Count number of binary strings without consecutive 1’s : Set 2
Given a positive integer N, the task is to count all possible distinct binary strings of length N such that there are no consecutive 1’s.
Input: N = 5
The non-negative integers <= 5 with their corresponding binary representations are:
0 : 0
1 : 1
2 : 10
3 : 11
4 : 100
5 : 101
Among them, only 3 has two consecutive 1’s. Therefore required count = 5
Input: N = 12
Approach: In this article, an approach using the concept of digit-dp is discussed.
- Similar to the digit-dp problem, a 3-dimensional table is created here to store the computed values. It is assumed that the N < 231 – 1, and the range of every number is only 2 (Either 0 or 1). Therefore, the dimensions of the table are taken as 32 x 2 x 2.
- After constructing the table, the given number is converted to a binary string.
- Then, the number is iterated. For every iteration:
- Check if the previous digit is a 0 or 1.
- If it is a 0, then the present number can either be a 0 or 1.
- But if the previous number is 1, then the present number has to be 0 because we can’t have two consecutive 1’s in the binary representation.
- Now, the table is exactly filled like the digit-dp problem.
Below is the implementation of the above approach
Time Complexity: O(L * log(N))
- O(log(N)) to convert the number from Decimal to binary.
- O(L) to fill the table, where L is the length of the binary form.
Auxiliary Space: O(32 * 2 * 2)