run for approximately n times; however, I could not manage to describe it mathematically. Thanks in advance.
int sum = 0;
for (int i = 1; i < n; i++) {
for (int j = 1 ; j < i*i; j++) {
if (j % i == 0) {
for (int k = 0; k < j; k++) {
sum++;
}
}
}
}
Outer loop
Well, it's clear that it's O(n) because i is bounded by n.
Inner loops
If we take a look at the second loop alone, then it looks as follows:
...
for (int j = 1 ; j < i*i; j++){
...
j is bounded by i*i or simply n^2.
However, the innermost loop won't be executed for every j, but only for js that are divisible by i because that's what the constraint j % i == 0 means. Since j ~ i*i, there will be only i cases, when the innermost loop is executed. So, the number of iterations in the inner loops is bounded by i^3 or simply n^3.
Result
Hence, the overall time complexity is O(n4).
for the following code:
for(i=0;i<5;i++)
for(j=2;j<n;j++)
{
c[i][j]=0;
for(k=0;k<n;k++)
c[i][j]=a[i][k]*b[k][j];
}
I would say the run time is theta(n^3), as I see in the k loop, there is two n (n^3) and on the other loop, another n, making it n^3. Would this be right or what did I miss.
Thank you
Here is your code, formatted:
for (i=0; i < 5; i++) {
for (j=2; j < n; j++) {
c[i][j] = 0;
for (k=0; k < n;k++)
c[i][j] = a[i][k]*b[k][j];
}
}
The outer loop in i only iterates 5 times, and so can just be treated as a constant penalty as far as complexity is concerned. The inner two loops in j and k are independent of each other, and both are O(n). We can therefore just multiple the complexities to get O(n^2) for the overall running time as a function of n.
I was asked to find the big-O and Big-Omega if I know that function f has O(log(n)), Ω(1) and function g has O(n), Ω((log(n))^2)
for (int i = n; i >= 0; i/=2)
if (f(i) <= g(i))
for (int j = f(i)*f(i); j < n; j++)
f(j);
The big problem that I have is that I don't know how to incorporate the complexity of the funstions in the calculation. I mean I know how to calculate the complexity of loops that looks like this:
for(int i =0 ; i< n*2; i++) {
....
}
or like this
for(int i = 0; i < n; i++) {
for(int j = 0; j < n; j++) {
}
}
Thank you in advance.
This is what I've tried:
for (int i = n; i >= 0; i/=2)// this is aproximatly O(log(n))
if (f(i) <= g(i))// because O(n) < O(log(n)) this line is O(n)
for (int j = f(i)*f(i); j < n; j++)// O(n*log(n))
f(j);// O(log(n))
So by my calculation I get O(log(n)*n *n *log(n)*log(n))=O(n^2*log^3(n))
This is tricky question, because the loop execution depends on values, returned by functions f and g. However remember, that you need to estimate the worst case - so you need to assume two things:
f(i) <= g(i) is always true, so the internal loop always executes
the internal loop starts from 0, because it's a minimal value, which you can get as a result of squaring the f(i) value
So, your piece of code becomes much simpler:
for (int i = n; i >= 0; i/=2)
{
f(i);
g(i);
f(i);
f(i);
for (int j = 0; j < n; j++)
f(j);
}
I think you can take over from here.
I have been searching through the forums on big O notation and learned quite a bit. My problem is pretty specific and I think a unique case will better help me understand big O, I am ignoring constants.
To my understanding if a loop goes through all elements than it's O(n).
for(int i = 0; i < n; i++)
{
}
If a loop goes through all of n, inside another loop that goes through all of n, it's multiplied n * n = n^2
for(int i = 0; i < n; i++)
{
for(int j = 0; j < n; j++)
{
}
}
Lastly if a loop is followed by another loop that goes through all elements it is n + n = 2n
for(int j = 0; j < n; j++)
{
}
for(int k = 0; k < n; k++)
{
}
My question directly proceeds these lines of code
for(int i = 0; i < n; i++)
{
for(int j = 0; j < n; j++)
{
}
for(int k = 0; k < n; k++)
{
}
for(int l = 0; l < n; l++)
{
for(int m = 0; m < n; m++)
{
}
}
}
So based on the rules above I am calculating big O to be n * (n + n + n * n), which is n^3 + 2n^2. So would that make my big O(n^3) or would my big O be O(n^3 +2n^2). Am I going about this all wrong? Or am I somewhere close in the ballpark? Mainly I'm trying to figure out if these loops would be less than O(n^4). Thanks in advance.
The big-O notation is used to characterize the asymptotic behavior of an algorithm depending on some value n that indicates the data volume, but independent of any constant, e.g. processor speed.
In your example, n^3 grown faster than 2n^2, i.e., for large n, 2n^2 can be neglected compared to n^3. The asymptotic behavior of your nested loops thus have order O(n^3).