Advanced Dynamic Programming Solver Calculator

Problem	Sample Input	Expected Output	Purpose
0/1 Knapsack	Capacity 15; Weights 2,3,5,7,1,4; Values 10,5,15,7,6,18	Maximum value 54; Selected A, B, C, E, F	Find the best value under a weight limit.
Coin Change	Target 11; Coins 1,2,5	Combinations 11; Minimum coins 3; Best 5×2, 1×1	Measure both counting and optimization outcomes.
Matrix Chain	30,35,15,5,10,20,25	Minimum multiplications 15125	Choose the cheapest multiplication order.

Formula Used

Knapsack recurrence: DP[i,w] = max(DP[i−1,w], DP[i−1,w−weight_i] + value_i). This compares excluding and including the current item whenever the remaining capacity allows inclusion.

Coin change ways: Ways[a] = Ways[a] + Ways[a−coin]. This accumulates the number of distinct combinations that form each amount using available denominations.

Coin change minimum coins: Min[a] = min(Min[a], Min[a−coin] + 1). This keeps the smallest number of coins needed for each reachable amount.

Matrix chain multiplication: m[i,j] = min(m[i,k] + m[k+1,j] + p[i−1]p[k]p[j]). This chooses the split point producing the fewest scalar multiplications across every subchain.

FAQs

1. What does this calculator solve?

It solves three classic dynamic programming tasks: 0/1 knapsack, coin change, and matrix chain multiplication. Each mode returns optimized results, supporting details, and a state-table view when the grid size stays manageable.

2. Why are inputs limited to positive integers?

These solver modes usually rely on discrete states. Positive integers keep the tables consistent, simplify state transitions, and match the standard recurrences used in introductory and advanced dynamic programming examples.

3. What is the difference between combinations and minimum coins?

Combinations count how many valid ways reach the target. Minimum coins finds the smallest number of denominations needed to reach that same target exactly. Both metrics are useful, but they answer different questions.

4. Why might a full state table not appear?

Large dynamic programming grids can become difficult to read and expensive to display in a browser. The calculator hides oversized tables and keeps the final optimized result visible instead.

5. Can I use repeated coin denominations?

Yes, but repeated values are consolidated before solving. This prevents duplicate denominations from distorting counts and keeps the recurrence aligned with the intended unlimited-coin formulation.

6. How is the knapsack choice recovered?

The solver walks backward through the dynamic programming table. Whenever a value changes from one row to the previous row, the related item is marked as part of the optimal selection.

7. What does the matrix chain result mean?

It reports the smallest number of scalar multiplications needed to multiply the full matrix sequence. The displayed parenthesization shows the order that achieves that minimum cost.

8. Are the CSV and PDF files based on current results?

Yes. The export buttons use the solution currently shown near the top of the page. Run the solver again after changing inputs if you want updated files.