You are given a description of a brute-force algorithm that finds the longest palindromic substring in O(n^3) time. Analyze the logic and identify the computational bottlenecks. Refactor the algorithm conceptually to achieve O(n^2) or O(n) time complexity (Manacher's algorithm). Describe the changes in data structures or iteration logic required to achieve this optimization. Do not write code; instead, provide a pseudocode representation of the optimized logic and explain the mathematical proof of its improved complexity.