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Dynamic Programming: Master the Patterns
Demystifying DP: Overlapping subproblems, optimal substructure, and the must-know patterns from 1D sequences to 2D grids and Knapsacks.
90 minutes
7Detailed Sections
Senior Level
- Dynamic Programming (DP) is an optimization over plain recursion. It is used when a problem can be broken down into subproblems, and these subproblems are solved multiple times.
- Two Requirements for DP:
- Overlapping Subproblems: You find yourself solving the same small problem again and again.
- Optimal Substructure: The optimal solution to the large problem can be constructed from the optimal solutions of its subproblems.
- Two Approaches:
- Top-Down (Memoization): Standard recursion + a cache (usually an array or map) to store results.
- Bottom-Up (Tabulation): Start from the base cases and fill up a table (typically iterative).
Key Takeaways
1
DP = Recursion + Re-using solutions2
Memoization: Top-down; cache recursive results3
Tabulation: Bottom-up; iterative table filling4
Overlapping subproblems: solves "re-work" in recursion5
Optimal substructure: builds final answer from sub-answers6
Senior takeaway: If recursion looks like a tree with repeating nodes, use DPVisual Diagram
┌─────────────────────────────────────────────────────────┐ │ Fibonacci: Recursion vs. DP │ ├─────────────────────────────────────────────────────────┤ │ Plain Recursion: O(2^n) DP (Memoization): O(n) │ │ │ │ fib(5) fib(5) │ │ / \ / \ │ │ fib(4) fib(3) fib(4) fib(3) │ │ / \ / \ / \ (memoized)│ │ fib(3) fib(2) fib(2) fib(1) fib(3) fib(2) │ │ (repeated) (repeated) │ └─────────────────────────────────────────────────────────┘