3234. Count the Number of Substrings With Dominant Ones
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You are given a binary string s.
Return the number of substrings with dominant ones.
A string has dominant ones if the number of ones in the string is greater than or equal to the square of the number of zeros in the string.
Example 1:
Input: s = "00011"
Output: 5
Explanation:
The substrings with dominant ones are shown in the table below.
| i | j | s[i..j] | Number of Zeros | Number of Ones |
|---|---|---|---|---|
| 3 | 3 | 1 | 0 | 1 |
| 4 | 4 | 1 | 0 | 1 |
| 2 | 3 | 01 | 1 | 1 |
| 3 | 4 | 11 | 0 | 2 |
| 2 | 4 | 011 | 1 | 2 |
Example 2:
Input: s = "101101"
Output: 16
Explanation:
The substrings with non-dominant ones are shown in the table below.
Since there are 21 substrings total and 5 of them have non-dominant ones, it follows that there are 16 substrings with dominant ones.
| i | j | s[i..j] | Number of Zeros | Number of Ones |
|---|---|---|---|---|
| 1 | 1 | 0 | 1 | 0 |
| 4 | 4 | 0 | 1 | 0 |
| 1 | 4 | 0110 | 2 | 2 |
| 0 | 4 | 10110 | 2 | 3 |
| 1 | 5 | 01101 | 2 | 3 |
Constraints:
1 <= s.length <= 4 * 104sconsists only of characters'0'and'1'.
Hints
Hint 1
Let us fix the starting index
l of the substring and count the number of indices r such that l <= r and the substring s[l..r] has dominant ones.Hint 2
A substring with dominant ones has at most
sqrt(n) zeros.Hint 3
We cannot iterate over every
r and check if the s[l..r] has dominant ones. Instead, we iterate over the next sqrt(n) zeros to the left of l and count the number of substrings with dominant ones where the current zero is the rightmost zero of the substring.