Act as a senior software engineer with 10+ years of experience in algorithm optimization. Explain step-by-step how to implement a dynamic programming solution for [PROBLEM_TYPE], such as the knapsack problem or Fibonacci sequence. Cover the following: 1) Identifying overlapping subproblems and optimal substructure, 2) Choosing between top-down (memoization) and bottom-up (tabulation) approaches, 3) Writing the recurrence relation for [SPECIFIC_PROBLEM], and 4) Optimizing space complexity for [CONSTRAINTS]. Provide [LANGUAGE]-specific code snippets and highlight common pitfalls like incorrect base cases or redundant calculations. Tailor the explanation for [AUDIENCE_LEVEL], whether beginners or advanced coders.