3501 - Maximize Active Section with Trade II.

Hard

You are given a binary string s of length n, where:

• '1' represents an active section.

• '0' represents an inactive section.

You can perform at most one trade to maximize the number of active sections in s. In a trade, you:

• Convert a contiguous block of '1's that is surrounded by '0's to all '0's.

• Afterward, convert a contiguous block of '0's that is surrounded by '1's to all '1's.

Additionally, you are given a 2D array queries, where <code>queriesi = l_i, r_i</code> represents a substring <code>sl_i...r_i</code>.

For each query, determine the maximum possible number of active sections in s after making the optimal trade on the substring <code>sl_i...r_i</code>.

Return an array answer, where answer[i] is the result for queries[i].

Note

• For each query, treat <code>sl_i...r_i</code> as if it is augmented with a '1' at both ends, forming <code>t = '1' + sl_i...r_i + '1'</code>. The augmented '1's do not contribute to the final count.

• The queries are independent of each other.

Example 1:

Input: s = "01", queries = [0,1]

Output: 1

Explanation:

Because there is no block of '1's surrounded by '0's, no valid trade is possible. The maximum number of active sections is 1.

Example 2:

Input: s = "0100", queries = [0,3,0,2,1,3,2,3]

Output: 4,3,1,1

Explanation:

• Query [0, 3] → Substring "0100" → Augmented to "101001" Choose "0100", convert "0100" → "0000" → "1111". The final string without augmentation is "1111". The maximum number of active sections is 4.

• Query [0, 2] → Substring "010" → Augmented to "10101" Choose "010", convert "010" → "000" → "111". The final string without augmentation is "1110". The maximum number of active sections is 3.

• Query [1, 3] → Substring "100" → Augmented to "11001" Because there is no block of '1's surrounded by '0's, no valid trade is possible. The maximum number of active sections is 1.

• Query [2, 3] → Substring "00" → Augmented to "1001" Because there is no block of '1's surrounded by '0's, no valid trade is possible. The maximum number of active sections is 1.

Example 3:

Input: s = "1000100", queries = [1,5,0,6,0,4]

Output: 6,7,2

Explanation:

• Query [1, 5] → Substring "00010" → Augmented to "1000101" Choose "00010", convert "00010" → "00000" → "11111". The final string without augmentation is "1111110". The maximum number of active sections is 6.

• Query [0, 6] → Substring "1000100" → Augmented to "110001001" Choose "000100", convert "000100" → "000000" → "111111". The final string without augmentation is "1111111". The maximum number of active sections is 7.

• Query [0, 4] → Substring "10001" → Augmented to "1100011" Because there is no block of '1's surrounded by '0's, no valid trade is possible. The maximum number of active sections is 2.

Example 4:

Input: s = "01010", queries = [0,3,1,4,1,3]

Output: 4,4,2

Explanation:

• Query [0, 3] → Substring "0101" → Augmented to "101011" Choose "010", convert "010" → "000" → "111". The final string without augmentation is "11110". The maximum number of active sections is 4.

• Query [1, 4] → Substring "1010" → Augmented to "110101" Choose "010", convert "010" → "000" → "111". The final string without augmentation is "01111". The maximum number of active sections is 4.

• Query [1, 3] → Substring "101" → Augmented to "11011" Because there is no block of '1's surrounded by '0's, no valid trade is possible. The maximum number of active sections is 2.

Constraints:

• <code>1 <= n == s.length <= 10⁵</code>

• <code>1 <= queries.length <= 10⁵</code>

• s[i] is either '0' or '1'.

• <code>queriesi = l_i, r_i</code>

• <code>0 <= l_i<= r_i< n</code>