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
Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.
- Count number of binary strings without consecutive 1’s : Set 2
- Count of binary strings of length N with even set bit count and at most K consecutive 1s
- Count of Binary strings of length N having atmost M consecutive 1s or 0s alternatively exactly K times
- Find the number of binary strings of length N with at least 3 consecutive 1s
- Count number of binary strings such that there is no substring of length greater than or equal to 3 with all 1's
- Count strings with consecutive 1's
- Count of sub-strings with equal consecutive 0's and 1's
- Count of non-overlapping sub-strings "101" and "010" in the given binary string
- Count binary strings with twice zeros in first half
- Count of binary strings of given length consisting of at least one 1
- Python | Check if there are K consecutive 1's in a binary number
- Count of strings possible by replacing two consecutive same character with new character
- Count binary strings with k times appearing adjacent two set bits
- Count of distinct XORs formed by rearranging two Binary strings
- Maximum number of consecutive 1's in binary representation of all the array elements
- Count the number of ordered sets not containing consecutive numbers
- Count minimum number of subsets (or subsequences) with consecutive numbers
- Minimum number of Binary strings to represent a Number
- Number of sub-strings in a given binary string divisible by 2
- Number of binary strings such that there is no substring of length ≥ 3