Week 35 of 43
Advanced DSA
WEEK 35 LEARNING MODULE
Dynamic Programming I: Top-Down Memoization
Introduces Dynamic Programming fundamentals, Overlapping Subproblems, Optimal Substructure, Top-Down Memoization, state space identification, and recurrence equations.
5 Practice Problems4 Core ConceptsVerified Curriculum: high Confidence
What You'll Learn
- ✓Master core DP requirements: Overlapping Subproblems and Optimal Substructure.
- ✓Implement Top-Down DP: combining recursion with memoization caching (hash tables or arrays).
- ✓Define state parameters and formulate recurrence relations (state transition equations).
- ✓Transform exponential brute-force recursion ($O(2^n)$) into polynomial time ($O(n)$ or $O(n \cdot W)$).
- ✓Solve classic DP problems: Climbing Stairs, Partition Equal Subset Sum (0/1 Knapsack), Target Sum, and Coin Change.
Core Concepts
Overlapping Subproblems
Problem re-evaluates the exact same smaller subproblems repeatedly during recursive exploration.
Optimal Substructure
Optimal solution to the overall problem can be constructed from optimal solutions to its subproblems.
Top-Down Memoization Cache
O(states * transitions)Caching computed state results in an array or dictionary, returning cached results on duplicate state visits.
0/1 Knapsack Subsets Formulation
O(n * target)State transition: dp(index, remaining_sum) = dp(index+1, sum) or dp(index+1, sum - nums[index]).
Algorithms Covered
Top-Down Memoized RecursionClimbing StairsPartition Equal Subset Sum (0/1 Knapsack)Target Sum DPCoin Change (Min Coins DP)
Data Structures Applied
Memoization Cache (Array / Hash Map)Recursion Call Stack
“Those who cannot remember the past are condemned to repeat it.”
— George Santayana