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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
This commit is contained in:
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_dynamic_programming/dp_solution_pipeline.md
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@@ -35,7 +35,7 @@ To illustrate the problem-solving steps more vividly, we use a classic problem,
Given an $n \times m$ two-dimensional grid `grid`, each cell in the grid contains a non-negative integer representing the cost of that cell. The robot starts from the top-left cell and can only move down or right at each step until it reaches the bottom-right cell. Return the minimum path sum from the top-left to the bottom-right.
The following figure shows an example, where the given grid's minimum path sum is $13$.
The figure below shows an example, where the given grid's minimum path sum is $13$.
![Minimum Path Sum Example Data](dp_solution_pipeline.assets/min_path_sum_example.png)
@@ -45,7 +45,7 @@ Each round of decisions in this problem is to move one step down or right from t
The state $[i, j]$ corresponds to the subproblem: the minimum path sum from the starting point $[0, 0]$ to $[i, j]$, denoted as $dp[i, j]$.
Thus, we obtain the two-dimensional $dp$ matrix shown below, whose size is the same as the input grid $grid$.
Thus, we obtain the two-dimensional $dp$ matrix shown in the figure below, whose size is the same as the input grid $grid$.
![State definition and DP table](dp_solution_pipeline.assets/min_path_sum_solution_state_definition.png)
@@ -59,7 +59,7 @@ Thus, we obtain the two-dimensional $dp$ matrix shown below, whose size is the s
For the state $[i, j]$, it can only be derived from the cell above $[i-1, j]$ or the cell to the left $[i, j-1]$. Therefore, the optimal substructure is: the minimum path sum to reach $[i, j]$ is determined by the smaller of the minimum path sums of $[i, j-1]$ and $[i-1, j]$.
Based on the above analysis, the state transition equation shown in the following figure can be derived:
Based on the above analysis, the state transition equation shown in the figure below can be derived:
$$
dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
@@ -104,7 +104,7 @@ Implementation code as follows:
[file]{min_path_sum}-[class]{}-[func]{min_path_sum_dfs}
```
The following figure shows the recursive tree rooted at $dp[2, 1]$, which includes some overlapping subproblems, the number of which increases sharply as the size of the grid `grid` increases.
The figure below shows the recursive tree rooted at $dp[2, 1]$, which includes some overlapping subproblems, the number of which increases sharply as the size of the grid `grid` increases.
Essentially, the reason for overlapping subproblems is: **there are multiple paths to reach a certain cell from the top-left corner**.
@@ -132,7 +132,7 @@ Implement the dynamic programming solution iteratively, code as shown below:
[file]{min_path_sum}-[class]{}-[func]{min_path_sum_dp}
```
The following figures show the state transition process of the minimum path sum, traversing the entire grid, **thus the time complexity is $O(nm)$**.
The figure below show the state transition process of the minimum path sum, traversing the entire grid, **thus the time complexity is $O(nm)$**.
The array `dp` is of size $n \times m$, **therefore the space complexity is $O(nm)$**.