Module Review: Dynamic Programming
[!NOTE] This module explores the core principles of Module Review: Dynamic Programming, deriving solutions from first principles and hardware constraints to build world-class, production-ready expertise.
1. Key Takeaways
- Dynamic Programming (DP) solves complex problems by breaking them down into simpler subproblems and storing their solutions to avoid redundant work.
- Two Approaches:
- Top-Down (Memoization): Recursive + Caching. Good when not all subproblems need to be solved.
- Bottom-Up (Tabulation): Iterative + Table. Good for space optimization and avoiding recursion depth limits.
- Core Properties: A problem must have Overlapping Subproblems and Optimal Substructure to be solved with DP.
- Space Optimization: Always look for opportunities to reduce space complexity (e.g., using a rolling array) after solving the problem with a full table.
- State Design: Defining the correct state variables is often the hardest part of the problem.
2. Interactive Flashcards
Test your knowledge of key DP concepts.
Memoization
(Tap to flip)
The Top-Down approach to DP. It involves caching the results of recursive calls to avoid re-computing them.
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3. DP Cheat Sheet
| Problem | Time Complexity | Space Complexity (Optimized) | Recurrence Pattern |
|---|---|---|---|
| Fibonacci | O(N) | O(1) | dp[i] = dp[i-1] + dp[i-2] |
| Climbing Stairs | O(N) | O(1) | dp[i] = dp[i-1] + dp[i-2] |
| 0/1 Knapsack | O(N × W) | O(W) | max(dp[w], val + dp[w-wt]) |
| Unbounded Knapsack | O(N × W) | O(W) | max(dp[w], val + dp[w-wt]) (iterate forward) |
| LCS | O(N × M) | O(min(N, M)) | if c1==c2: 1+prev else max(top, left) |
| Subset Sum | O(N × Sum) | O(Sum) | dp[s] = dp[s] \|\| dp[s-num] |
| Min Path Sum | O(N × M) | O(min(N, M)) | grid[i][j] += min(top, left) |
| LIS (Longest Increasing Subsequence) | O(N2) | O(N) | if num[i] > num[j]: max(dp[i], 1+dp[j]) |
4. Practice in the Vault
Looking to solidify your understanding? Head over to the Problem Vault to solve curated problems related to this module with detailed walkthroughs and optimal solutions.