Given a binary string S of length N, the task is to find the number of pairs of integers [L, R] 1 ≤ L < R ≤ N such that S[L . . . R] (the substring of S from L to R) can be reduced to 1 length string by replacing substrings “01” or “10” with “1” and “0” respectively.
Input: S = “0110”
Explanation: The 4 substrings are 01, 10, 110, 0110.
Input: S = “00000”
Approach: The solution is based on the following mathematical idea:
We can solve this based on the exclusion principle. Instead of finding possible pairs find the number of impossible cases and subtract that from all possible substrings (i.e. N*(N+1)/2 ).
How to find impossible cases?
When s[i] and s[i-1] are same, then after reduction it will either become “00” or “11”. In both cases, the substring cannot be reduced to length 1. So substring from 0 to i, from 1 to i, . . . cannot be made to have length 1. That count of substrings is i.
Follow the below steps to solve the problem:
- Initialize answer ans = N * (N + 1) / 2
- Run a loop from i = 1 to N – 1
- If S[i] is equal to S[i – 1], then subtract i from ans.
- Return ans – N (because there are N substrings having length 1).
Below is the implementation of the above approach.
Time Complexity: O(N)
Auxiliary Space: O(1)