Differences between time complexity and space complexity?

Question

I have seen that in most cases the time complexity is related to the space complexity and vice versa. For example in an array traversal:

for i=1 to length(v)
    print (v[i])
endfor

Here it is easy to see that the algorithm complexity in terms of time is O(n), but it looks to me like the space complexity is also n (also represented as O(n)?).

My question: is it possible that an algorithm has different time complexity than space complexity?

Answer 1

The time and space complexities are not related to each other. They are used to describe how much space/time your algorithm takes based on the input.

• For example when the algorithm has space complexity of:

  • O(1) - constant - the algorithm uses a fixed (small) amount of space which doesn't depend on the input. For every size of the input the algorithm will take the same (constant) amount of space. This is the case in your example as the input is not taken into account and what matters is the time/space of the print command.
  • O(n), O(n^2), O(log(n))... - these indicate that you create additional objects based on the length of your input. For example creating a copy of each object of v storing it in an array and printing it after that takes O(n) space as you create n additional objects.

• In contrast the time complexity describes how much time your algorithm consumes based on the length of the input. Again:

  • O(1) - no matter how big is the input it always takes a constant time - for example only one instruction. Like

    function(list l) {
        print("i got a list");
    }
    

  • O(n), O(n^2), O(log(n)) - again it's based on the length of the input. For example

    function(list l) {
         for (node in l) {
            print(node);
        }
    }
    

Note that both last examples take O(1) space as you don't create anything. Compare them to

function(list l) {
    list c;
    for (node in l) {
        c.add(node);
    }
}

which takes O(n) space because you create a new list whose size depends on the size of the input in linear way.

Your example shows that time and space complexity might be different. It takes v.length * print.time to print all the elements. But the space is always the same - O(1) because you don't create additional objects. So, yes, it is possible that an algorithm has different time and space complexity, as they are not dependent on each other.