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Bug fixes and improvements (#1348)

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* Add "reference" for EN version. Bug fixes.

* Unify the figure reference as "the figure below" and "the figure above".
Bug fixes.

* Format the EN markdown files.

* Replace "" with <u></u> for EN version and bug fixes

* Fix biary_tree_dfs.png

* Fix biary_tree_dfs.png

* Fix zh-hant/biary_tree_dfs.png

* Fix heap_sort_step1.png

* Sync zh and zh-hant versions.

* Bug fixes

* Fix EN figures

* Bug fixes

* Fix the figure labels for EN version
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Yudong Jin
2024-05-06 14:44:48 +08:00
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parent 8e60d12151
commit c4a7966882
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en/docs/chapter_dynamic_programming/knapsack_problem.md
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@@ -8,7 +8,7 @@ In this section, we will first solve the most common 0-1 knapsack problem.
Given $n$ items, the weight of the $i$-th item is $wgt[i-1]$ and its value is $val[i-1]$, and a knapsack with a capacity of $cap$. Each item can be chosen only once. What is the maximum value of items that can be placed in the knapsack under the capacity limit?
Observe the following figure, since the item number $i$ starts counting from 1, and the array index starts from 0, thus the weight of item $i$ corresponds to $wgt[i-1]$ and the value corresponds to $val[i-1]$.
Observe the figure below, since the item number $i$ starts counting from 1, and the array index starts from 0, thus the weight of item $i$ corresponds to $wgt[i-1]$ and the value corresponds to $val[i-1]$.
![Example data of the 0-1 knapsack](knapsack_problem.assets/knapsack_example.png)
@@ -76,19 +76,19 @@ After introducing memoization, **the time complexity depends on the number of su
[file]{knapsack}-[class]{}-[func]{knapsack_dfs_mem}
```
The following figure shows the search branches that are pruned in memoized search.
The figure below shows the search branches that are pruned in memoized search.
![The memoized search recursive tree of the 0-1 knapsack problem](knapsack_problem.assets/knapsack_dfs_mem.png)
### Method three: Dynamic programming
Dynamic programming essentially involves filling the $dp$ table during the state transition, the code is shown below:
Dynamic programming essentially involves filling the $dp$ table during the state transition, the code is shown in the figure below:
```src
[file]{knapsack}-[class]{}-[func]{knapsack_dp}
```
As shown in the figures below, both the time complexity and space complexity are determined by the size of the array `dp`, i.e., $O(n \times cap)$.
As shown in the figure below, both the time complexity and space complexity are determined by the size of the array `dp`, i.e., $O(n \times cap)$.
=== "<1>"
![The dynamic programming process of the 0-1 knapsack problem](knapsack_problem.assets/knapsack_dp_step1.png)