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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
committed by GitHub
parent 8e60d12151
commit c4a7966882
99 changed files with 615 additions and 259 deletions
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en/docs/chapter_backtracking/subset_sum_problem.md
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@@ -23,7 +23,7 @@ Unlike the permutation problem, **elements in this problem can be chosen an unli
Inputting the array $[3, 4, 5]$ and target element $9$ into the above code yields the results $[3, 3, 3], [4, 5], [5, 4]$. **Although it successfully finds all subsets with a sum of $9$, it includes the duplicate subset $[4, 5]$ and $[5, 4]$**.
This is because the search process distinguishes the order of choices, however, subsets do not distinguish the choice order. As shown in the following figure, choosing $4$ before $5$ and choosing $5$ before $4$ are different branches, but correspond to the same subset.
This is because the search process distinguishes the order of choices, however, subsets do not distinguish the choice order. As shown in the figure below, choosing $4$ before $5$ and choosing $5$ before $4$ are different branches, but correspond to the same subset.
![Subset search and pruning out of bounds](subset_sum_problem.assets/subset_sum_i_naive.png)
@@ -34,7 +34,7 @@ To eliminate duplicate subsets, **a straightforward idea is to deduplicate the r
### Duplicate subset pruning
**We consider deduplication during the search process through pruning**. Observing the following figure, duplicate subsets are generated when choosing array elements in different orders, for example in the following situations.
**We consider deduplication during the search process through pruning**. Observing the figure below, duplicate subsets are generated when choosing array elements in different orders, for example in the following situations.
1. When choosing $3$ in the first round and $4$ in the second round, all subsets containing these two elements are generated, denoted as $[3, 4, \dots]$.
2. Later, when $4$ is chosen in the first round, **the second round should skip $3$** because the subset $[4, 3, \dots]$ generated by this choice completely duplicates the subset from step `1.`.
@@ -62,7 +62,7 @@ Besides, we have made the following two optimizations to the code.
[file]{subset_sum_i}-[class]{}-[func]{subset_sum_i}
```
The following figure shows the overall backtracking process after inputting the array $[3, 4, 5]$ and target element $9$ into the above code.
The figure below shows the overall backtracking process after inputting the array $[3, 4, 5]$ and target element $9$ into the above code.
![Subset sum I backtracking process](subset_sum_problem.assets/subset_sum_i.png)
@@ -74,7 +74,7 @@ The following figure shows the overall backtracking process after inputting the
Compared to the previous question, **this question's input array may contain duplicate elements**, introducing new problems. For example, given the array $[4, \hat{4}, 5]$ and target element $9$, the existing code's output results in $[4, 5], [\hat{4}, 5]$, resulting in duplicate subsets.
**The reason for this duplication is that equal elements are chosen multiple times in a certain round**. In the following figure, the first round has three choices, two of which are $4$, generating two duplicate search branches, thus outputting duplicate subsets; similarly, the two $4$s in the second round also produce duplicate subsets.
**The reason for this duplication is that equal elements are chosen multiple times in a certain round**. In the figure below, the first round has three choices, two of which are $4$, generating two duplicate search branches, thus outputting duplicate subsets; similarly, the two $4$s in the second round also produce duplicate subsets.
![Duplicate subsets caused by equal elements](subset_sum_problem.assets/subset_sum_ii_repeat.png)
@@ -90,6 +90,6 @@ At the same time, **this question stipulates that each array element can only be
[file]{subset_sum_ii}-[class]{}-[func]{subset_sum_ii}
```
The following figure shows the backtracking process for the array $[4, 4, 5]$ and target element $9$, including four types of pruning operations. Please combine the illustration with the code comments to understand the entire search process and how each type of pruning operation works.
The figure below shows the backtracking process for the array $[4, 4, 5]$ and target element $9$, including four types of pruning operations. Please combine the illustration with the code comments to understand the entire search process and how each type of pruning operation works.
![Subset sum II backtracking process](subset_sum_problem.assets/subset_sum_ii.png)