From 1aff6d6cc2c471505a4dfb4bc6e36f810f3e0c75 Mon Sep 17 00:00:00 2001 From: krahets Date: Mon, 21 Aug 2023 03:57:09 +0800 Subject: [PATCH] Polish the content. --- docs/assets/covers/chapter_hashing.jpg | Bin 140434 -> 140215 bytes .../linked_list.md | 4 ++-- docs/chapter_backtracking/n_queens_problem.md | 2 +- .../subset_sum_problem.md | 14 +++++++------- .../chapter_computational_complexity/index.md | 6 +++--- .../time_complexity.md | 12 +++++++++--- docs/chapter_data_structure/summary.md | 2 +- .../intro_to_dynamic_programming.md | 2 +- docs/chapter_greedy/max_capacity_problem.md | 2 +- docs/chapter_hashing/hash_algorithm.md | 8 ++++---- docs/chapter_hashing/hash_collision.md | 2 +- docs/chapter_hashing/index.md | 6 +++--- docs/chapter_heap/build_heap.md | 8 ++++---- docs/chapter_heap/top_k.md | 2 +- docs/chapter_preface/about_the_book.md | 2 +- docs/chapter_sorting/bubble_sort.md | 2 +- docs/chapter_sorting/insertion_sort.md | 2 +- docs/chapter_sorting/selection_sort.md | 2 +- 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-![复杂度](../assets/covers/chapter_complexity_analysis.jpg){ width="600" } +![时空复杂度](../assets/covers/chapter_complexity_analysis.jpg){ width="600" } !!! abstract - 复杂度犹如浩瀚的算法宇宙中的时空向导。 + 复杂度分析犹如浩瀚的算法宇宙中的时空向导。 它带领我们在时间与空间这两个维度上深入探索,寻找更优雅的解决方案。 diff --git a/docs/chapter_computational_complexity/time_complexity.md b/docs/chapter_computational_complexity/time_complexity.md index e9f77c4a8..8153cbab1 100755 --- a/docs/chapter_computational_complexity/time_complexity.md +++ b/docs/chapter_computational_complexity/time_complexity.md @@ -1210,7 +1210,7 @@ $$ ![常数阶、线性阶和平方阶的时间复杂度](time_complexity.assets/time_complexity_constant_linear_quadratic.png) -以冒泡排序为例,外层循环执行 $n - 1$ 次,内层循环执行 $n-1, n-2, \cdots, 2, 1$ 次,平均为 $n / 2$ 次,因此时间复杂度为 $O((n - 1) n / 2) = O(n^2)$ 。 +以冒泡排序为例,外层循环执行 $n - 1$ 次,内层循环执行 $n-1, n-2, \dots, 2, 1$ 次,平均为 $n / 2$ 次,因此时间复杂度为 $O((n - 1) n / 2) = O(n^2)$ 。 === "Java" @@ -1596,7 +1596,13 @@ $$ !!! tip - “一分为 $m$”对应的时间复杂度 $O(\log_m n)$ 。我们通常会省略底数 $m$ ,直接将其记为 $O(\log n)$ 。 + 准确来说,“一分为 $m$”对应的时间复杂度是 $O(\log_m n)$ 。而通过对数换底公式,我们可以得到具有不同底数的、相等的时间复杂度: + + $$ + O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n) + $$ + + 因此我们通常会省略底数 $m$ ,将对数阶直接记为 $O(\log n)$ 。 ### 线性对数阶 $O(n \log n)$ @@ -1683,7 +1689,7 @@ $$ 阶乘阶对应数学上的“全排列”问题。给定 $n$ 个互不重复的元素,求其所有可能的排列方案,方案数量为: $$ -n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1 +n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1 $$ 阶乘通常使用递归实现。例如在以下代码中,第一层分裂出 $n$ 个,第二层分裂出 $n - 1$ 个,以此类推,直至第 $n$ 层时停止分裂: diff --git a/docs/chapter_data_structure/summary.md b/docs/chapter_data_structure/summary.md index 2f17c078b..571a6ba8d 100644 --- a/docs/chapter_data_structure/summary.md +++ b/docs/chapter_data_structure/summary.md @@ -15,7 +15,7 @@ !!! question "为什么哈希表同时包含线性数据结构和非线性数据结构?" - 哈希表底层是数组,而为了解决哈希冲突,我们可能会使用“链式地址”(后续散列表章节会讲)。在拉链法中,数组中每个地址(桶)指向一个链表;当这个链表长度超过一定阈值时,又可能被转化为树(通常为红黑树)。因此,哈希表可能同时包含线性(数组、链表)和非线性(树)数据结构。 + 哈希表底层是数组,而为了解决哈希冲突,我们可能会使用“链式地址”(后续哈希表章节会讲)。在拉链法中,数组中每个地址(桶)指向一个链表;当这个链表长度超过一定阈值时,又可能被转化为树(通常为红黑树)。因此,哈希表可能同时包含线性(数组、链表)和非线性(树)数据结构。 !!! question "`char` 类型的长度是 1 byte 吗?" diff --git a/docs/chapter_dynamic_programming/intro_to_dynamic_programming.md b/docs/chapter_dynamic_programming/intro_to_dynamic_programming.md index 910b80b90..7b2726658 100644 --- a/docs/chapter_dynamic_programming/intro_to_dynamic_programming.md +++ b/docs/chapter_dynamic_programming/intro_to_dynamic_programming.md @@ -117,7 +117,7 @@ 我们可以尝试从问题分解的角度分析这道题。设爬到第 $i$ 阶共有 $dp[i]$ 种方案,那么 $dp[i]$ 就是原问题,其子问题包括: $$ -dp[i-1] , dp[i-2] , \cdots , dp[2] , dp[1] +dp[i-1] , dp[i-2] , \dots , dp[2] , dp[1] $$ 由于每轮只能上 $1$ 阶或 $2$ 阶,因此当我们站在第 $i$ 阶楼梯上时,上一轮只可能站在第 $i - 1$ 阶或第 $i - 2$ 阶上。换句话说,我们只能从第 $i -1$ 阶或第 $i - 2$ 阶前往第 $i$ 阶。 diff --git a/docs/chapter_greedy/max_capacity_problem.md b/docs/chapter_greedy/max_capacity_problem.md index 8a0d3d005..2eead7c93 100644 --- a/docs/chapter_greedy/max_capacity_problem.md +++ b/docs/chapter_greedy/max_capacity_problem.md @@ -156,7 +156,7 @@ $$ 比如在状态 $cap[i, j]$ 下,$i$ 为短板、$j$ 为长板。若贪心地将短板 $i$ 向内移动一格,会导致以下状态被“跳过”。**这意味着之后无法验证这些状态的容量大小**。 $$ -cap[i, i+1], cap[i, i+2], \cdots, cap[i, j-2], cap[i, j-1] +cap[i, i+1], cap[i, i+2], \dots, cap[i, j-2], cap[i, j-1] $$ ![移动短板导致被跳过的状态](max_capacity_problem.assets/max_capacity_skipped_states.png) diff --git a/docs/chapter_hashing/hash_algorithm.md b/docs/chapter_hashing/hash_algorithm.md index fb42cf844..387a5ee54 100644 --- a/docs/chapter_hashing/hash_algorithm.md +++ b/docs/chapter_hashing/hash_algorithm.md @@ -198,8 +198,8 @@ index = hash(key) % capacity $$ \begin{aligned} \text{modulus} & = 9 \newline -\text{key} & = \{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \cdots \} \newline -\text{hash} & = \{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\cdots \} +\text{key} & = \{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \dots \} \newline +\text{hash} & = \{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\dots \} \end{aligned} $$ @@ -208,8 +208,8 @@ $$ $$ \begin{aligned} \text{modulus} & = 13 \newline -\text{key} & = \{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \cdots \} \newline -\text{hash} & = \{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \cdots \} +\text{key} & = \{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \dots \} \newline +\text{hash} & = \{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \dots \} \end{aligned} $$ diff --git a/docs/chapter_hashing/hash_collision.md b/docs/chapter_hashing/hash_collision.md index facc8f21c..0de9c3c4c 100644 --- a/docs/chapter_hashing/hash_collision.md +++ b/docs/chapter_hashing/hash_collision.md @@ -204,7 +204,7 @@ ### 多次哈希 -顾名思义,多次哈希方法是使用多个哈希函数 $f_1(x)$ , $f_2(x)$ , $f_3(x)$ , $\cdots$ 进行探测。 +顾名思义,多次哈希方法是使用多个哈希函数 $f_1(x)$ , $f_2(x)$ , $f_3(x)$ , $\dots$ 进行探测。 - **插入元素**:若哈希函数 $f_1(x)$ 出现冲突,则尝试 $f_2(x)$ ,以此类推,直到找到空位后插入元素。 - **查找元素**:在相同的哈希函数顺序下进行查找,直到找到目标元素时返回;或遇到空位或已尝试所有哈希函数,说明哈希表中不存在该元素,则返回 $\text{None}$ 。 diff --git a/docs/chapter_hashing/index.md b/docs/chapter_hashing/index.md index dbcbe3035..dd2235ebd 100644 --- a/docs/chapter_hashing/index.md +++ b/docs/chapter_hashing/index.md @@ -1,13 +1,13 @@ -# 散列表 +# 哈希表

-![散列表](../assets/covers/chapter_hashing.jpg){ width="600" } +![哈希表](../assets/covers/chapter_hashing.jpg){ width="600" }
!!! abstract - 在计算机世界中,散列表如同一位智能的图书管理员。 + 在计算机世界中,哈希表如同一位智能的图书管理员。 他知道如何计算索书号,从而可以快速找到目标书籍。 diff --git a/docs/chapter_heap/build_heap.md b/docs/chapter_heap/build_heap.md index d63541a54..b07bdacf1 100644 --- a/docs/chapter_heap/build_heap.md +++ b/docs/chapter_heap/build_heap.md @@ -102,22 +102,22 @@ 因此,我们可以将各层的“节点数量 $\times$ 节点高度”求和,**从而得到所有节点的堆化迭代次数的总和**。 $$ -T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{(h-1)}\times1 +T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \dots + 2^{(h-1)}\times1 $$ 化简上式需要借助中学的数列知识,先对 $T(h)$ 乘以 $2$ ,得到 $$ \begin{aligned} -T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{h-1}\times1 \newline -2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \cdots + 2^{h}\times1 \newline +T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \dots + 2^{h-1}\times1 \newline +2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \dots + 2^{h}\times1 \newline \end{aligned} $$ 使用错位相减法,用下式 $2 T(h)$ 减去上式 $T(h)$ ,可得 $$ -2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \cdots + 2^{h-1} + 2^h +2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \dots + 2^{h-1} + 2^h $$ 观察上式,发现 $T(h)$ 是一个等比数列,可直接使用求和公式,得到时间复杂度为 diff --git a/docs/chapter_heap/top_k.md b/docs/chapter_heap/top_k.md index 9ea00b3e6..f12f585b1 100644 --- a/docs/chapter_heap/top_k.md +++ b/docs/chapter_heap/top_k.md @@ -8,7 +8,7 @@ ## 方法一:遍历选择 -我们可以进行 $k$ 轮遍历,分别在每轮中提取第 $1$ , $2$ , $\cdots$ , $k$ 大的元素,时间复杂度为 $O(nk)$ 。 +我们可以进行 $k$ 轮遍历,分别在每轮中提取第 $1$ , $2$ , $\dots$ , $k$ 大的元素,时间复杂度为 $O(nk)$ 。 该方法只适用于 $k \ll n$ 的情况,因为当 $k$ 与 $n$ 比较接近时,其时间复杂度趋向于 $O(n^2)$ ,非常耗时。 diff --git a/docs/chapter_preface/about_the_book.md b/docs/chapter_preface/about_the_book.md index 4bfdceeb6..9785784d4 100644 --- a/docs/chapter_preface/about_the_book.md +++ b/docs/chapter_preface/about_the_book.md @@ -23,7 +23,7 @@ 本书主要内容包括: - **复杂度分析**:数据结构和算法的评价维度与方法。时间复杂度、空间复杂度的推算方法、常见类型、示例等。 -- **数据结构**:基本数据类型,数据结构的分类方法。数组、链表、栈、队列、散列表、树、堆、图等数据结构的定义、优缺点、常用操作、常见类型、典型应用、实现方法等。 +- **数据结构**:基本数据类型,数据结构的分类方法。数组、链表、栈、队列、哈希表、树、堆、图等数据结构的定义、优缺点、常用操作、常见类型、典型应用、实现方法等。 - **算法**:搜索、排序、分治、回溯、动态规划、贪心等算法的定义、优缺点、效率、应用场景、解题步骤、示例题目等。 ![Hello 算法内容结构](about_the_book.assets/hello_algo_mindmap.png) diff --git a/docs/chapter_sorting/bubble_sort.md b/docs/chapter_sorting/bubble_sort.md index b3c4d86e5..51869050c 100755 --- a/docs/chapter_sorting/bubble_sort.md +++ b/docs/chapter_sorting/bubble_sort.md @@ -188,6 +188,6 @@ ## 算法特性 -- **时间复杂度为 $O(n^2)$ 、自适应排序** :各轮“冒泡”遍历的数组长度依次为 $n - 1$ , $n - 2$ , $\cdots$ , $2$ , $1$ ,总和为 $(n - 1) n / 2$ 。在引入 `flag` 优化后,最佳时间复杂度可达到 $O(n)$ 。 +- **时间复杂度为 $O(n^2)$ 、自适应排序** :各轮“冒泡”遍历的数组长度依次为 $n - 1$ , $n - 2$ , $\dots$ , $2$ , $1$ ,总和为 $(n - 1) n / 2$ 。在引入 `flag` 优化后,最佳时间复杂度可达到 $O(n)$ 。 - **空间复杂度为 $O(1)$ 、原地排序**:指针 $i$ , $j$ 使用常数大小的额外空间。 - **稳定排序**:由于在“冒泡”中遇到相等元素不交换。 diff --git a/docs/chapter_sorting/insertion_sort.md b/docs/chapter_sorting/insertion_sort.md index 6fbcd9c26..dbcaed7f0 100755 --- a/docs/chapter_sorting/insertion_sort.md +++ b/docs/chapter_sorting/insertion_sort.md @@ -93,7 +93,7 @@ ## 算法特性 -- **时间复杂度 $O(n^2)$ 、自适应排序** :最差情况下,每次插入操作分别需要循环 $n - 1$ , $n-2$ , $\cdots$ , $2$ , $1$ 次,求和得到 $\frac{(n - 1) n}{2}$ ,因此时间复杂度为 $O(n^2)$ 。在遇到有序数据时,插入操作会提前终止。当输入数组完全有序时,插入排序达到最佳时间复杂度 $O(n)$ 。 +- **时间复杂度 $O(n^2)$ 、自适应排序** :最差情况下,每次插入操作分别需要循环 $n - 1$ , $n-2$ , $\dots$ , $2$ , $1$ 次,求和得到 $(n - 1) n / 2$ ,因此时间复杂度为 $O(n^2)$ 。在遇到有序数据时,插入操作会提前终止。当输入数组完全有序时,插入排序达到最佳时间复杂度 $O(n)$ 。 - **空间复杂度 $O(1)$ 、原地排序** :指针 $i$ , $j$ 使用常数大小的额外空间。 - **稳定排序**:在插入操作过程中,我们会将元素插入到相等元素的右侧,不会改变它们的顺序。 diff --git a/docs/chapter_sorting/selection_sort.md b/docs/chapter_sorting/selection_sort.md index 549c32726..a621355fa 100644 --- a/docs/chapter_sorting/selection_sort.md +++ b/docs/chapter_sorting/selection_sort.md @@ -119,7 +119,7 @@ ## 算法特性 -- **时间复杂度为 $O(n^2)$ 、非自适应排序**:外循环共 $n - 1$ 轮,第一轮的未排序区间长度为 $n$ ,最后一轮的未排序区间长度为 $2$ ,即各轮外循环分别包含 $n$ , $n - 1$ , $\cdots$ , $2$ 轮内循环,求和为 $\frac{(n - 1)(n + 2)}{2}$ 。 +- **时间复杂度为 $O(n^2)$ 、非自适应排序**:外循环共 $n - 1$ 轮,第一轮的未排序区间长度为 $n$ ,最后一轮的未排序区间长度为 $2$ ,即各轮外循环分别包含 $n$ , $n - 1$ , $\dots$ , $2$ 轮内循环,求和为 $\frac{(n - 1)(n + 2)}{2}$ 。 - **空间复杂度 $O(1)$ 、原地排序**:指针 $i$ , $j$ 使用常数大小的额外空间。 - **非稳定排序**:在交换元素时,有可能将 `nums[i]` 交换至其相等元素的右边,导致两者的相对顺序发生改变。 diff --git a/mkdocs.yml b/mkdocs.yml index a121f8a1f..3c72ecabf 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -144,7 +144,7 @@ nav: - 1.1   算法无处不在: chapter_introduction/algorithms_are_everywhere.md - 1.2   算法是什么: chapter_introduction/what_is_dsa.md - 1.3   小结: chapter_introduction/summary.md - - 第 2 章   复杂度: + - 第 2 章   时空复杂度: # [icon: material/timer-sand] - chapter_computational_complexity/index.md - 2.1   算法效率评估: chapter_computational_complexity/performance_evaluation.md @@ -173,7 +173,7 @@ nav: - 5.2   队列: chapter_stack_and_queue/queue.md - 5.3   双向队列: chapter_stack_and_queue/deque.md - 5.4   小结: chapter_stack_and_queue/summary.md - - 第 6 章   散列表: + - 第 6 章   哈希表: # [icon: material/table-search] - chapter_hashing/index.md - 6.1   哈希表: chapter_hashing/hash_map.md