You are given a theoretical algorithm that processes a graph with n nodes and e edges using a brute-force approach with a time complexity of O(n^3). Your task is to propose a refactorized algorithm that reduces the time complexity to O(n^2) or better. Explain the data structures you would use (e.g., adjacency lists, priority queues) and provide a pseudocode implementation. Additionally, analyze the trade-offs between time complexity and space complexity in your solution.