Given a positive integer N, count all possible distinct binary strings of length N such that there are no consecutive 1’s.
Input: N = 2 Output: 3 // The 3 strings are 00, 01, 10 Input: N = 3 Output: 5 // The 5 strings are 000, 001, 010, 100, 101
This problem can be solved using Dynamic Programming. Let a[i] be the number of binary strings of length i which do not contain any two consecutive 1’s and which end in 0. Similarly, let b[i] be the number of such strings which end in 1. We can append either 0 or 1 to a string ending in 0, but we can only append 0 to a string ending in 1. This yields the recurrence relation:
a[i] = a[i - 1] + b[i - 1] b[i] = a[i - 1]
The base cases of above recurrence are a = b = 1. The total number of strings of length i is just a[i] + b[i].
Following is the implementation of above solution. In the following implementation, indexes start from 0. So a[i] represents the number of binary strings for input length i+1. Similarly, b[i] represents binary strings for input length i+1.
If we take a closer look at the pattern, we can observe that the count is actually (n+2)’th Fibonacci number for n >= 1. The Fibonacci Numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 141, ….
n = 1, count = 2 = fib(3) n = 2, count = 3 = fib(4) n = 3, count = 5 = fib(5) n = 4, count = 8 = fib(6) n = 5, count = 13 = fib(7) ................
Therefore we can count the strings in O(Log n) time also using the method 5 here.
This article is contributed by Rahul Jain. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
- Count strings with consecutive 1's
- Count binary strings with twice zeros in first half
- Count binary strings with k times appearing adjacent two set bits
- Python | Check if there are K consecutive 1's in a binary number
- Count minimum number of subsets (or subsequences) with consecutive numbers
- Number of Binary Strings of length N with K adjacent Set Bits
- Given a binary string, count number of substrings that start and end with 1.
- 1 to n bit numbers with no consecutive 1s in binary representation.
- Maximum consecutive one’s (or zeros) in a binary array
- Count of arrays having consecutive element with different values
- Count ways to reach a score using 1 and 2 with no consecutive 2s
- Count Subarrays with Consecutive elements differing by 1
- Length of longest consecutive ones by at most one swap in a Binary String
- Maximum Consecutive Zeroes in Concatenated Binary String
- Add n binary strings