I need you to help me optimize the following algorithm implementation: ```[PASTE YOUR ALGORITHM CODE]``` Please analyze and improve this algorithm through this structured approach: 1) Current Performance Analysis: - Time complexity with detailed explanation - Space complexity with memory usage patterns - Bottlenecks and inefficiencies identified - Edge cases and potential failure points 2) Optimization Strategy: - Applicable algorithmic patterns for this problem - Data structure alternatives and tradeoffs - Potential algorithm paradigm shifts - Language-specific optimization opportunities 3) Implementation Improvements: - Provide refactored implementation with optimizations - Maintain code readability and maintainability - Add necessary error handling - Include comments explaining key optimizations 4) Performance Comparison: - Before/after time complexity analysis - Before/after space complexity analysis - Expected performance gains with examples - Tradeoffs made in the optimization process 5) Testing and Validation: - Edge case test scenarios - Performance validation approach - Correctness verification methodology - Benchmarking recommendations Format your response with clear sections for Analysis, Strategy, Refactored Code, Complexity Analysis, and Testing Plan. Ensure the optimized implementation maintains the original functionality while improving performance characteristics.