Dynamic Programming Interview
Master dynamic programming coding interviews with our AI-powered coach. Practice classic DP problems like knapsack, coin change, and LIS with detailed solutions, optimization techniques, and pattern recognition.
Dynamic Programming Interview in Action
Interviewer: "You have a knapsack with capacity W and n items, each with weight w[i] and value v[i]. Find the maximum value you can achieve without exceeding the weight limit. Each item can be used at most once."
Problem Analysis & Approach
Example Input:
- Capacity: W = 50
- Items: [(weight=10, value=60), (weight=20, value=100), (weight=30, value=120)]
- Expected Output: Maximum value = 220 (items 2 and 3)
DP State Definition:
dp[i][w]
= maximum value using first i items with weight limit w- Base case: dp[0][w] = 0 (no items = no value)
- Recurrence: dp[i][w] = max(exclude item i, include item i)
1. Recursive Solution (Brute Force)
Recursive Solution Analysis:
- Problem: Exponential time complexity due to overlapping subproblems
- Example: Computing knapsack(items=3, capacity=10) multiple times
- Solution: Use memoization to cache results and avoid recomputation
- Pattern Recognition: Choice at each step + optimal substructure = DP candidate
2. Top-Down DP (Memoization)
3. Bottom-Up DP (Tabulation)
DP Table Visualization
Knapsack DP Table Example (Weights=[1,3,4,5], Values=[1,4,5,7], Capacity=7): Items: 0 1 2 3 4 (item indices) [0] [1] [4] [5] [7] (values) [0] [1] [3] [4] [5] (weights) w: 0 1 2 3 4 5 6 7 i=0: 0 0 0 0 0 0 0 0 (no items) i=1: 0 1 1 1 1 1 1 1 (item 0: w=1,v=1) i=2: 0 1 1 4 5 5 5 5 (item 1: w=3,v=4) i=3: 0 1 1 4 5 6 6 9 (item 2: w=4,v=5) i=4: 0 1 1 4 5 7 8 9 (item 3: w=5,v=7) Final Answer: dp[4][7] = 9 Selected Items: [0, 1] (weights: 1+3=4, values: 1+4=5) or [2] (weight: 4, value: 5) or [3] (weight: 5, value: 7) Optimal: Item 3 alone gives value 7, or items 0+2 give value 6 Actually: Items 1+2 give value 4+5=9 with weight 3+4=7 ✓
4. Space-Optimized DP
DP Approach Comparison
Approach | Time Complexity | Space Complexity | Pros & Cons |
---|---|---|---|
Recursive | O(2^n) | O(n) | Easy to understand, exponential time |
Memoization | O(n × W) | O(n × W) + O(n) | Top-down, natural recursion + cache |
Tabulation | O(n × W) | O(n × W) | Bottom-up, can track solution path |
Space-Optimized | O(n × W) | O(W) | Minimal memory, harder to track items |
Note: n = number of items, W = knapsack capacity
Key DP Interview Insights:
1. Problem Recognition:
- Optimal Substructure: Optimal solution contains optimal solutions to subproblems
- Overlapping Subproblems: Same subproblems solved multiple times
- Choice at each step: Include/exclude item creates decision tree
2. State Definition Strategy:
- What varies? Item index (i) and remaining capacity (w)
- What do we want? Maximum value achievable
- State: dp[i][w] = max value with first i items, capacity w
3. Implementation Tips:
- Start simple: Write recursive solution first, then optimize
- Base cases: Handle edge cases (no items, no capacity)
- Direction: In 1D optimization, iterate backwards to avoid overwriting
- Backtracking: If solution path needed, keep 2D table
Follow-up Interview Questions:
- "What if we can use unlimited quantity of each item?" (Unbounded Knapsack)
- "How would you handle fractional knapsack?" (Greedy approach)
- "Can you reduce space complexity further?" (Use only two 1D arrays)
- "How would you handle negative weights or values?"
🧮 Classic DP Problems
Master essential DP problems: knapsack, coin change, longest increasing subsequence, edit distance, and more.
🔄 Multiple Approaches
Learn recursive, memoization, tabulation, and space-optimized solutions with complexity analysis.
📊 Pattern Recognition
Identify DP patterns: optimal substructure, overlapping subproblems, and state transition design.
âš¡ Optimization Techniques
Learn space optimization, rolling arrays, and advanced DP tricks to minimize memory usage.
🎯 State Definition
Master the art of defining DP states, transition functions, and base case identification.
💡 Interview Strategy
Learn systematic approach to DP problems: recognize, define state, find recurrence, optimize.
Master Dynamic Programming
Join thousands of developers who've mastered DP concepts and landed positions at top tech companies with our AI-powered coaching.
Get Your DP Interview AI CoachFree practice problems • Real-time optimization • All DP patterns covered
Related Algorithm Guides
Explore more algorithm interview guides powered by AI coaching