Knapsack Visualizer
Solve the 0/1 knapsack problem step by step with interactive DP table visualization.
Interactive 0/1 knapsack problem visualizer — watch the DP table fill step by step.
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The 0/1 knapsack problem is a classic dynamic programming problem. Given items with weights and values, choose items that maximize total value without exceeding capacity. This knapsack visualizer shows how the DP table is built step by step.
📌 How the 0/1 Knapsack Algorithm Works:
DP[i][w] = max(value[i] + DP[i-1][w-weight[i]], DP[i-1][w])
Each cell represents the maximum value achievable with first i items and capacity w.
Interactive 0/1 Knapsack Visualizer
Enter items, set capacity, and watch the knapsack problem dynamic programming solution in action.
Items
How the 0/1 Knapsack Algorithm Works
Understanding the knapsack algorithm visualization step by step.
Create DP Table
Create a table with (n+1) rows and (W+1) columns, where n = number of items, W = capacity. Initialize first row and column to 0.
Fill the Table
For each item i and capacity w, decide: include item (if fits) or skip it. Take the maximum of both options.
Formula: DP[i][w] = max(value[i] + DP[i-1][w-weight[i]], DP[i-1][w])
If item i doesn’t fit, DP[i][w] = DP[i-1][w]. The bottom-right cell gives the optimal value.
Backtrack Selection
Trace back from bottom-right to find which items were selected for the optimal solution.
Knapsack Problem Example
Walk through a complete example step by step.
Problem: Capacity = 10
Items: (Weight: 2, Value: 3), (Weight: 3, Value: 4), (Weight: 4, Value: 5), (Weight: 5, Value: 6)
Step 1: Create DP table (5 rows x 11 columns)
Step 2: Fill table using recurrence relation
Step 3: Optimal value = 13 (items with weight 3+5 or 2+4+5?)
Step 4: Selected items: Item 2 (wt:3, val:4) and Item 4 (wt:5, val:6)
Answer: Maximum value = 13
Frequently Asked Questions
Common questions about the knapsack problem and dynamic programming.
What is the 0/1 knapsack problem?
The 0/1 knapsack problem is a classic optimization problem where you have items with weights and values, and you must choose items to maximize total value without exceeding a given capacity. “0/1” means you can take each item at most once (cannot take fractions).
How does dynamic programming solve knapsack?
The knapsack problem dynamic programming approach builds a 2D table where DP[i][w] represents the maximum value achievable with first i items and capacity w. The recurrence relation decides whether to include or exclude each item.
What’s the time complexity of the knapsack algorithm?
The time complexity is O(n × W), where n is the number of items and W is the capacity. This is pseudo-polynomial, efficient for moderate capacities but can be slow for very large capacities.
What’s the difference between 0/1 and fractional knapsack?
The 0/1 knapsack problem requires taking an item entirely or not at all. The fractional knapsack allows taking fractions of items and can be solved greedily (by value/weight ratio).
Is this knapsack solver free?
Yes! This knapsack visualizer is completely free. No signup, no ads. Just open the tool and start learning dynamic programming.
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Free knapsack visualizer for computer science students learning dynamic programming.