I spent way too long on the brute force before realizing the substring count alone would wreck my runtime.
First, clarify the problem: for each substring, the minimum swaps to make it a palindrome equals the number of mismatched character pairs divided by 2. Then, design an efficient algorithm, likely using dynamic programming or prefix sums, to compute this sum over all substrings without enumerating all O(n^2) substrings naively.
Pro tip: Mention that the minimum swaps for a substring is exactly half the number of mismatched pairs, and that you can compute the total sum in O(n^2) time by expanding around centers or using DP, which is optimal for this problem.
Restate the problem to ensure understanding: for each substring, find the minimum adjacent swaps to make it a palindrome, then sum these minimums. Confirm that swaps are adjacent and that we only consider substrings of length >= 2.
Show that for a substring, the minimum number of swaps equals the number of mismatched character pairs divided by 2. Explain that each swap can fix at most one mismatched pair.
Propose an O(n^2) solution: iterate over all possible centers (for odd and even length palindromes) and expand outward, maintaining the count of mismatches. Alternatively, use DP to compute the sum of mismatches for all substrings.
State that the time complexity is O(n^2) and space complexity is O(1) or O(n) depending on implementation. Justify why this is optimal given the need to consider all substrings.
Walk through a small example, such as 'abba' or 'abc', to verify the formula and algorithm. Discuss edge cases like empty string, single character, and strings with all same characters.
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