maximize-active-section-with-trade-ii

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 queries[i] = [li, ri] represents a substring s[li...ri].

For each query, determine the maximum possible number of active sections in s after making the optimal trade on the substring s[li...ri].

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

Note

• For each query, treat s[li...ri] as if it is augmented with a '1' at both ends, forming t = '1' + s[li...ri] + '1'. 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:

• 1 <= n == s.length <= 105


• 1 <= queries.length <= 105


• s[i] is either '0' or '1'.


• queries[i] = [li, ri]


• 0 <= li <= ri < n