Minimum Length of String After Operations

Minimum Length of String After Operations

1. What is this problem about?

This problem describes a process where you can reduce the length of a string by removing characters. The rule is: if you have a character at index i, and you can find the same character at some index j < i (to its left) and some index k > i (to its right), you can remove both the character at j and the character at k.

You want to apply this operation as many times as possible to minimize the final length of the string.

2. Why is this asked in interviews?

This question, recently asked by IBM and Amazon, tests Pattern Recognition and Hash Table usage. It looks like a complex simulation, but there's a simple mathematical rule hidden beneath the surface. Interviewer's look for:

• Observation: Realizing that the positions don't matter as much as the count of each character.
• Mathematical Simplification: Can the candidate reduce the problem to "if count > 2, what is the final count?"
• Efficiency: Solving in $O(n)$ time with $O(1)$ extra space (for the alphabet count).

3. Algorithmic pattern used

The pattern is Frequency Counting and Parity Logic.

• If a character appears c times:
  • If c is odd, we can keep reducing until only 1 character remains. (e.g., 3 -> 1, 5 -> 3 -> 1).
  • If c is even (and c > 0), we can keep reducing until only 2 characters remain. (e.g., 4 -> 2, 6 -> 4 -> 2).
• The total minimum length is the sum of these final counts for all characters present in the string.

4. Example explanation

String: abaacbc

• a: 3 times. 3 is odd $\rightarrow$ reduced to 1.
• b: 2 times. 2 is even $\rightarrow$ remains 2.
• c: 2 times. 2 is even $\rightarrow$ remains 2. Total minimum length: $1 + 2 + 2 = 5$.

Wait, why 1 or 2?