I got a working solution but it was nowhere near O(n) and it showed.
First, clarify the cost definition: the minimum number of character replacements to make a substring a palindrome. Then, design an efficient algorithm, likely using dynamic programming to compute costs for all substrings in O(n^2) time, and sum them. Discuss trade-offs between brute force and optimized approaches.
Pro tip: Mention that the cost for a substring is the number of mismatched character pairs from the ends inward, which can be computed incrementally using DP. This shows you understand the problem deeply and can optimize beyond naive O(n^3).
Confirm that cost means the minimum number of character replacements to make the substring a palindrome, and that we need the sum over all substrings.
For a substring s[i..j], the cost is the number of mismatched pairs (s[i+k] != s[j-k]) for k from 0 to (j-i)/2. This can be computed in O(length) per substring.
Use DP to compute costs for all substrings efficiently: cost[i][j] = cost[i+1][j-1] + (s[i] != s[j] ? 1 : 0). This reduces time to O(n^2).
Iterate over all substrings, sum their costs, and return the total. Discuss potential overflow and use appropriate data types.
State time and space complexity (O(n^2) time, O(n^2) space, can be optimized to O(n) space). Handle empty string, single character, and large inputs.
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