03 Count even length

Count even length | Practice | GeeksforGeekspractice.geeksforgeeks.org

Question

Given a number n, find the count of all binary sequences of length 2n such that the sum of the first n bits is the same as the sum of the last n bits.

Example:

Input: n = 2
Output: 6
Explanation: There are 6 sequences of length 
2*n, the sequences are 0101, 0110, 1010, 1001, 
0000 and 1111.

Example:

Input: n = 1
Output: 2
Explanation: There are 2 sequences of length 
2*n, the sequence are 00 and 11.

Your Task: You don't need to read or print anything. Your task is to complete the function compute_value() which takes n as the input parameter and returns the count of all binary sequences of length 2*n such that the sum of the first n bits is the same as the sum of the last n bits modulo $10^9 + 7$.

Expected Time Complexity: $O(n \times \log n)$ Expected Space Complexity: $O(1)$

Constraints:

$1 \le n \le10^5$

Using DFS

Logic

From the left and the right end, there are 4 possible values that can be inserted:

0 0

0 1

1 0

1 1

After insertion of each of these combinations, the left sum and right sum vary.

Try all these 4 combinations and when the required size has been satisfied; check if the left and right sum are equal.

If so, count it in else ignore it.

Code

int res = 0;
void getCount(int leftSum, int rightSum, int p1, int p2) {
    if(p1 > p2) {
        res += leftSum == rightSum;
        return;
    }
    
    // 0 0
    getCount(leftSum, rightSum, p1 + 1, p2 - 1);
    // 0 1
    getCount(leftSum, rightSum + 1, p1 + 1, p2 - 1);
    // 1 0
    getCount(leftSum + 1, rightSum, p1 + 1, p2 - 1);
    // 1 1
    getCount(leftSum + 1, rightSum + 1, p1 + 1, p2 - 1);
}

int compute_value(int n)
{
    getCount(0, 0, 0, 2 * n - 1);
    return res;
}