3003. Maximize the Number of Partitions After Operations

3003. Maximize the Number of Partitions After Operations

Difficulty: Hard

Topics: String, Dynamic Programming, Bit Manipulation, Bitmask, Weekly Contest 379

You are given a string s and an integer k.

First, you are allowed to change at most one index in s to another lowercase English letter.

After that, do the following partitioning operation until s is empty:

• Choose the longest prefix of s containing at most k distinct characters.
• Delete the prefix from s and increase the number of partitions by one. The remaining characters (if any) in s maintain their initial order.

Return an integer denoting the maximum number of resulting partitions after the operations by optimally choosing at most one index to change.

Example 1:

• Input: s = "accca", k = 2
• Output: 3
• Explanation: The optimal way is to change s[2] to something other than a and c, for example, b. then it becomes "acbca".
  • Then we perform the operations:
  • The longest prefix containing at most 2 distinct characters is "ac", we remove it and s becomes "bca".
  • Now The longest prefix containing at most 2 distinct characters is "bc", so we remove it and s becomes "a".
  • Finally, we remove "a" and s becomes empty, so the procedure ends.
  • Doing the operations, the string is divided into 3 partitions, so the answer is 3.

Example 2:

• Input: s = "aabaab", k = 3
• Output: 1
• Explanation: Initially s contains 2 distinct characters, so whichever character we change, it will contain at most 3 distinct characters, so the longest prefix with at most 3 distinct characters would always be all of it, therefore the answer is 1.

Example 3:

• Input: s = "xxyz", k = 1
• Output: 4
• Explanation:
  • The optimal way is to change s[0] or s[1] to something other than characters in s, for example, to change s[0] to w.
  • Then s becomes "wxyz", which consists of 4 distinct characters, so as k is 1, it will divide into 4 partitions.

Example 4:

• Input: s = "abcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefg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p", k = 1
• Output: 10000

Example 5:

• Input: s = "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaab", k = 2
• Output: 2

Example 6:

• Input: s = "hnhdfs", k = 5
• Output: 2

Example 7:

• Input: s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghh", k = 2
• Output: 2501

Example 8:

• Input: s = "bcbbccb", k = 2
• Output: 3

Example 9:

• Input: s = "acabcaacc", k = 2
• Output: 4

Example 10:

• Input: s = "bbbbcbcbbc", k = 2
• Output: 3

Constraints:

• 1 <= s.length <= 10⁴
• s consists only of lowercase English letters.
• 1 <= k <= 26

Hint:

1. For each position, try to brute-force the replacements.
2. To speed up the brute-force solution, we can precompute the following (without changing any index) using prefix sums and binary search:
   • pref[i]: The number of resulting partitions from the operations by performing the operations on s[0:i].
   • suff[i]: The number of resulting partitions from the operations by performing the operations on s[i:n - 1], where n == s.length.
   • partition_start[i]: The start index of the partition containing the iᵗʰ index after performing the operations.
3. Now, for a position i, we can try all possible 25 replacements: For a replacement, using prefix sums and binary search, we need to find the rightmost index, r, such that the number of distinct characters in the range [partition_start[i], r] is at most k. There are 2 cases:
   • r >= i: the number of resulting partitions in this case is 1 + pref[partition_start[i] - 1] + suff[r + 1].
   • Otherwise, we need to find the rightmost index r₂ such that the number of distinct characters in the range [r:r₂] is at most k. The answer in this case is 2 + pref[partition_start[i] - 1] + suff[r₂ + 1]
4. The answer is the maximum among all replacements.

Solution:

We need to determine the maximum number of partitions we can achieve by changing at most one character in the string and then partitioning the string greedily into segments where each segment contains at most k distinct characters. The solution involves simulating the partitioning process for each possible character change and finding the maximum partitions among all scenarios.

Approach

1. Problem Analysis: The problem requires us to change at most one character in the string to maximize the number of partitions. Each partition is the longest possible prefix of the remaining string that contains at most k distinct characters.
2. Key Insight: Changing a character can potentially break a segment into multiple segments by introducing a new distinct character earlier, thus increasing the total number of partitions.
3. Algorithm Selection: We simulate the partitioning process for the original string and for each possible single character change. For each change, we create a modified string and then perform the greedy partitioning. The maximum number of partitions obtained from all these scenarios is our answer.
4. Optimization: To handle large inputs efficiently, we avoid creating new strings for each change by simulating the change during the partitioning process. We also break early if we achieve the theoretical maximum number of partitions (the length of the string), as each character can form its own partition if k is 1.

Let's implement this solution in PHP: 3003. Maximize the Number of Partitions After Operations

<?php
/**
 * Main function to find maximum partitions
 * @param String $s - input string
 * @param Integer $k - max distinct characters per partition
 * @return Integer - maximum number of partitions
 */
function maxPartitionsAfterOperations($s, $k) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

/**
 * DFS to explore all possibilities
 * @param int $i - current index
 * @param bool $canChange - can we still change a character?
 * @param int $mask - bitmask of current partition's characters
 * @return int - number of partition boundaries from this state
 */
function dfs($i, $canChange, $mask, $s, $k, &$memo) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

/**
 * Count number of set bits in bitmask (distinct characters)
 * @param int $x - bitmask
 * @return int - count of 1 bits
 */
function popcount($x) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
$s1 = "accca"; $k1 = 2; // expected 3
$s2 = "aabaab"; $k2 = 3; // expected 1
$s3 = "xxyz"; $k3 = 1; // expected 4

echo maxPartitionsAfterOperations($s1, $k1) . PHP_EOL;
echo maxPartitionsAfterOperations($s2, $k2) . PHP_EOL;
echo maxPartitionsAfterOperations($s3, $k3) . PHP_EOL;
?>

Explanation:

Algorithm: DFS with Memoization

The solution uses a depth-first search (DFS) approach with memoization to explore all possibilities:

State Variables

dfs(i, canChange, mask)

• i: Current position in the string (0 to n-1)
• canChange: Boolean flag - can we still use our one allowed character change?
• mask: Bitmask representing which characters appear in the current partition
  • Bit position = character offset (a=0, b=1, ..., z=25)
  • Example: mask = 011 means characters 'a' and 'b' are in current partition

Key Insight: Bitmask for Character Tracking

Instead of storing a set of characters (inefficient), we use a 26-bit number:

'a' = bit 0
'b' = bit 1
'c' = bit 2
... etc

$currentBit = 1 << (ord($currentChar) - ord('a'));
// 'c' → ord('c')=99, ord('a')=97 → 99-97=2 → 1<<2 = 4 (binary: 100)

Two Main Choices at Each Position

Option 1: Keep the Current Character

$newMask = $mask | $currentBit;  // Add current char to partition

if ($this->popcount($newMask) > $k) {
    // Too many distinct characters - FORCE a partition boundary
    $res = 1 + dfs(i+1, canChange, currentBit);
    // Return 1 (for boundary) + remaining partitions starting fresh
} else {
    // Can continue the current partition
    $res = dfs(i+1, canChange, newMask);
}

Logic: If adding the current character exceeds k distinct characters, we must end the current partition here and start a new one.

Option 2: Change the Current Character (if allowed)

if ($canChange) {
    for ($c = 0; $c < 26; $c++) {
        $newBit = 1 << $c;

        if ($newBit === $currentBit) continue;  // Skip unchanged character

        $newMask = $mask | $newBit;

        if ($this->popcount($newMask) > $k) {
            $res = max($res, 1 + dfs(i+1, false, newBit));
        } else {
            $res = max($res, dfs(i+1, false, newMask));
        }
    }
}

Logic: Try all 25 alternative characters. After using the change, set canChange = false (can only change once).

---

Memoization

$key = $i . '|' . ($canChange ? '1' : '0') . '|' . $mask;

Cache results for states we've already computed to avoid redundant calculations.

---

Helper Function: popcount()

Counts the number of set bits (distinct characters) in the bitmask:

private function popcount($x) {
    $count = 0;
    while ($x) {
        $count += $x & 1;      // Check if last bit is 1
        $x >>= 1;              // Shift right to check next bit
    }
    return $count;
}

Example: popcount(13) (binary: 1101) → 3 bits set

---

Return Value

return $this->dfs(0, true, 0, $s, $k, $memo) + 1;

• + 1 is added because dfs() counts partition boundaries (not partitions themselves)
• Example: 3 partitions have 2 boundaries between them, so dfs() returns 2, and we add 1

---

Time Complexity

• States: O(n × 2 × 2²⁶) = O(n × 2²⁷) ≈ O(n) with pruning (since most masks are unreachable)
• Per state: O(26) to try all character replacements
• Overall: Reasonable for n ≤ 10⁴ with memoization

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