Minimum Number of Operations to Make String Sorted

Minimum Number of Operations to Make String Sorted

1. What is this problem about?

The Minimum Number of Operations to Make String Sorted interview question is a rigorous "Hard" problem that combines string manipulation with high-level combinatorics. You are given a string and a specific "swap and reverse" operation that resembles one step of finding the next lexicographical permutation in reverse. You need to find how many such operations are needed to reach the alphabetically sorted version of the string.

2. Why is this asked in interviews?

Samsung and other companies use this to test a candidate's mastery of math and modular arithmetic. The problem is essentially asking: "What is the lexicographical rank of this string among all its permutations?". This requires calculating permutations with repetitions (multiset permutations) using factorials and modular inverse.

3. Algorithmic pattern used

This problem follows the Math, Combinatorics, String interview pattern. The rank of a string can be found using the formula for permutations: For each position i, count how many characters S[j] (where j > i) are smaller than S[i]. If you place one of those smaller characters at position i, the number of ways to arrange the remaining characters is (n - 1 - i)! / (freq[a]! * freq[b]! * ...). You sum these counts for all positions.

4. Example explanation

String: "cba"

1. 'c': Smaller chars are 'b' and 'a' (2 chars).
   • If we pick 'b' or 'a' for 1st spot, remaining 2 spots can be filled in 2! ways.
   • Count = 2 * 2! = 4.
2. 'b': Smaller char is 'a' (1 char).
   • Remaining 1 spot can be filled in 1! ways.
   • Count = 1 * 1! = 1.
3. 'a': No smaller chars. Count = 0. Total: 4 + 1 = 5 operations.

5. Common mistakes candidates make

The biggest challenge is handling the large numbers. The answer must be returned modulo 10^9 + 7, so candidates must implement modular inverse (using Fermat's Little Theorem) to handle the division in the multiset permutation formula. Another mistake is not correctly counting the "smaller" characters when there are duplicates.

6. Interview preparation tip

Brush up on "lexicographical rank" algorithms. It's a standard competitive programming topic that occasionally appears in "Hard" interview rounds. Familiarize yourself with pre-computing factorials and their modular inverses to make the O(N * 26) solution efficient.