如何计算这个简单算法的最坏情况运行时间复杂度

Question

I am trying to learn how to work out worst case run time complexity and I was wondering if someone could explain in detail how to work it out for the code below by highlighting all the operations etc.... Sorry for the awful code example, I'm very aware it is the most pointless method ever!

public int two(int[] n){
    int twosFound=0;
    int other=0;

    for (int i = 0; i < n.length; i++) {

        if (n[i]==2) {
            System.out.println("2 is found");
            twosFound++;
        }

        else other++;

    }

    return twosFound;
}

Answer 1

Start with the basic building blocks, and then work your way upwards:

• initializing an int variable does not depend on the array length, therefore it is O(1).
• (By the way, it is a terrible idea to call the array n when you want to talk about O(n). This can easily lead to confusion.)
• Accessing n.length takes constant time. On the other hand, if n were a linked list or some other data structure, you would have to further analyze it.
• Accessing n[i] takes constant time.
• System.out.println takes constant time, since it neither depends on the array length nor on the array contents.
• Incrementing an int variable takes constant time.
• The if statement takes the worst of whatever its components take; in this case it is constant time.
• The for loop iterates linearly over the array, therefore it takes O(n.length), multiplied with O( whatever happens inside the body of the for statement ).
• In this case, this is O(n) * O(1) = O(n).

Answer 2

The time complexity if O(n) (actually, exactly n ) because control accesses all n elements of the array. (plus there are no extra conditions to terminate the loop)

Answer 3

Your worst case complexity does not differs from the other cases. Its because of your algorithm. Its running time is solely dependent on the number of elements in the array. You test each element to satisfy a condition, the more elements you have the longer it will take the algo to run.

In fact it clear that time complexity is O(number_of_elements) . It means that time is linear dependent on the number of elements. If you take twice bigger array the time will also increase twice.