Translate all code to English (#1836)

* Review the EN heading format. * Fix pythontutor headings. * Fix pythontutor headings. * bug fixes * Fix headings in **/summary.md * Revisit the CN-to-EN translation for Python code using Claude-4.5 * Revisit the CN-to-EN translation for Java code using Claude-4.5 * Revisit the CN-to-EN translation for Cpp code using Claude-4.5. * Fix the dictionary. * Fix cpp code translation for the multipart strings. * Translate Go code to English. * Update workflows to test EN code. * Add EN translation for C. * Add EN translation for CSharp. * Add EN translation for Swift. * Trigger the CI check. * Revert. * Update en/hash_map.md * Add the EN version of Dart code. * Add the EN version of Kotlin code. * Add missing code files. * Add the EN version of JavaScript code. * Add the EN version of TypeScript code. * Fix the workflows. * Add the EN version of Ruby code. * Add the EN version of Rust code. * Update the CI check for the English version code. * Update Python CI check. * Fix cmakelists for en/C code. * Fix Ruby comments
2026-07-06 04:34:20 +00:00 · 2025-12-31 07:44:52 +08:00
parent 45e1295241
commit 2778a6f9c7
1284 changed files with 71557 additions and 3275 deletions
@@ -1,8 +1,8 @@
-# Unbounded knapsack problem
+# Unbounded Knapsack 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.

-## Unbounded knapsack problem
+## Unbounded Knapsack Problem

 !!! question

@@ -10,7 +10,7 @@ In this section, we first solve another common knapsack problem: the unbounded k

 ![Example data for unbounded knapsack problem](unbounded_knapsack_problem.assets/unbounded_knapsack_example.png)

-### Dynamic programming approach
+### Dynamic Programming Approach

 The unbounded knapsack problem is very similar to the 0-1 knapsack problem, **differing only in that there is no limit on the number of times an item can be selected**.

@@ -28,7 +28,7 @@ $$
 dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 $$

-### Code implementation
+### Code Implementation

 Comparing the code for the two problems, there is one change in state transition from $i-1$ to $i$, with everything else identical:

@@ -36,7 +36,7 @@ Comparing the code for the two problems, there is one change in state transition
 [file]{unbounded_knapsack}-[class]{}-[func]{unbounded_knapsack_dp}
 ```

-### Space optimization
+### Space Optimization

 Since the current state is transferred from states on the left and above, **after space optimization, each row in the $dp$ table should be traversed in forward order**.

@@ -66,7 +66,7 @@ The code implementation is relatively simple, just delete the first dimension of
 [file]{unbounded_knapsack}-[class]{}-[func]{unbounded_knapsack_dp_comp}
 ```

-## Coin change problem
+## Coin Change Problem

 The knapsack problem represents a large class of dynamic programming problems and has many variants, such as the coin change problem.

@@ -76,7 +76,7 @@ The knapsack problem represents a large class of dynamic programming problems an

 ![Example data for coin change problem](unbounded_knapsack_problem.assets/coin_change_example.png)

-### Dynamic programming approach
+### Dynamic Programming Approach

 **The coin change problem can be viewed as a special case of the unbounded knapsack problem**, with the following connections and differences.

@@ -107,7 +107,7 @@ When the target amount is $0$, the minimum number of coins needed to make it up

 When there are no coins, **it is impossible to make up any amount $> 0$**, which is an invalid solution. To enable the $\min()$ function in the state transition equation to identify and filter out invalid solutions, we consider using $+ \infty$ to represent them, i.e., set all $dp[0, a]$ in the first row to $+ \infty$.

-### Code implementation
+### Code Implementation

 Most programming languages do not provide a $+ \infty$ variable, and can only use the maximum value of integer type `int` as a substitute. However, this can lead to large number overflow: the $+ 1$ operation in the state transition equation may cause overflow.

@@ -164,7 +164,7 @@ The figure below shows the dynamic programming process for coin change, which is
 === "<15>"
    ![coin_change_dp_step15](unbounded_knapsack_problem.assets/coin_change_dp_step15.png)

-### Space optimization
+### Space Optimization

 The space optimization for the coin change problem is handled in the same way as the unbounded knapsack problem:

@@ -172,7 +172,7 @@ The space optimization for the coin change problem is handled in the same way as
 [file]{coin_change}-[class]{}-[func]{coin_change_dp_comp}
 ```

-## Coin change problem II
+## Coin Change Problem Ii

 !!! question

@@ -180,7 +180,7 @@ The space optimization for the coin change problem is handled in the same way as

 ![Example data for coin change problem II](unbounded_knapsack_problem.assets/coin_change_ii_example.png)

-### Dynamic programming approach
+### Dynamic Programming Approach

 Compared to the previous problem, this problem's goal is to find the number of combinations, so the subproblem becomes: **the number of combinations among the first $i$ types of coins that can make up amount $a$**. The $dp$ table remains a two-dimensional matrix of size $(n+1) \times (amt + 1)$.

@@ -192,13 +192,13 @@ $$

 When the target amount is $0$, no coins need to be selected to make up the target amount, so all $dp[i, 0]$ in the first column should be initialized to $1$. When there are no coins, it is impossible to make up any amount $>0$, so all $dp[0, a]$ in the first row equal $0$.

-### Code implementation
+### Code Implementation

 ```src
 [file]{coin_change_ii}-[class]{}-[func]{coin_change_ii_dp}
 ```

-### Space optimization
+### Space Optimization

 The space optimization is handled in the same way, just delete the coin dimension: