Bug fixes and improvements (#1078)
* Fix the logo in the en version * Optimize header color and fix body background color * Update theme switch's name * Fix backfrop-filter on Safari * Update some animation's file name for adding egde when cropping * Re-count the comments number * A bug fix in n_queens_problem.md
Yudong Jin
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[file]{n_queens}-[class]{}-[func]{n_queens}
|
||||
```
|
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|
||||
逐行放置 $n$ 次，考虑列约束，则从第一行到最后一行分别有 $n$、$n-1$、$\dots$、$2$、$1$ 个选择，**因此时间复杂度为 $O(n!)$** 。实际上，根据对角线约束的剪枝也能够大幅缩小搜索空间，因而搜索效率往往优于以上时间复杂度。
|
||||
逐行放置 $n$ 次，考虑列约束，则从第一行到最后一行分别有 $n$、$n-1$、$\dots$、$2$、$1$ 个选择，使用 $O(n!)$ 时间；当保存解时，需要复制矩阵 `state` 并添加进 `res` ，复制操作使用 $O(n^2)$ 时间；因此总体时间复杂度为 $O(n! \cdot n^2)$ 。实际上，根据对角线约束的剪枝也能够大幅缩小搜索空间，因而搜索效率往往优于以上时间复杂度。
|
||||
|
||||
数组 `state` 使用 $O(n^2)$ 空间，数组 `cols`、`diags1` 和 `diags2` 皆使用 $O(n)$ 空间。最大递归深度为 $n$ ，使用 $O(n)$ 栈帧空间。因此，**空间复杂度为 $O(n^2)$** 。
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至此，我们就得到了下图所示的二维 $dp$ 矩阵，其尺寸与输入网格 $grid$ 相同。
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!!! note
|
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||||
@@ -65,7 +65,7 @@ $$
|
||||
dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
|
||||
$$
|
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!!! note
|
||||
|
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@@ -79,7 +79,7 @@ $$
|
||||
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||||
如下图所示，由于每个格子是由其左方格子和上方格子转移而来，因此我们使用循环来遍历矩阵，外循环遍历各行，内循环遍历各列。
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!!! note
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@@ -12,19 +12,19 @@
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- **初始化**：传入 $n$ 个顶点，初始化长度为 $n$ 的顶点列表 `vertices` ，使用 $O(n)$ 时间；初始化 $n \times n$ 大小的邻接矩阵 `adjMat` ，使用 $O(n^2)$ 时间。
|
||||
|
||||
=== "初始化邻接矩阵"
|
||||
|
||||
|
||||
|
||||
=== "添加边"
|
||||
|
||||
|
||||
|
||||
=== "删除边"
|
||||
|
||||
|
||||
|
||||
=== "添加顶点"
|
||||
|
||||
|
||||
|
||||
=== "删除顶点"
|
||||
|
||||
|
||||
|
||||
以下是基于邻接矩阵表示图的实现代码：
|
||||
|
||||
@@ -43,19 +43,19 @@
|
||||
- **初始化**：在邻接表中创建 $n$ 个顶点和 $2m$ 条边，使用 $O(n + m)$ 时间。
|
||||
|
||||
=== "初始化邻接表"
|
||||
|
||||
|
||||
|
||||
=== "添加边"
|
||||
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||||
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||||
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=== "删除边"
|
||||
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||||
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||||
|
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=== "添加顶点"
|
||||
|
||||
|
||||
|
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=== "删除顶点"
|
||||
|
||||
|
||||
|
||||
以下是邻接表的代码实现。对比上图，实际代码有以下不同。
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@@ -347,19 +347,19 @@
|
||||
如下图所示，我们将双向链表的头节点和尾节点视为双向队列的队首和队尾，同时实现在两端添加和删除节点的功能。
|
||||
|
||||
=== "LinkedListDeque"
|
||||
|
||||
|
||||
|
||||
=== "push_last()"
|
||||
|
||||
|
||||
|
||||
=== "push_first()"
|
||||
|
||||
|
||||
|
||||
=== "pop_last()"
|
||||
|
||||
|
||||
|
||||
=== "pop_first()"
|
||||
|
||||
|
||||
|
||||
实现代码如下所示：
|
||||
|
||||
@@ -372,19 +372,19 @@
|
||||
如下图所示，与基于数组实现队列类似，我们也可以使用环形数组来实现双向队列。
|
||||
|
||||
=== "ArrayDeque"
|
||||
|
||||
|
||||
|
||||
=== "push_last()"
|
||||
|
||||
|
||||
|
||||
=== "push_first()"
|
||||
|
||||
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||||
|
||||
=== "pop_last()"
|
||||
|
||||
|
||||
|
||||
=== "pop_first()"
|
||||
|
||||
|
||||
|
||||
在队列的实现基础上，仅需增加“队首入队”和“队尾出队”的方法：
|
||||
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@@ -321,13 +321,13 @@
|
||||
如下图所示，我们可以将链表的“头节点”和“尾节点”分别视为“队首”和“队尾”，规定队尾仅可添加节点，队首仅可删除节点。
|
||||
|
||||
=== "LinkedListQueue"
|
||||
|
||||
|
||||
|
||||
=== "push()"
|
||||
|
||||
|
||||
|
||||
=== "pop()"
|
||||
|
||||
|
||||
|
||||
以下是用链表实现队列的代码：
|
||||
|
||||
@@ -349,13 +349,13 @@
|
||||
可以看到，入队和出队操作都只需进行一次操作，时间复杂度均为 $O(1)$ 。
|
||||
|
||||
=== "ArrayQueue"
|
||||
|
||||
|
||||
|
||||
=== "push()"
|
||||
|
||||
|
||||
|
||||
=== "pop()"
|
||||
|
||||
|
||||
|
||||
你可能会发现一个问题：在不断进行入队和出队的过程中，`front` 和 `rear` 都在向右移动，**当它们到达数组尾部时就无法继续移动了**。为了解决此问题，我们可以将数组视为首尾相接的“环形数组”。
|
||||
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||||
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@@ -319,13 +319,13 @@
|
||||
如下图所示，对于入栈操作，我们只需将元素插入链表头部，这种节点插入方法被称为“头插法”。而对于出栈操作，只需将头节点从链表中删除即可。
|
||||
|
||||
=== "LinkedListStack"
|
||||
|
||||
|
||||
|
||||
=== "push()"
|
||||
|
||||
|
||||
|
||||
=== "pop()"
|
||||
|
||||
|
||||
|
||||
以下是基于链表实现栈的示例代码：
|
||||
|
||||
@@ -338,13 +338,13 @@
|
||||
使用数组实现栈时，我们可以将数组的尾部作为栈顶。如下图所示，入栈与出栈操作分别对应在数组尾部添加元素与删除元素，时间复杂度都为 $O(1)$ 。
|
||||
|
||||
=== "ArrayStack"
|
||||
|
||||
|
||||
|
||||
=== "push()"
|
||||
|
||||
|
||||
|
||||
=== "pop()"
|
||||
|
||||
|
||||
|
||||
由于入栈的元素可能会源源不断地增加，因此我们可以使用动态数组，这样就无须自行处理数组扩容问题。以下为示例代码：
|
||||
|
||||
|
||||
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@@ -148,7 +148,7 @@ hide:
|
||||
<img src="https://img.shields.io/badge/C-snow?logo=c&logoColor=A8B9CC">
|
||||
<img src="https://img.shields.io/badge/Zig-snow?logo=zig&logoColor=F7A41D">
|
||||
</div>
|
||||
<p style="margin-top: 2em;">500 幅动画图解、12 种编程语言代码、2000 条社区问答，助你快速入门数据结构与算法</p>
|
||||
<p style="margin-top: 2em;">500 幅动画图解、12 种编程语言代码、3000 条社区问答，助你快速入门数据结构与算法</p>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||