Explain dynamic programming. How do you identify when to use it, and walk through a classic DP problem.
Dynamic programming is one of the most challenging and commonly tested topics in coding interviews. Interviewers want to see if you can recognize overlapping subproblems, define a recurrence relation, and implement an efficient solution using memoization or tabulation.
Define DP: an optimization technique for problems with overlapping subproblems and optimal substructure.
Explain the two approaches: top-down (memoization with recursion) and bottom-up (tabulation with iteration).
Walk through a classic example like Fibonacci or climbing stairs to illustrate both approaches.
Discuss how to identify DP problems: counting ways, finding min/max, or making decisions at each step.
Dynamic programming solves problems by breaking them into overlapping subproblems and storing results to avoid redundant computation. There are two approaches: top-down memoization, where I add a cache to a recursive solution, and bottom-up tabulation, where I fill a table iteratively from the smallest subproblem. Take the climbing stairs problem: you can climb 1 or 2 steps at a time; how many ways to reach step n? The recurrence is dp[i] = dp[i-1] + dp[i-2], with dp[0]=1 and dp[1]=1. Naive recursion is O(2^n), but with DP it becomes O(n) time. For tabulation, I only need the last two values, so I optimize space to O(1). I identify DP candidates by looking for these signals: the problem asks for count, min, or max; there are choices at each step; and solving smaller versions of the same problem helps solve the larger one. Classic DP problems include knapsack, longest common subsequence, and edit distance.
Start by writing the recursive brute-force solution, then add memoization — this is the easiest path to a DP solution.
Always look for opportunities to reduce space complexity by only keeping needed previous states.
Practice the top 15 DP patterns (knapsack, LCS, LIS, matrix chain, etc.) — most problems are variations.
Memoization is top-down: recursive with a cache. Tabulation is bottom-up: iterative, filling a table from base cases. Both achieve the same time complexity.
Look for overlapping subproblems (same sub-computations repeated) and optimal substructure (optimal solution built from optimal sub-solutions).
No. Greedy is faster when it works (O(n) vs O(n²) for some problems), but it only works when local optimal choices lead to global optimal. DP is more general.
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