Can I get some help in understanding how to solve this tutorial question! I still do not understand my professors explanation. I am unsure of how to count the big 0 for the third/most inner loop. She explains that the answer for this algorithm is O(n^2) and that the 2nd and third loop has to be seen as one loop with the big 0 of O(n). Can someone please explain to me the big O notation for the 2nd / third loop in basic layman terms
Assuming n = 2^m
for ( int i = n; i > 0; i --) {
for (int j =1; j < n; j *= 2){
for (int k =0; k < j; k++){
}
}
}
As far as I understand, the first loop has a big O notation of O(n)
Second loop = log(n)
Third loop = log (n) (since the number of times it will be looped has been reduced by logn) * 2^(2^m-1)( to represent the increase in j? )
lets add print statement to the innermost loop.
for (int j =1; j < n; j *= 2){
for (int k =0; k < j; k++){
print(1)
}
}
output for
j = 1, 1 1
j = 2, 1 1 1
j = 4, 1 1 1 1 1
...
j = n, 1 1 1 1 1 ... n+1 times.
The question boils down to how many 1s will this print.
That number is
(2^0 + 1) + (2^1 + 1) + (2^2 + 1) + ... + (n + 1)
= (2^0 + 1) + (2^1 + 1) + (2^2 + 1) + ... + (n + 1)
= log n + (1 + 2 + 4 + ... + n)
= O(log n + n)
= O(n).
assuming you know why (1 + 2 + 4 + ... + n) = O(n)
O-notation is an upperbound. You can say it has O(n^2). For least upperbound, I believe it should be O(n*log(n)*log(n)) which belongs to O(n^2).
It’s because of the logarithm. If you have log(16) raised to the power 2 is 16. So log(n) raised to the power of 2 is n. That is why your teacher says to view the second and third loop as O(n) together.
If the max iterations for the second loop are O(log(n)) then the second and third loops will be: O(1 + 2 + 3 + ... + log(n)) = O(log(n)(log(n) + 1)/2) = O((log(n)^2 + log(n))/2) = O(n)
for ( int i = n; i > 0; i --) { // This runs n times
for (int j =1; j < n; j *= 2){ // This runs atmost log(n) times, i.e m times.
for (int k =0; k < j; k++){ // This will run atmost m times, when the value of j is m.
}
}
}
Hence, the overall complexity will be the product of all three, as mentioned in the comments under the question.
Upper bound can be loose or tight.
You can say that it is loosely bound under O(n^2) or tightly bound under O(n * m^2).
Related
Hi I'm trying to analyze the time complexity of this algorithm but I'm having difficult unraveling and counting how many times the final loop will execute. I realize that the first loop is log(n) but after that I can't seem to get to a sum which evaluates well. Here is the algorithm:
for(int i = 1; i <= n; i = 2*i){
for(int j = 1; j <= i; j = 2*j){
for(int k = 0; k <= j; k++){
// Some elementary operation here.
}
}
}
I cannot figure out how many times the loop k executes in general w.r.t to n
Thanks for any help!
It is O(n).
1 + 2 + 4 + ... + 2^N == 2^(N + 1) - 1.
The last loop, for a specific j, executes j times.
And for a specific i, the inner 2 loops execute 1 + 2 + 4 + ... + i times, which is equal to about 2 * i.
So the total execution times is 1 * 2 + 2 * 2 + 4 * 2 + ... + N * 2, which is about 4 * N.
I'm working on a data structures course and I'm not sure how to proceed w/ this Big O analysis:
sum = 0;
for(i = 1; i < n; i++)
for(j = 1; j < i*i; j++)
if(j % i == 0)
for(k = 0; k < j; k++)
sum++;
My initial idea is that this is O(n^3) after reduction, because the innermost loop will only run when j/i has no remainder and the multiplication rule is inapplicable. Is my reasoning correct here?
Let's ignore the outer loop for a second here, and let's analyze it in terms of i.
The mid loop runs i^2 times, and is invoking the inner loop whenever j%i == 0, that means you run it on i, 2i, 3i, ...,i^2, and at each time you run until the relevant j, this means that the inner loop summation of running time is:
i + 2i + 3i + ... + (i-1)*i = i(1 + 2 + ... + i-1) = i* [i*(i-1)/2]
The last equality comes from sum of arithmetic progression.
The above is in O(i^3).
repeat this to the outer loop which runs from 1 to n and you will get running time of O(n^4), since you actually have:
C*1^3 + C*2^3 + ... + C*(n-1)^3 = C*(1^3 + 2^3 + ... + (n-1)^3) =
= C/4 * (n^4 - 2n^3 + n^2)
The last equation comes from sum of cubes
And the above is in O(n^4), which is your complexity.
Where n is the input to the function can be any integer.
i = n, total = 0;
while (i > 0) {
for (j=0; j<i; j++)
for (k=0; k<i; k++)
total++;
i = i/4;
}
What is the time complexity of this function?
One way to think about this is to look at the loops independently.
This inner loop nest:
for (j=0; j<i; j++)
for (k=0; k<i; k++)
total++;
will execute a total of Θ(i2) operations, since each loop independently runs i times.
Now, let's look at the outer loop:
while (i > 0) {
/* do Theta(i^2) work */
i = i/4;
}
This loop will run a total of at most 1 + log4 i times, since on each iteration i is cut by a factor of 1/4, and this can only happen 1 + log4 i times before i drops to zero. The question, then, is how much work will be done.
One way to solve this is to write a simple recurrence relation for the total work done. We can think of the loop as a tail-recursive function, where each iteration does Θ(i2) work and then makes a recursive call on a subproblem of size 4. This gives this recurrence:
T(n) = T(n / 4) + Θ(n2).
Applying the Master Theorem, we see that a = 1, b = 4, and c = 2. Since logb a = log4 1 = 0 and 0 < c, the Master Theorem says (by Case 3) that the runtime solves to Θ(n2). Therefore, the total work done is Θ(n2).
Here's another way to think about this. The first iteration of the loop does n2 work. The next does (n / 4)2 = n2 / 16 work. The next does (n / 64)2 = n2 / 256 work. In fact, iteration x of the loop will do n2 / 16x work. Therefore, the total work done is given by
n2(1 + 1 / 16 + 1 / 162 + 1 / 163 + ...)
≤ n2(1 / (1 - 1/16))
= Θ(n2)
(This uses the formula for the sum of an infinite geometric series).
Hope this helps!
The running time is O(n^2), and the number of times that total is incremented is asymptotic to n^2/(1-1/16) which is about 1.067 n^2.
The recurrence is going to be
T(n) = n^2 + T(n/4)
= n^2 + n^2/16 + T(n/16)
= n^2 (1 + 1/16 + 1/16^2 + ...)
= n^2 / (1 - 1/16)
This code fragment:
i = n, total = 0;
while (i > 0) {
for (j=0; j<i; j++)
for (k=0; k<i; k++)
total++;
i = i/4;
}
is equivalent to this one:
for ( i = n ; i > 0 ; i = i / 4 )
for ( j = 0 ; j < i ; j ++)
for ( k = 0 ; k < i ; k ++)
total ++;
Therefore, methodically (empirically verified), you may obtain the following using Sigma Notation:*
With many thanks to WolframAlpha.
I have the following algorithm below :
for(i = 1; i < n; i++){
SmallPos = i;
Smallest = Array[SmallPos];
for(j = i + 1; j <= n; j++)
if (Array[j] < Smallest) {
SmallPos = j;
Smallest = Array[SmallPos];
}
Array[SmallPos] = Array[i];
Array[i] = Smallest;
}
Here is my calculation :
For the nested loop, I find a time complexity of
1 ("int i = 0") + n+1 ("i < n") + n ("i++")
* [1 ("j = i + 1") + n+1 ("j < n") + n ("j++")]
+ 3n (for the if-statement and the statements in it)
+ 4n (the 2 statements before the inner for-loop and the final 2 statements after the inner for-loop).
This is (1 + n + 1 + n)(1 + 1 + n + n) + 7n = (2 + 2n)(2 + 2n) + 7n = 4n^2 + 15n + 4.
But unfortunately, the text book got T(n) = 2n^2 +4n -5.
Please, anyone care to explain to me where I got it wrong?
Here is a formal manner to represent your algorithm, mathematically (Sigma Notation):
Replace c by the number of operations in the outer loop, and c' by the number of operations in the inner loop.
Hello I have an algorthm in C++ and I want to find the instructions executed. The code is below
cin >> n;
for(i=1;i<=n;i++)
for (j = 1; j <= n; j ++)
A[i][j] = 0;
for(i=1;i<=n;i++)
A[i][i] = 1;
now, after my calculation, I got this T(n) = n^2+8n-5. I just need someone else to verify if I am correct. Thanks
Ok, let's do an analysis step by step.
The first instruction
cin >> n
counts as one operation: 1.
Then the loop
for(i=1;i<=n;i++)
for (j = 1; j <= n; j ++)
A[i][j] = 0;
Let's go from inside out. The j loop performs n array assignments (A[i][j] = 0), (n + 1) j <= n comparisons and n j ++ assignments. It also performs once the assignment j = 1. So in total this gives: n + (n +1) + n + 1 = 3n + 2.
Then the outer i loop performs (n + 1) i <= n comparisons, n i ++ assignments and executes n times the j loop. It also performs one i = 1 assignment. This results in: n + (n + 1) + n * (3n + 2) + 1 = 3n^2 + 4n + 2.
Finally the last for loop performs n array assignments, (n + 1) i <= n comparisons and n i ++ assignments. It also performs one assignment i = 1. This results in: n + (n + 1) + n + 1 = 3n + 2.
Now, adding up three operations we get:
(1) + (3n^2 + 4n + 2) + (3n + 2) = 3n^2 + 7n + 5 = T(n) total operations.
The time function is equivalent, assuming that assignments, comparisons, additions and cin are all done in constant time. That would yield an algorithm of complexity O(n^2).
This is of curse assuming that n >= 1.