Given two integers n and k, count the number of binary strings of length n with k as number of times adjacent 1’s appear.
Input : n = 5, k = 2 Output : 6 Explanation: Binary strings of length 5 in which k number of times two adjacent set bits appear. 00111 01110 11100 11011 10111 11101 Input : n = 4, k = 1 Output : 3 Explanation: Binary strings of length 3 in which k number of times two adjacent set bits appear. 0011 1100 0110
Lets try writing the recursive function for the above problem statement:
1) n = 1, only two binary strings exist with length 1, not having any adjacent 1’s
String 1 : “0”
String 2 : “1”
2) For all n > 1 and all k, two cases arise
a) Strings ending with 0 : String of length n can be created by appending 0 to all strings of length n-1 having k times two adjacent 1’s ending with both 0 and 1 (Having 0 at n’th position will not change the count of adjacent 1’s).
b) Strings ending with 1 : String of length n can be created by appending 1 to all strings of length n-1 having k times adjacent 1’s and ending with 0 and to all strings of length n-1 having k-1 adjacent 1’s and ending with 1.
Example: let s = 011 i.e. a string ending with 1 having adjacent count as 1. Adding 1 to it, s = 0111 increase the count of adjacent 1.
Let there be an array dp[i][j] where dp[i][j] denotes number of binary strings with length i having j number of two adjacent 1's and ending with 0. Similarly dp[i][j] denotes the same binary strings with length i and j adjacent 1's but ending with 1. Then: dp = 1 and dp = 1 For all other i and j, dp[i][j] = dp[i-1][j] + dp[i-1][j] dp[i][j] = dp[i-1][j] + dp[i-1][j-1] Then, output dp[n][k] + dp[n][k]
Time Complexity : O(n2)
This article is contributed by Ekta Goel. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
- Number of Binary Strings of length N with K adjacent Set Bits
- Generate all binary strings of length n with sub-string "01" appearing exactly twice
- Count of strings where adjacent characters are of difference one
- Generate all the binary strings of N bits
- Count even length binary sequences with same sum of first and second half bits
- Count binary strings with twice zeros in first half
- Count number of binary strings without consecutive 1's
- Count of arrays in which all adjacent elements are such that one of them divide the another
- Maximum sum of nodes in Binary tree such that no two are adjacent | Dynamic Programming
- Minimum adjacent swaps required to Sort Binary array
- Count of Numbers in a Range where digit d occurs exactly K times
- Count pairs (A, B) such that A has X and B has Y number of set bits and A+B = C
- Find all even length binary sequences with same sum of first and second half bits
- Add n binary strings
- Bitwise AND of N binary strings