Complexity in a recursion algorithm

Question

I am currently studying data structures at university and stumbled across a question about complexity in recursion.

Given this code:

Unsigned func (unsigned n)
{
 if (n ==0) return 1;
 if(n==1) return 2;

 \\do somthing(in O(1) )

 return func(n-1) + func(n-1);
} 

I know what the code does. I know that in the form it is now, the time complexity is O(2^n).

My question however is: will the time complexity change if instead of the last return call of the code I will write: return 2*func(n-1) ?

I know that as far as memory complexity, we are talking about a significant reduction in space taken by the recursion, but as far as time complexity, will there be any change?

I did the math using a recursive function and came to the understanding that there will be no change in time complexity, am I right?

Answer 1

This method is having only O(n), because if you run it with 5, it goes to recursion with 4, then with 3 etc.

Unsigned func (unsigned n)
{
 if (n ==0) return 1;
 if(n==1) return 2;

 \\do somthing(in O(1) )

 return 2*func(n-1);
}

However what about this :

Unsigned func (unsigned n)
{
 if (n ==0) return 1;
 if(n==1) return 2;

 \\do somthing(in O(1) )

 return func(n-1) + func(n-1);
}

At for example func(5) it will execute first as following

5 -> 4 -> 3 -> 2 -> 1

Then it returns to 2, but there it will execute the "second" part, so the whole process looks like

5 -> 4 -> 3 -> 2-> 1; 2-> 1; 3->2->1; etc.

Therefore it DOES change complexity drastically from O(n) to O(2^n)

---

Lets try practical example this code :

public class Complexity {
    private static int counter;
    public static void main(String[] args) {
        for (int i = 0; i < 20; i++) {
            counter = 0;
            func(i);
            System.out.println("For n="+i+" of 2*func the number of cycles is " + counter);
            counter = 0;
            func2(i);
            System.out.println("For n="+i+" of func + func the number of cycles is " + counter);
        }
    }

    public static int func(int n) {
        counter++;
        if (n == 0) {
            return 1;
        }
        if (n == 1) {
            return 2;
        }
        return 2 * func(n - 1);
    }

    public static int func2(int n) {
        counter++;
        if (n == 0) {
            return 1;
        }
        if (n == 1) {
            return 2;
        }
        return func2(n - 1) + func2(n - 1);
    }    
}

Having this output :

For n=0 of 2*func the number of cycles is 1
For n=0 of func + func the number of cycles is 1
For n=1 of 2*func the number of cycles is 1
For n=1 of func + func the number of cycles is 1
For n=2 of 2*func the number of cycles is 2
For n=2 of func + func the number of cycles is 3
For n=3 of 2*func the number of cycles is 3
For n=3 of func + func the number of cycles is 7
For n=4 of 2*func the number of cycles is 4
For n=4 of func + func the number of cycles is 15
For n=5 of 2*func the number of cycles is 5
For n=5 of func + func the number of cycles is 31
For n=6 of 2*func the number of cycles is 6
For n=6 of func + func the number of cycles is 63
For n=7 of 2*func the number of cycles is 7
For n=7 of func + func the number of cycles is 127
For n=8 of 2*func the number of cycles is 8
For n=8 of func + func the number of cycles is 255
For n=9 of 2*func the number of cycles is 9
For n=9 of func + func the number of cycles is 511
For n=10 of 2*func the number of cycles is 10
For n=10 of func + func the number of cycles is 1023
For n=11 of 2*func the number of cycles is 11
For n=11 of func + func the number of cycles is 2047
For n=12 of 2*func the number of cycles is 12
For n=12 of func + func the number of cycles is 4095
For n=13 of 2*func the number of cycles is 13
For n=13 of func + func the number of cycles is 8191
For n=14 of 2*func the number of cycles is 14
For n=14 of func + func the number of cycles is 16383
For n=15 of 2*func the number of cycles is 15
For n=15 of func + func the number of cycles is 32767
For n=16 of 2*func the number of cycles is 16
For n=16 of func + func the number of cycles is 65535
For n=17 of 2*func the number of cycles is 17
For n=17 of func + func the number of cycles is 131071
For n=18 of 2*func the number of cycles is 18
For n=18 of func + func the number of cycles is 262143
For n=19 of 2*func the number of cycles is 19
For n=19 of func + func the number of cycles is 524287

---

But if you remember already computed results, the complexity is still O(n) even with the second approach :

public class Complexity {
    private static int counter;
    private static int[] results;

    public static void main(String[] args) {
        for (int i = 0; i < 20; i++) {
            counter = 0;
            func(i);
            System.out.println("For n="+i+" of 2*func the number of cycles is " + counter);
            counter = 0;
            func2(i);
            System.out.println("For n="+i+" of func + func the number of cycles is " + counter);
            counter = 0;
            results = new int[i+1];
            func3(i);
            System.out.println("For n="+i+" of func + func with remembering the number of cycles is " + counter);
        }
    }

    public static int func(int n) {
        counter++;
        if (n == 0) {
            return 1;
        }
        if (n == 1) {
            return 2;
        }
        return 2 * func(n - 1);
    }

    public static int func2(int n) {
        counter++;
        if (n == 0) {
            return 1;
        }
        if (n == 1) {
            return 2;
        }        
        return func2(n - 1) + func2(n - 1);
    }

    public static int func3(int n) {
        counter++;
        if (n == 0) {
            return 1;
        }
        if (n == 1) {
            return 2;
        }

        if (results[n] == 0){
            results[n] = func3(n - 1) + func3(n - 1);
        }

        return results[n];
    }        
}

Having this output :

For n=0 of 2*func the number of cycles is 1
For n=0 of func + func the number of cycles is 1
For n=0 of func + func with remembering the number of cycles is 1
For n=1 of 2*func the number of cycles is 1
For n=1 of func + func the number of cycles is 1
For n=1 of func + func with remembering the number of cycles is 1
For n=2 of 2*func the number of cycles is 2
For n=2 of func + func the number of cycles is 3
For n=2 of func + func with remembering the number of cycles is 3
For n=3 of 2*func the number of cycles is 3
For n=3 of func + func the number of cycles is 7
For n=3 of func + func with remembering the number of cycles is 5
For n=4 of 2*func the number of cycles is 4
For n=4 of func + func the number of cycles is 15
For n=4 of func + func with remembering the number of cycles is 7
For n=5 of 2*func the number of cycles is 5
For n=5 of func + func the number of cycles is 31
For n=5 of func + func with remembering the number of cycles is 9
For n=6 of 2*func the number of cycles is 6
For n=6 of func + func the number of cycles is 63
For n=6 of func + func with remembering the number of cycles is 11
For n=7 of 2*func the number of cycles is 7
For n=7 of func + func the number of cycles is 127
For n=7 of func + func with remembering the number of cycles is 13
For n=8 of 2*func the number of cycles is 8
For n=8 of func + func the number of cycles is 255
For n=8 of func + func with remembering the number of cycles is 15
For n=9 of 2*func the number of cycles is 9
For n=9 of func + func the number of cycles is 511
For n=9 of func + func with remembering the number of cycles is 17
For n=10 of 2*func the number of cycles is 10
For n=10 of func + func the number of cycles is 1023
For n=10 of func + func with remembering the number of cycles is 19
For n=11 of 2*func the number of cycles is 11
For n=11 of func + func the number of cycles is 2047
For n=11 of func + func with remembering the number of cycles is 21
For n=12 of 2*func the number of cycles is 12
For n=12 of func + func the number of cycles is 4095
For n=12 of func + func with remembering the number of cycles is 23
For n=13 of 2*func the number of cycles is 13
For n=13 of func + func the number of cycles is 8191
For n=13 of func + func with remembering the number of cycles is 25
For n=14 of 2*func the number of cycles is 14
For n=14 of func + func the number of cycles is 16383
For n=14 of func + func with remembering the number of cycles is 27
For n=15 of 2*func the number of cycles is 15
For n=15 of func + func the number of cycles is 32767
For n=15 of func + func with remembering the number of cycles is 29
For n=16 of 2*func the number of cycles is 16
For n=16 of func + func the number of cycles is 65535
For n=16 of func + func with remembering the number of cycles is 31
For n=17 of 2*func the number of cycles is 17
For n=17 of func + func the number of cycles is 131071
For n=17 of func + func with remembering the number of cycles is 33
For n=18 of 2*func the number of cycles is 18
For n=18 of func + func the number of cycles is 262143
For n=18 of func + func with remembering the number of cycles is 35
For n=19 of 2*func the number of cycles is 19
For n=19 of func + func the number of cycles is 524287
For n=19 of func + func with remembering the number of cycles is 37