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Revisit the English version (#1885)

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Yudong Jin
2026-04-10 23:03:03 +08:00
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parent ae03a167a4
commit b01036b09e
132 changed files with 1702 additions and 1508 deletions
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en/docs/chapter_backtracking/permutations_problem.md
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@@ -30,7 +30,7 @@ As shown in the figure below, we can unfold the search process into a recursion
To ensure that each element is chosen only once, we consider introducing a boolean array `selected`, where `selected[i]` indicates whether `choices[i]` has been chosen. We implement the following pruning operation based on it.
- After making a choice `choice[i]`, we set `selected[i]` to $\text{True}$, indicating that it has been chosen.
- After making a choice `choices[i]`, we set `selected[i]` to $\text{True}$, indicating that it has been chosen.
- When traversing the candidate list `choices`, we skip all nodes that have been chosen, which is pruning.
As shown in the figure below, suppose we choose $1$ in the first round, $3$ in the second round, and $2$ in the third round. Then we need to prune the branch of element $1$ in the second round and prune the branches of elements $1$ and $3$ in the third round.
@@ -55,13 +55,13 @@ After understanding the above information, we can fill in the blanks in the temp
Suppose the input array is $[1, 1, 2]$. To distinguish the two duplicate elements $1$, we denote the second $1$ as $\hat{1}$.
As shown in the figure below, the method described above generates permutations where half are duplicates.
As shown in the figure below, half of the permutations generated by the above method are duplicates.
![Duplicate permutations](permutations_problem.assets/permutations_ii.png)
So how do we remove duplicate permutations? The most direct approach is to use a hash set to directly deduplicate the permutation results. However, this is not elegant because **the search branches that generate duplicate permutations are unnecessary and should be identified and pruned early**, which can further improve algorithm efficiency.
### Pruning Duplicate Elements
### Pruning Equal Elements
Observe the figure below. In the first round, choosing $1$ or choosing $\hat{1}$ is equivalent. All permutations generated under these two choices are duplicates. Therefore, we should prune $\hat{1}$.
@@ -73,7 +73,7 @@ Essentially, **our goal is to ensure that multiple equal elements are chosen onl
### Code Implementation
Building on the code from the previous problem, we consider opening a hash set `duplicated` in each round of choices to record which elements have been tried in this round, and prune duplicate elements:
Building on the code from the previous problem, we initialize a hash set `duplicated` in each round of choices to record which elements have already been tried in that round, and prune equal elements:
```src
[file]{permutations_ii}-[class]{}-[func]{permutations_ii}
@@ -88,7 +88,7 @@ The maximum recursion depth is $n$, using $O(n)$ stack frame space. `selected` u
Note that although both `selected` and `duplicated` are used for pruning, they have different objectives.
- **Pruning duplicate choices**: There is only one `selected` throughout the entire search process. It records which elements are included in the current state, and its purpose is to prevent an element from appearing repeatedly in `state`.
- **Pruning duplicate elements**: Each round of choices (each `backtrack` function call) contains a `duplicated` set. It records which elements have been chosen in this round's iteration (the `for` loop), and its purpose is to ensure that equal elements are chosen only once.
- **Pruning equal elements**: Each round of choices (each `backtrack` function call) contains a `duplicated` set. It records which elements have been chosen in this round's iteration (the `for` loop), and its purpose is to ensure that equal elements are chosen only once.
The figure below shows the effective scope of the two pruning conditions. Note that each node in the tree represents a choice, and the nodes on the path from the root to a leaf node form a permutation.