Algorithm analysis: Am I analyzing these algorithms correctly? How to approach problems like these [closed]

Question

1)

  x = 25;
    for (int i = 0; i < myArray.length; i++)
    {
        if (myArray[i] == x)
            System.out.println("found!");
    }

I think this one is O(n).

2)

for (int r = 0; r < 10000; r++)
    for (int c = 0; c < 10000; c++)
        if (c % r == 0)
            System.out.println("blah!");

I think this one is O(1), because for any input n, it will run 10000 * 10000 times. Not sure if this is right.

3)

a = 0
for (int i = 0; i < k; i++)
{
    for (int j = 0; j < i; j++)
        a++;
}

I think this one is O(i * k). I don't really know how to approach problems like this where the inner loop is affected by variables being incremented in the outer loop. Some key insights here would be much appreciated. The outer loop runs k times, and the inner loop runs 1 + 2 + 3 + ... + k times. So that sum should be (k/2) * (k+1), which would be order of k^2. So would it actually be O(k^3)? That seems too large. Again, don't know how to approach this.

4)

int key = 0;    //key may be any value
int first = 0;
int last = intArray.length-1;;
int mid = 0;
boolean found = false;

while( (!found) && (first <= last) )
{
    mid = (first + last) / 2;

    if(key == intArray[mid]) 
        found = true;
    if(key < intArray[mid])
        last = mid - 1;
    if(key > intArray[mid]) 
        first = mid + 1;
}

This one, I think is O(log n). But, I came to this conclusion because I believe it is a binary search and I know from reading that the runtime is O(log n). I think it's because you divide the input size by 2 for each iteration of the loop. But, I don't know if this is the correct reasoning or how to approach similar algorithms that I haven't seen and be able to deduce that they run in logarithmic time in a more verifiable or formal way.

5)

int currentMinIndex = 0;

for (int front = 0; front < intArray.length; front++)
{
    currentMinIndex = front;

    for (int i = front; i < intArray.length; i++)
    {
        if (intArray[i] < intArray[currentMinIndex])
        {
            currentMinIndex = i;
        }
    }

    int tmp = intArray[front];
    intArray[front] = intArray[currentMinIndex];
    intArray[currentMinIndex] = tmp;
}

I am confused about this one. The outer loop runs n times. And the inner for loop runs n + (n-1) + (n-2) + ... (n - k) + 1 times? So is that O(n^3) ??

Answer 1

More or less, yes.

1 is correct - it seems you are searching for a specific element in what I assume is an un-sorted collection. If so, the worst case is that the element is at the very end of the list, hence O(n).

2 is correct, though a bit strange. It is O(1) assuming r and c are constants and the bounds are not variables. If they are constant, then yes O(1) because there is nothing to input.

3 I believe that is considered O(n^2) still. There would be some constant factor like k * n^2, drop the constant and you got O(n^2).

4 looks a lot like a binary search algorithm for a sorted collection. O(logn) is correct. It is log because at each iteration you are essentially halving the # of possible choices in which the element you are looking for could be in.

5 is looking like a bubble sort, O(n^2), for similar reasons to 3.