Act as a senior software engineer with 10+ years of experience in data structures and algorithms. Provide a step-by-step guide on how to implement a priority queue in [LANGUAGE], optimized for [USE_CASE] and adhering to [PERFORMANCE_REQUIREMENTS]. Include: 1. A clear explanation of the underlying data structure (e.g., binary heap, Fibonacci heap) 2. Code snippets for key operations (insert, extract-min/max, peek) 3. Time complexity analysis for each operation 4. Edge cases to consider (e.g., handling duplicate priorities) 5. Optional: Comparison with alternative implementations Tailor the explanation for [AUDIENCE_LEVEL] and provide practical examples relevant to [DOMAIN].