<p>You are given a <strong>binary</strong> string <code>s</code> and an integer <code>k</code>.</p>

<p>You are also given a 2D integer array <code>queries</code>, where <code>queries[i] = [l<sub>i</sub>, r<sub>i</sub>]</code>.</p>

<p>A <strong>binary string</strong> satisfies the <strong>k-constraint</strong> if <strong>either</strong> of the following conditions holds:</p>

<ul>
	<li>The number of <code>0</code>&#39;s in the string is at most <code>k</code>.</li>
	<li>The number of <code>1</code>&#39;s in the string is at most <code>k</code>.</li>
</ul>

<p>Return an integer array <code>answer</code>, where <code>answer[i]</code> is the number of <span data-keyword="substring-nonempty">substrings</span> of <code>s[l<sub>i</sub>..r<sub>i</sub>]</code> that satisfy the <strong>k-constraint</strong>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">s = &quot;0001111&quot;, k = 2, queries = [[0,6]]</span></p>

<p><strong>Output:</strong> <span class="example-io">[26]</span></p>

<p><strong>Explanation:</strong></p>

<p>For the query <code>[0, 6]</code>, all substrings of <code>s[0..6] = &quot;0001111&quot;</code> satisfy the k-constraint except for the substrings <code>s[0..5] = &quot;000111&quot;</code> and <code>s[0..6] = &quot;0001111&quot;</code>.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">s = &quot;010101&quot;, k = 1, queries = [[0,5],[1,4],[2,3]]</span></p>

<p><strong>Output:</strong> <span class="example-io">[15,9,3]</span></p>

<p><strong>Explanation:</strong></p>

<p>The substrings of <code>s</code> with a length greater than 3 do not satisfy the k-constraint.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= s.length &lt;= 10<sup>5</sup></code></li>
	<li><code>s[i]</code> is either <code>&#39;0&#39;</code> or <code>&#39;1&#39;</code>.</li>
	<li><code>1 &lt;= k &lt;= s.length</code></li>
	<li><code>1 &lt;= queries.length &lt;= 10<sup>5</sup></code></li>
	<li><code>queries[i] == [l<sub>i</sub>, r<sub>i</sub>]</code></li>
	<li><code>0 &lt;= l<sub>i</sub> &lt;= r<sub>i</sub> &lt; s.length</code></li>
	<li>All queries are distinct.</li>
</ul>

class Solution:
    def _lower_bound(self, nums: list[int], left: int, right: int, key: int) -> int:
        result = right + 1

        while left <= right:
            mid = left + ((right - left) >> 1)
            if nums[mid] >= key:
                result = mid
                right = mid - 1
            else:
                left = mid + 1

        return result

    def countKConstraintSubstrings(
        self, s: str, k: int, queries: List[List[int]]
    ) -> List[int]:
        n = len(s)
        first_index = [0] * n
        length_prefix_sum = [0] * n
        answer = []
        left = 0
        right = 0
        zeroes_cnt = 0
        ones_cnt = 0

        for right in range(n):
            if ord(s[right]) == ord("0"):
                zeroes_cnt += 1
            else:
                ones_cnt += 1
            while ones_cnt > k and zeroes_cnt > k:
                if ord(s[left]) == ord("0"):
                    zeroes_cnt -= 1
                else:
                    ones_cnt -= 1
                left += 1
            first_index[right] = left
            length_prefix_sum[right] = (
                right - left + 1 + (length_prefix_sum[right - 1] if right > 0 else 0)
            )
        
        for left_end, right_end in queries:
            idx = self._lower_bound(nums=first_index, left=left_end, right=right_end, key=left_end)
            result = length_prefix_sum[right_end] - (length_prefix_sum[idx - 1] if idx > 0 else 0)
            idx -= left_end
            result += idx * (idx + 1) // 2
            answer.append(result)
        
        return answer
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