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Add "reference" for EN version. Bug fixes. (#1326)

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Yudong Jin
2024-05-01 06:49:33 +08:00
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en/docs/chapter_dynamic_programming/unbounded_knapsack_problem.md
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# Complete knapsack problem
# Unbounded knapsack problem
In this section, we first solve another common knapsack problem: the complete knapsack, and then explore a special case of it: the coin change problem.
In this section, we first solve another common knapsack problem: the unbounded knapsack, and then explore a special case of it: the coin change problem.
## Complete knapsack problem
## Unbounded knapsack problem
!!! question
Given $n$ items, where the weight of the $i^{th}$ item is $wgt[i-1]$ and its value is $val[i-1]$, and a backpack with a capacity of $cap$. **Each item can be selected multiple times**. What is the maximum value of the items that can be put into the backpack without exceeding its capacity? See the example below.
![Example data for the complete knapsack problem](unbounded_knapsack_problem.assets/unbounded_knapsack_example.png)
![Example data for the unbounded knapsack problem](unbounded_knapsack_problem.assets/unbounded_knapsack_example.png)
### Dynamic programming approach
The complete knapsack problem is very similar to the 0-1 knapsack problem, **the only difference being that there is no limit on the number of times an item can be chosen**.
The unbounded knapsack problem is very similar to the 0-1 knapsack problem, **the only difference being that there is no limit on the number of times an item can be chosen**.
- In the 0-1 knapsack problem, there is only one of each item, so after placing item $i$ into the backpack, you can only choose from the previous $i-1$ items.
- In the complete knapsack problem, the quantity of each item is unlimited, so after placing item $i$ in the backpack, **you can still choose from the previous $i$ items**.
- In the unbounded knapsack problem, the quantity of each item is unlimited, so after placing item $i$ in the backpack, **you can still choose from the previous $i$ items**.
Under the rules of the complete knapsack problem, the state $[i, c]$ can change in two ways.
Under the rules of the unbounded knapsack problem, the state $[i, c]$ can change in two ways.
- **Not putting item $i$ in**: As with the 0-1 knapsack problem, transition to $[i-1, c]$.
- **Putting item $i$ in**: Unlike the 0-1 knapsack problem, transition to $[i, c-wgt[i-1]]$.
@@ -43,7 +43,7 @@ Since the current state comes from the state to the left and above, **the space-
This traversal order is the opposite of that for the 0-1 knapsack. Please refer to the following figures to understand the difference.
=== "<1>"
![Dynamic programming process for the complete knapsack problem after space optimization](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step1.png)
![Dynamic programming process for the unbounded knapsack problem after space optimization](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step1.png)
=== "<2>"
![unbounded_knapsack_dp_comp_step2](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step2.png)
@@ -78,11 +78,11 @@ The knapsack problem is a representative of a large class of dynamic programming
### Dynamic programming approach
**The coin change can be seen as a special case of the complete knapsack problem**, sharing the following similarities and differences.
**The coin change can be seen as a special case of the unbounded knapsack problem**, sharing the following similarities and differences.
- The two problems can be converted into each other: "item" corresponds to "coin", "item weight" corresponds to "coin denomination", and "backpack capacity" corresponds to "target amount".
- The optimization goals are opposite: the complete knapsack problem aims to maximize the value of items, while the coin change problem aims to minimize the number of coins.
- The complete knapsack problem seeks solutions "not exceeding" the backpack capacity, while the coin change seeks solutions that "exactly" make up the target amount.
- The optimization goals are opposite: the unbounded knapsack problem aims to maximize the value of items, while the coin change problem aims to minimize the number of coins.
- The unbounded knapsack problem seeks solutions "not exceeding" the backpack capacity, while the coin change seeks solutions that "exactly" make up the target amount.
**First step: Think through each round's decision-making, define the state, and thus derive the $dp$ table**
@@ -92,7 +92,7 @@ The two-dimensional $dp$ table is of size $(n+1) \times (amt+1)$.
**Second step: Identify the optimal substructure and derive the state transition equation**
This problem differs from the complete knapsack problem in two aspects of the state transition equation.
This problem differs from the unbounded knapsack problem in two aspects of the state transition equation.
- This problem seeks the minimum, so the operator $\max()$ needs to be changed to $\min()$.
- The optimization is focused on the number of coins, so simply add $+1$ when a coin is chosen.
@@ -117,7 +117,7 @@ For this reason, we use the number $amt + 1$ to represent an invalid solution, b
[file]{coin_change}-[class]{}-[func]{coin_change_dp}
```
The following images show the dynamic programming process for the coin change problem, which is very similar to the complete knapsack problem.
The following images show the dynamic programming process for the coin change problem, which is very similar to the unbounded knapsack problem.
=== "<1>"
![Dynamic programming process for the coin change problem](unbounded_knapsack_problem.assets/coin_change_dp_step1.png)
@@ -166,7 +166,7 @@ The following images show the dynamic programming process for the coin change pr
### Space optimization
The space optimization for the coin change problem is handled in the same way as for the complete knapsack problem:
The space optimization for the coin change problem is handled in the same way as for the unbounded knapsack problem:
```src
[file]{coin_change}-[class]{}-[func]{coin_change_dp_comp}