From 0c18198c0120f90dc09b56f55615b196c1724793 Mon Sep 17 00:00:00 2001 From: krahets Date: Sun, 13 Aug 2023 19:32:11 +0800 Subject: [PATCH] Fix the figures. Polish the chapter of data structure. --- .../chapter_backtracking/subset_sum_ii.dart | 1 + docs/chapter_array_and_linkedlist/array.md | 14 +++--- .../linked_list.md | 2 +- .../basic_data_types.md | 8 +-- .../character_encoding.md | 46 ++++++++++-------- .../classification_of_data_structure.md | 22 ++++----- .../chapter_data_structure/number_encoding.md | 36 +++++++------- docs/chapter_data_structure/summary.md | 13 ++--- .../unbounded_knapsack_dp_comp_step2.png | Bin 67442 -> 67137 bytes .../unbounded_knapsack_dp_comp_step3.png | Bin 68946 -> 68649 bytes .../unbounded_knapsack_dp_comp_step4.png | Bin 69343 -> 69032 bytes .../unbounded_knapsack_dp_comp_step5.png | Bin 68784 -> 68480 bytes .../unbounded_knapsack_problem.md | 8 +-- docs/chapter_greedy/greedy_algorithm.md | 2 +- 14 files changed, 79 insertions(+), 73 deletions(-) diff --git a/codes/dart/chapter_backtracking/subset_sum_ii.dart b/codes/dart/chapter_backtracking/subset_sum_ii.dart index a545c52ce..877a4a6a5 100644 --- a/codes/dart/chapter_backtracking/subset_sum_ii.dart +++ b/codes/dart/chapter_backtracking/subset_sum_ii.dart @@ -49,6 +49,7 @@ List> subsetSumII(List nums, int target) { return res; } +/* Driver Code */ void main() { List nums = [4, 4, 5]; int target = 9; diff --git a/docs/chapter_array_and_linkedlist/array.md b/docs/chapter_array_and_linkedlist/array.md index fba6faa35..4a348c6d0 100755 --- a/docs/chapter_array_and_linkedlist/array.md +++ b/docs/chapter_array_and_linkedlist/array.md @@ -4,7 +4,7 @@ ![数组定义与存储方式](array.assets/array_definition.png) -**数组初始化**。通常有无初始值和给定初始值两种方式,我们可根据需求选择合适的方法。在未给定初始值的情况下,数组的所有元素通常会被初始化为默认值 $0$ 。 +**数组初始化**。通常有无初始值和给定初始值两种方式,我们可根据需求选择合适的方法。在大多数编程语言中,若未指定初始值,数组的所有元素通常会被默认初始化为 $0$ 。 === "Java" @@ -123,7 +123,7 @@ elementAddr = firtstElementAddr + elementLength * elementIndex 观察上图,我们发现数组首个元素的索引为 $0$ ,这似乎有些反直觉,因为从 $1$ 开始计数会更自然。 - 然而,从地址计算公式的角度看,**索引本质上表示的是内存地址的偏移量**。首个元素的地址偏移量是 $0$ ,因此索引为 $0$ 也是合理的。 + 然而从地址计算公式的角度看,**索引本质上表示的是内存地址的偏移量**。首个元素的地址偏移量是 $0$ ,因此索引为 $0$ 也是合理的。 访问元素的高效性带来了诸多便利。例如,我们可以在 $O(1)$ 时间内随机获取数组中的任意一个元素。 @@ -201,7 +201,7 @@ elementAddr = firtstElementAddr + elementLength * elementIndex ## 数组缺点 -**数组在初始化后长度不可变**。由于系统无法保证数组之后的内存空间是可用的,因此数组长度无法扩展。而若希望扩容数组,则需新建一个数组,然后把原数组元素依次拷贝到新数组,在数组很大的情况下,这是非常耗时的。 +**数组在初始化后长度不可变**。系统无法保证数组之后的内存空间是可用的,因此数组长度无法扩展。而若希望扩容数组,则需新建一个数组,然后把原数组元素依次拷贝到新数组。在数组很大的情况下,这是非常耗时的。 === "Java" @@ -275,7 +275,7 @@ elementAddr = firtstElementAddr + elementLength * elementIndex [class]{}-[func]{extend} ``` -**数组中插入或删除元素效率低下**。如果我们想要在数组中间插入一个元素,由于数组元素在内存中是“紧挨着的”,它们之间没有空间再放任何数据。因此,我们不得不将此索引之后的所有元素都向后移动一位,然后再把元素赋值给该索引。 +**数组中插入或删除元素效率低下**。数组元素在内存中是“紧挨着的”,它们之间没有空间再放任何数据。这意味着如果我们想要在数组中间插入一个元素,就不得不将此索引之后的所有元素都向后移动一位,然后再把元素赋值给该索引。 ![数组插入元素](array.assets/array_insert_element.png) @@ -351,7 +351,7 @@ elementAddr = firtstElementAddr + elementLength * elementIndex [class]{}-[func]{insert} ``` -删除元素也类似,如果我们想要删除索引 $i$ 处的元素,则需要把索引 $i$ 之后的元素都向前移动一位。值得注意的是,删除元素后,原先末尾的元素变得“无意义”了,我们无需特意去修改它。 +删除元素也类似,如果我们想要删除索引 $i$ 处的元素,则需要把索引 $i$ 之后的元素都向前移动一位。值得注意的是,删除元素后,原先末尾的元素变得“无意义”了,所以我们无需特意去修改它。 ![数组删除元素](array.assets/array_remove_element.png) @@ -589,6 +589,6 @@ elementAddr = firtstElementAddr + elementLength * elementIndex - **随机访问**:如果我们想要随机抽取一些样本,那么可以用数组存储,并生成一个随机序列,根据索引实现样本的随机抽取。 - **排序和搜索**:数组是排序和搜索算法最常用的数据结构。例如,快速排序、归并排序、二分查找等都需要在数组上进行。 -- **查找表**:当我们需要快速查找一个元素或者需要查找一个元素的对应关系时,可以使用数组作为查找表。例如,我们有一个字符到其 ASCII 码的映射,可以将字符的 ASCII 码值作为索引,对应的元素存放在数组中的对应位置。 +- **查找表**:当我们需要快速查找一个元素或者需要查找一个元素的对应关系时,可以使用数组作为查找表。假如我们想要实现字符到 ASCII 码的映射,则可以将字符的 ASCII 码值作为索引,对应的元素存放在数组中的对应位置。 - **机器学习**:神经网络中大量使用了向量、矩阵、张量之间的线性代数运算,这些数据都是以数组的形式构建的。数组是神经网络编程中最常使用的数据结构。 -- **数据结构实现**:数组可以用于实现栈、队列、哈希表、堆、图等数据结构。例如,邻接矩阵是图的常见表示之一,它实质上是一个二维数组。 +- **数据结构实现**:数组可以用于实现栈、队列、哈希表、堆、图等数据结构。例如,图的邻接矩阵表示实际上是一个二维数组。 diff --git a/docs/chapter_array_and_linkedlist/linked_list.md b/docs/chapter_array_and_linkedlist/linked_list.md index 2bb2ce5c6..918a774f5 100755 --- a/docs/chapter_array_and_linkedlist/linked_list.md +++ b/docs/chapter_array_and_linkedlist/linked_list.md @@ -4,7 +4,7 @@ 「链表 Linked List」是一种线性数据结构,其每个元素都是一个节点对象,各个节点之间通过指针连接,从当前节点通过指针可以访问到下一个节点。**由于指针记录了下个节点的内存地址,因此无需保证内存地址的连续性**,从而可以将各个节点分散存储在内存各处。 -链表中的「节点 Node」包含两项数据,一是节点「值 Value」,二是指向下一节点的「指针 Pointer」,或称「引用 Reference」。 +链表中的「节点 Node」包含两项数据,一是节点「值 Value」,二是指向下一节点的「引用 Reference」,或称「指针 Pointer」。 ![链表定义与存储方式](linked_list.assets/linkedlist_definition.png) diff --git a/docs/chapter_data_structure/basic_data_types.md b/docs/chapter_data_structure/basic_data_types.md index f17e722ca..4648ea31d 100644 --- a/docs/chapter_data_structure/basic_data_types.md +++ b/docs/chapter_data_structure/basic_data_types.md @@ -36,14 +36,14 @@ 对于上表,需要注意以下几点: - C, C++ 未明确规定基本数据类型大小,而因实现和平台各异。上表遵循 LP64 [数据模型](https://en.cppreference.com/w/cpp/language/types#Properties),其用于 Unix 64 位操作系统(例如 Linux , macOS)。 -- 字符 `char` 的大小在 C, C++ 中为 1 字节,在大多数编程语言中取决于特定的字符编码方法,详见「字符编码」章节。 -- 现代计算机 CPU 通常将 1 字节作为最小寻址内存单元。因此即使表示布尔量仅需 1 位($0$ 或 $1$),它在内存中通常被存储为 1 字节。 +- 字符 `char` 的大小在 C, C++ 中为 1 字节,在大多数编程语言中取决于特定的字符编码方法,详见“字符编码”章节。 +- 即使表示布尔量仅需 1 位($0$ 或 $1$),它在内存中通常被存储为 1 字节。这是因为现代计算机 CPU 通常将 1 字节作为最小寻址内存单元。 -那么,基本数据类型与数据结构之间有什么联系呢?我们知道,数据结构是在计算机中组织与存储数据的方式。它的主语是“结构”,而非“数据”。 +那么,基本数据类型与数据结构之间有什么联系呢?我们知道,数据结构是在计算机中组织与存储数据的方式。它的主语是“结构”而非“数据”。 如果想要表示“一排数字”,我们自然会想到使用数组。这是因为数组的线性结构可以表示数字的相邻关系和顺序关系,但至于存储的内容是整数 `int` 、小数 `float` 、还是字符 `char` ,则与“数据结构”无关。 -换句话说,**基本数据类型提供了数据的“内容类型”,而数据结构提供了数据的“组织方式”**。例如在以下代码中,我们用相同的数据结构(数组)来记录与表示不同的基本数据类型(`int` , `float` , `chat`, `bool`)。 +换句话说,**基本数据类型提供了数据的“内容类型”,而数据结构提供了数据的“组织方式”**。例如以下代码,我们用相同的数据结构(数组)来存储与表示不同的基本数据类型(`int` , `float` , `chat`, `bool`)。 === "Java" diff --git a/docs/chapter_data_structure/character_encoding.md b/docs/chapter_data_structure/character_encoding.md index 380b66d3d..346554c23 100644 --- a/docs/chapter_data_structure/character_encoding.md +++ b/docs/chapter_data_structure/character_encoding.md @@ -1,6 +1,6 @@ # 字符编码 * -在计算机中,所有数据都是以二进制数的形式存储的,字符 `char` 也不例外。为了表示字符,我们需要建立一套「字符集」,规定每个字符和二进制数之间的一一对应关系。有了字符集之后,计算机就可以通过查表完成二进制数到字符的转换。 +在计算机中,所有数据都是以二进制数的形式存储的,字符 `char` 也不例外。为了表示字符,我们需要建立一套“字符集”,规定每个字符和二进制数之间的一一对应关系。有了字符集之后,计算机就可以通过查表完成二进制数到字符的转换。 ## ASCII 字符集 @@ -8,31 +8,33 @@ ![ASCII 码](character_encoding.assets/ascii_table.png) -然而,**ASCII 码仅能够表示英文**。随着计算机的全球化,诞生了一种能够表示更多语言的字符集「EASCII」。它在 ASCII 的 7 位基础上扩展到 8 位,能够表示 256 个不同的字符。在世界范围内,陆续出现了一批适用于不同地区的 EASCII 字符集。这些字符集的前 128 个字符统一为 ASCII 码,后 128 个字符定义不同,以适应不同语言的需求。 +然而,**ASCII 码仅能够表示英文**。随着计算机的全球化,诞生了一种能够表示更多语言的字符集「EASCII」。它在 ASCII 的 7 位基础上扩展到 8 位,能够表示 256 个不同的字符。 + +在世界范围内,陆续出现了一批适用于不同地区的 EASCII 字符集。这些字符集的前 128 个字符统一为 ASCII 码,后 128 个字符定义不同,以适应不同语言的需求。 ## GBK 字符集 -后来人们发现,**EASCII 码仍然无法满足许多语言的字符数量要求**。例如,汉字大约有近十万个,光日常使用的就有几千个。中国国家标准总局于 1980 年发布了「GB2312」字符集,其收录了 6763 个汉字,基本满足了汉字的计算机处理需要。 +后来人们发现,**EASCII 码仍然无法满足许多语言的字符数量要求**。比如汉字大约有近十万个,光日常使用的就有几千个。中国国家标准总局于 1980 年发布了「GB2312」字符集,其收录了 6763 个汉字,基本满足了汉字的计算机处理需要。 -然而,GB2312 无法处理部分的罕见字和繁体字。之后在 GB2312 的基础上,扩展得到了「GBK」字符集,它共收录了 21886 个汉字。在 GBK 编码方案中,ASCII 字符使用一个字节表示,汉字使用两个字节表示。 +然而,GB2312 无法处理部分的罕见字和繁体字。「GBK」字符集是在 GB2312 的基础上扩展得到的,它共收录了 21886 个汉字。在 GBK 的编码方案中,ASCII 字符使用一个字节表示,汉字使用两个字节表示。 ## Unicode 字符集 -随着计算机的蓬勃发展,字符集与编码标准百花齐放,而这带来了许多问题。一方面,这些字符集一般只定义了特定语言的字符,无法在多语言环境下正常工作;另一方面,同一种语言也存在多种字符集标准,如果两台电脑安装的是不同的编码标准,则在信息传递时就会出现乱码。 +随着计算机的蓬勃发展,字符集与编码标准百花齐放,而这带来了许多问题。一方面,这些字符集一般只定义了特定语言的字符,无法在多语言环境下正常工作。另一方面,同一种语言也存在多种字符集标准,如果两台电脑安装的是不同的编码标准,则在信息传递时就会出现乱码。 -那个时代的人们就在想:**如果推出一个足够完整的字符集,将世界范围内的所有语言和符号都收录其中,不就可以解决跨语言环境和乱码问题了吗**?在这种想法的驱动下,一个大而全的字符集 Unicode 应运而生。 +那个时代的研究人员就在想:**如果推出一个足够完整的字符集,将世界范围内的所有语言和符号都收录其中,不就可以解决跨语言环境和乱码问题了吗**?在这种想法的驱动下,一个大而全的字符集 Unicode 应运而生。 「Unicode」的全称为“统一字符编码”,理论上能容纳一百多万个字符。它致力于将全球范围内的字符纳入到统一的字符集之中,提供一种通用的字符集来处理和显示各种语言文字,减少因为编码标准不同而产生的乱码问题。 自 1991 年发布以来,Unicode 不断扩充新的语言与字符。截止 2022 年 9 月,Unicode 已经包含 149186 个字符,包括各种语言的字符、符号、甚至是表情符号等。在庞大的 Unicode 字符集中,常用的字符占用 2 字节,有些生僻的字符占 3 字节甚至 4 字节。 -Unicode 是一种字符集标准,本质上是给每个字符分配一个编号(称为“码点”),**但它并没有规定在计算机中如何存储这些字符码点**。我们不禁会问:当多种长度的 Unicode 码点同时出现在同一个文本中时,系统如何解析字符?例如,给定一个长度为 2 字节的编码,系统如何确认它是一个 2 字节的字符还是两个 1 字节的字符? +Unicode 是一种字符集标准,本质上是给每个字符分配一个编号(称为“码点”),**但它并没有规定在计算机中如何存储这些字符码点**。我们不禁会问:当多种长度的 Unicode 码点同时出现在同一个文本中时,系统如何解析字符?例如给定一个长度为 2 字节的编码,系统如何确认它是一个 2 字节的字符还是两个 1 字节的字符? 对于以上问题,**一种直接的解决方案是将所有字符存储为等长的编码**。如下图所示,“Hello”中的每个字符占用 1 字节,“算法”中的每个字符占用 2 字节。我们可以通过高位填 0 ,将“Hello 算法”中的所有字符都编码为 2 字节长度。这样系统就可以每隔 2 字节解析一个字符,恢复出这个短语的内容了。 ![Unicode 编码示例](character_encoding.assets/unicode_hello_algo.png) -然而,ASCII 码已经向我们证明,编码英文只需要 1 字节。若采用上述方案,英文文本占用空间的大小将会是 ASCII 编码下大小的 2 倍,非常浪费内存空间。因此,我们需要一种更加高效的 Unicode 编码方法。 +然而 ASCII 码已经向我们证明,编码英文只需要 1 字节。若采用上述方案,英文文本占用空间的大小将会是 ASCII 编码下大小的两倍,非常浪费内存空间。因此,我们需要一种更加高效的 Unicode 编码方法。 ## UTF-8 编码 @@ -40,35 +42,41 @@ Unicode 是一种字符集标准,本质上是给每个字符分配一个编号 UTF-8 的编码规则并不复杂,分为两种情况: -- 对于长度为 1 字节的字符,将最高位设置为 $0$ 、其余 7 位设置为 Unicode 码点。值得注意的是,ASCII 字符在 Unicode 字符集中占据了前 128 个码点。也就是说,**UTF-8 编码可以向下兼容 ASCII 码**。这意味着我们可以使用 UTF-8 来解析年代久远的 ASCII 码文本。 -- 对于长度为 $n$ 字节的字符(其中 $n > 1$),将首个字节的高 $n$ 位都设置为 $1$ 、第 $n + 1$ 位设置为 $0$ ;从第二个字节开始,将每个字节的高 2 位都设置为 $10$ ;其余所有位用于填充字符的 Unicode 码点。 +1. 对于长度为 1 字节的字符,将最高位设置为 $0$ 、其余 7 位设置为 Unicode 码点。值得注意的是,ASCII 字符在 Unicode 字符集中占据了前 128 个码点。也就是说,**UTF-8 编码可以向下兼容 ASCII 码**。这意味着我们可以使用 UTF-8 来解析年代久远的 ASCII 码文本。 +2. 对于长度为 $n$ 字节的字符(其中 $n > 1$),将首个字节的高 $n$ 位都设置为 $1$ 、第 $n + 1$ 位设置为 $0$ ;从第二个字节开始,将每个字节的高 2 位都设置为 $10$ ;其余所有位用于填充字符的 Unicode 码点。 + +下图展示了“Hello算法”对应的 UTF-8 编码。观察发现,由于最高 $n$ 位都被设置为 $1$ ,因此系统可以通过读取最高位 $1$ 的个数来解析出字符的长度为 $n$ 。 -下图展示了“Hello算法”对应的 UTF-8 编码。将最高 $n$ 位设置为 $1$ 比较容易理解,可以向系统指出字符的长度为 $n$ 。那么,为什么要将其余所有字节的高 2 位都设置为 $10$ 呢?实际上,这个 $10$ 能够起到校验符的作用,因为在 UTF-8 编码规则下,不可能有字符的最高两位是 $10$ 。这是因为长度为 1 字节的字符的最高一位是 $0$ 。假设系统从一个错误的字节开始解析文本,字节头部的 $10$ 能够帮助系统快速的判断出异常。 +但为什么要将其余所有字节的高 2 位都设置为 $10$ 呢?实际上,这个 $10$ 能够起到校验符的作用。假设系统从一个错误的字节开始解析文本,字节头部的 $10$ 能够帮助系统快速的判断出异常。 + +之所以将 $10$ 当作校验符,是因为在 UTF-8 编码规则下,不可能有字符的最高两位是 $10$ 。这个结论可以用反证法来证明:假设一个字符的最高两位是 $10$ ,说明该字符的长度为 $1$ ,对应 ASCII 码。而 ASCII 码的最高位应该是 $0$ ,与假设矛盾。 ![UTF-8 编码示例](character_encoding.assets/utf-8_hello_algo.png) -除了 UTF-8 之外,常见的编码方式还包括 UTF-16 和 UTF-32 。它们为 Unicode 字符集提供了不同的编码方法。 +除了 UTF-8 之外,常见的编码方式还包括: + +- **UTF-16 编码**:使用 2 或 4 个字节来表示一个字符。所有的 ASCII 字符和常用的非英文字符,都用 2 个字节表示;少数字符需要用到 4 个字节表示。对于 2 字节的字符,UTF-16 编码与 Unicode 码点相等。 +- **UTF-32 编码**:每个字符都使用 4 个字节。这意味着 UTF-32 会比 UTF-8 和 UTF-16 更占用空间,特别是对于 ASCII 字符占比较高的文本。 -- **UTF-16 编码**:使用 2 或 4 个字节来表示一个字符。所有的 ASCII 字符和很多常用的非英文字符,都用 2 个字节表示;少数字符需要用到 4 个字节表示。对于 2 字节的字符,UTF-16 编码与 Unicode 码点相等。 -- **UTF-32 编码**:每个字符都使用 4 个字节。这意味着 UTF-32 会比 UTF-8 和 UTF-16 更占用空间,特别是对于主要使用 ASCII 字符的文本。 +从存储空间的角度看,使用 UTF-8 表示英文字符非常高效,因为它仅需 1 个字节;使用 UTF-16 编码某些非英文字符(例如中文)会更加高效,因为它只需要 2 个字节,而 UTF-8 可能需要 3 个字节。 -从存储空间的角度看,使用 UTF-8 表示英文字符非常高效,因为它仅需 1 个字节;使用 UTF-16 编码某些非英文字符(例如中文)会更加高效,因为它只需要 2 个字节,而 UTF-8 可能需要 3 个字节。从兼容性的角度看,UTF-8 的通用性最佳,许多工具和库都优先支持 UTF-8 。 +从兼容性的角度看,UTF-8 的通用性最佳,许多工具和库都优先支持 UTF-8 。 ## 编程语言的字符编码 -对于以往的大多数编程语言,程序运行中的字符串都采用 UTF-16 或 UTF-32 这类等长的编码。这是因为在等长编码下,我们可以将字符串看作数组来处理,具体来说: +对于以往的大多数编程语言,程序运行中的字符串都采用 UTF-16 或 UTF-32 这类等长的编码。这是因为在等长编码下,我们可以将字符串看作数组来处理,其优点包括: - **随机访问**: UTF-16 编码的字符串可以很容易地进行随机访问。UTF-8 是一种变长编码,要找到第 $i$ 个字符,我们需要从字符串的开始处遍历到第 $i$ 个字符,这需要 $O(n)$ 的时间。 - **字符计数**: 与随机访问类似,计算 UTF-16 字符串的长度也是 $O(1)$ 的操作。但是,计算 UTF-8 编码的字符串的长度需要遍历整个字符串。 - **字符串操作**: 在 UTF-16 编码的字符串中,很多字符串操作(如分割、连接、插入、删除等)都更容易进行。在 UTF-8 编码的字符串上进行这些操作通常需要额外的计算,以确保不会产生无效的 UTF-8 编码。 -编程语言的字符编码方案设计是一个很有趣的话题,涉及到许多因素: +实际上,编程语言的字符编码方案设计是一个很有趣的话题,其涉及到许多因素: - Java 的 `String` 类型使用 UTF-16 编码,每个字符占用 2 字节。这是因为 Java 语言设计之初,人们认为 16 位足以表示所有可能的字符。然而,这是一个不正确的判断。后来 Unicode 规范扩展到了超过 16 位,所以 Java 中的字符现在可能由一对 16 位的值(称为“代理对”)表示。 - JavaScript 和 TypeScript 的字符串使用 UTF-16 编码的原因与 Java 类似。当 JavaScript 语言在 1995 年被 Netscape 公司首次引入时,Unicode 还处于相对早期的阶段,那时候使用 16 位的编码就足够表示所有的 Unicode 字符了。 - C# 使用 UTF-16 编码,主要因为 .NET 平台是由 Microsoft 设计的,而 Microsoft 的很多技术,包括 Windows 操作系统,都广泛地使用 UTF-16 编码。 -由于以上编程语言对字符数量的低估,它们不得不采取“代理对”的方式来表示超过 16 位长度的 Unicode 字符。这是一个不得已为之的无奈之举。一方面,包含代理对的字符串中,一个字符可能占用 2 字节或 4 字节,因此丧失了等长编码的优势。另一方面,处理代理对需要增加额外代码,这增加了编程的复杂性和 Debug 难度。 +由于以上编程语言对字符数量的低估,它们不得不采取“代理对”的方式来表示超过 16 位长度的 Unicode 字符。这是一个不得已为之的无奈之举。一方面,包含代理对的字符串中,一个字符可能占用 2 字节或 4 字节,从而丧失了等长编码的优势。另一方面,处理代理对需要增加额外代码,这增加了编程的复杂性和 Debug 难度。 出于以上原因,部分编程语言提出了不同的编码方案: @@ -76,4 +84,4 @@ UTF-8 的编码规则并不复杂,分为两种情况: - Go 语言的 `string` 类型在内部使用 UTF-8 编码。Go 语言还提供了 `rune` 类型,它用于表示单个 Unicode 码点。 - Rust 语言的 str 和 String 类型在内部使用 UTF-8 编码。Rust 也提供了 char 类型,用于表示单个 Unicode 码点。 -需要注意的是,以上讨论的都是字符串在编程语言中的存储方式,**这和字符串如何在文件中存储或在网络中传输是两个不同的问题**。在文件存储或网络传输中,我们一般会将字符串编码为 UTF-8 格式,以达到最优的兼容性和空间效率。 +需要注意的是,以上讨论的都是字符串在编程语言中的存储方式,**这和字符串如何在文件中存储或在网络中传输是两个不同的问题**。在文件存储或网络传输中,我们通常会将字符串编码为 UTF-8 格式,以达到最优的兼容性和空间效率。 diff --git a/docs/chapter_data_structure/classification_of_data_structure.md b/docs/chapter_data_structure/classification_of_data_structure.md index aaa037985..40bdef27f 100644 --- a/docs/chapter_data_structure/classification_of_data_structure.md +++ b/docs/chapter_data_structure/classification_of_data_structure.md @@ -1,12 +1,12 @@ # 数据结构分类 -数据结构可以从逻辑结构和物理结构两个维度进行分类。 +常见的数据结构包括数组、链表、栈、队列、哈希表、树、堆、图,它们可以从“逻辑结构”和“物理结构”两个维度进行分类。 ## 逻辑结构:线性与非线性 **「逻辑结构」揭示了数据元素之间的逻辑关系**。在数组和链表中,数据按照顺序依次排列,体现了数据之间的线性关系;而在树中,数据从顶部向下按层次排列,表现出祖先与后代之间的派生关系;图则由节点和边构成,反映了复杂的网络关系。 -逻辑结构通常分为“线性”和“非线性”两类。线性结构比较直观,指数据在逻辑关系上呈线性排列;非线性结构则相反,呈非线性排列。 +逻辑结构可被分为“线性”和“非线性”两大类。线性结构比较直观,指数据在逻辑关系上呈线性排列;非线性结构则相反,呈非线性排列。 - **线性数据结构**:数组、链表、栈、队列、哈希表。 - **非线性数据结构**:树、堆、图、哈希表。 @@ -15,27 +15,27 @@ 非线性数据结构可以进一步被划分为树形结构和网状结构。 -- **线性结构**:数组、链表、队列、栈、哈希表,元素存在一对一的顺序关系。 -- **树形结构**:树、堆、哈希表,元素存在一对多的关系。 -- **网状结构**:图,元素存在多对多的关系。 +- **线性结构**:数组、链表、队列、栈、哈希表,元素之间是一对一的顺序关系。 +- **树形结构**:树、堆、哈希表,元素之间是一对多的关系。 +- **网状结构**:图,元素之间是多对多的关系。 ## 物理结构:连续与离散 在计算机中,内存和硬盘是两种主要的存储硬件设备。硬盘主要用于长期存储数据,容量较大(通常可达到 TB 级别)、速度较慢。内存用于运行程序时暂存数据,速度较快,但容量较小(通常为 GB 级别)。 -**在算法运行过程中,相关数据都存储在内存中**。下图展示了一个计算机内存条,其中每个黑色方块都包含一块内存空间。我们可以将内存想象成一个巨大的 Excel 表格,其中每个单元格都可以存储 1 byte 的数据,在算法运行时,所有数据都被存储在这些单元格中。 +**在算法运行过程中,相关数据都存储在内存中**。下图展示了一个计算机内存条,其中每个黑色方块都包含一块内存空间。我们可以将内存想象成一个巨大的 Excel 表格,其中每个单元格都可以存储一定大小的数据,在算法运行时,所有数据都被存储在这些单元格中。 -**系统通过「内存地址 Memory Location」来访问目标内存位置的数据**。计算机根据特定规则为表格中的每个单元格分配编号,确保每个内存空间都有唯一的内存地址。有了这些地址,程序便可以访问内存中的数据。 +**系统通过内存地址来访问目标位置的数据**。计算机根据特定规则为表格中的每个单元格分配编号,确保每个内存空间都有唯一的内存地址。有了这些地址,程序便可以访问内存中的数据。 ![内存条、内存空间、内存地址](classification_of_data_structure.assets/computer_memory_location.png) -内存是所有程序的共享资源,当内存被某个程序占用时,其他程序无法同时使用。**因此,在数据结构与算法的设计中,内存资源是一个重要的考虑因素**。例如,算法所占用的内存峰值不应超过系统剩余空闲内存;如果运行的程序很多并且缺少大量连续的内存空间,那么所选用的数据结构必须能够存储在离散的内存空间内。 +内存是所有程序的共享资源,当某块内存被某个程序占用时,则无法被其他程序同时使用了。**因此在数据结构与算法的设计中,内存资源是一个重要的考虑因素**。比如,算法所占用的内存峰值不应超过系统剩余空闲内存;如果缺少连续大块的内存空间,那么所选用的数据结构必须能够存储在离散的内存空间内。 -**「物理结构」反映了数据在计算机内存中的存储方式**,可分为数组的连续空间存储和链表的离散空间存储。物理结构从底层决定了数据的访问、更新、增删等操作方法,同时在时间效率和空间效率方面呈现出互补的特点。 +**「物理结构」反映了数据在计算机内存中的存储方式**,可分为连续空间存储(数组)和离散空间存储(链表)。物理结构从底层决定了数据的访问、更新、增删等操作方法,同时在时间效率和空间效率方面呈现出互补的特点。 ![连续空间存储与离散空间存储](classification_of_data_structure.assets/classification_phisical_structure.png) -**所有数据结构都是基于数组、链表或二者的组合实现的**。例如,栈和队列既可以使用数组实现,也可以使用链表实现;而哈希表的实现可能同时包含数组和链表。 +值得说明的是,**所有数据结构都是基于数组、链表或二者的组合实现的**。例如,栈和队列既可以使用数组实现,也可以使用链表实现;而哈希表的实现可能同时包含数组和链表。 - **基于数组可实现**:栈、队列、哈希表、树、堆、图、矩阵、张量(维度 $\geq 3$ 的数组)等。 - **基于链表可实现**:栈、队列、哈希表、树、堆、图等。 @@ -44,4 +44,4 @@ !!! tip - 如若感觉理解物理结构有困难,建议先阅读下一章“数组与链表”,然后再回头理解物理结构的含义。数组与链表是其他所有数据结构的基石,建议你投入更多时间深入了解这两种基本数据结构。 + 如若感觉理解物理结构有困难,建议先阅读下一章“数组与链表”,然后再回顾本节内容。 diff --git a/docs/chapter_data_structure/number_encoding.md b/docs/chapter_data_structure/number_encoding.md index 557608098..b95c19eb2 100644 --- a/docs/chapter_data_structure/number_encoding.md +++ b/docs/chapter_data_structure/number_encoding.md @@ -2,11 +2,13 @@ !!! note - 在本书中,标题带有的 * 符号的是选读章节。如果你时间有限或感到理解困难,建议先跳过,等学完必读章节后再单独攻克。 + 在本书中,标题带有的 * 符号的是选读章节。如果你时间有限或感到理解困难,可以先跳过,等学完必读章节后再单独攻克。 ## 原码、反码和补码 -从上一节的表格中我们发现,所有整数类型能够表示的负数都比正数多一个。例如,`byte` 的取值范围是 $[-128, 127]$ 。这个现象比较反直觉,它的内在原因涉及到原码、反码、补码的相关知识。在展开分析之前,我们首先给出三者的定义: +从上一节的表格中我们发现,所有整数类型能够表示的负数都比正数多一个。例如,`byte` 的取值范围是 $[-128, 127]$ 。这个现象比较反直觉,它的内在原因涉及到原码、反码、补码的相关知识。 + +在展开分析之前,我们首先给出三者的定义: - **原码**:我们将数字的二进制表示的最高位视为符号位,其中 $0$ 表示正数,$1$ 表示负数,其余位表示数字的值。 - **反码**:正数的反码与其原码相同,负数的反码是对其原码除符号位外的所有位取反。 @@ -14,7 +16,7 @@ ![原码、反码与补码之间的相互转换](number_encoding.assets/1s_2s_complement.png) -显然,「原码」最为直观,**然而数字却是以「补码」的形式存储在计算机中的**。这是因为原码存在一些局限性。 +显然「原码」最为直观。但实际上,**数字是以「补码」的形式存储在计算机中的**。这是因为原码存在一些局限性。 一方面,**负数的原码不能直接用于运算**。例如,我们在原码下计算 $1 + (-2)$ ,得到的结果是 $-3$ ,这显然是不对的。 @@ -27,20 +29,20 @@ $$ \end{aligned} $$ -为了解决此问题,计算机引入了「反码」。例如,我们先将原码转换为反码,并在反码下计算 $1 + (-2)$ ,并将结果从反码转化回原码,则可得到正确结果 $-1$ 。 +为了解决此问题,计算机引入了「反码」。如果我们先将原码转换为反码,并在反码下计算 $1 + (-2)$ ,最后将结果从反码转化回原码,则可得到正确结果 $-1$ 。 $$ \begin{aligned} & 1 + (-2) \newline -& = 0000 \space 0001 \space \text{(原码)} + 1000 \space 0010 \space \text{(原码)} \newline +& \rightarrow 0000 \space 0001 \space \text{(原码)} + 1000 \space 0010 \space \text{(原码)} \newline & = 0000 \space 0001 \space \text{(反码)} + 1111 \space 1101 \space \text{(反码)} \newline & = 1111 \space 1110 \space \text{(反码)} \newline & = 1000 \space 0001 \space \text{(原码)} \newline -& = -1 +& \rightarrow -1 \end{aligned} $$ -另一方面,**数字零的原码有 $+0$ 和 $-0$ 两种表示方式**。这意味着数字零对应着两个不同的二进制编码,而这可能会带来歧义问题。例如,在条件判断中,如果没有区分正零和负零,可能会导致错误的判断结果。如果我们想要处理正零和负零歧义,则需要引入额外的判断操作,其可能会降低计算机的运算效率。 +另一方面,**数字零的原码有 $+0$ 和 $-0$ 两种表示方式**。这意味着数字零对应着两个不同的二进制编码,其可能会带来歧义。比如在条件判断中,如果没有区分正零和负零,则可能会导致判断结果出错。而如果我们想要处理正零和负零歧义,则需要引入额外的判断操作,其可能会降低计算机的运算效率。 $$ \begin{aligned} @@ -49,7 +51,7 @@ $$ \end{aligned} $$ -与原码一样,反码也存在正负零歧义问题。为此,计算机进一步引入了「补码」。那么,补码有什么作用呢?我们先来分析一下负零的补码的计算过程: +与原码一样,反码也存在正负零歧义问题,因此计算机进一步引入了「补码」。我们先来观察一下负零的原码、反码、补码的转换过程: $$ \begin{aligned} @@ -59,30 +61,30 @@ $$ \end{aligned} $$ -在负零的反码基础上加 $1$ 会产生进位,而由于 byte 的长度只有 8 位,因此溢出到第 9 位的 $1$ 会被舍弃。**从而得到负零的补码为 $0000 \space 0000$ ,与正零的补码相同**。这意味着在补码表示中只存在一个零,从而解决了正负零歧义问题。 +在负零的反码基础上加 $1$ 会产生进位,但 `byte` 类型的长度只有 8 位,因此溢出到第 9 位的 $1$ 会被舍弃。也就是说,**负零的补码为 $0000 \space 0000$ ,与正零的补码相同**。这意味着在补码表示中只存在一个零,正负零歧义从而得到解决。 -还剩余最后一个疑惑:byte 的取值范围是 $[-128, 127]$ ,多出来的一个负数 $-128$ 是如何得到的呢?我们注意到,区间 $[-127, +127]$ 内的所有整数都有对应的原码、反码和补码,并且原码和补码之间是可以互相转换的。 +还剩余最后一个疑惑:`byte` 类型的取值范围是 $[-128, 127]$ ,多出来的一个负数 $-128$ 是如何得到的呢?我们注意到,区间 $[-127, +127]$ 内的所有整数都有对应的原码、反码和补码,并且原码和补码之间是可以互相转换的。 然而,**补码 $1000 \space 0000$ 是一个例外,它并没有对应的原码**。根据转换方法,我们得到该补码的原码为 $0000 \space 0000$ 。这显然是矛盾的,因为该原码表示数字 $0$ ,它的补码应该是自身。计算机规定这个特殊的补码 $1000 \space 0000$ 代表 $-128$ 。实际上,$(-1) + (-127)$ 在补码下的计算结果就是 $-128$ 。 $$ \begin{aligned} & (-127) + (-1) \newline -& = 1111 \space 1111 \space \text{(原码)} + 1000 \space 0001 \space \text{(原码)} \newline +& \rightarrow 1111 \space 1111 \space \text{(原码)} + 1000 \space 0001 \space \text{(原码)} \newline & = 1000 \space 0000 \space \text{(反码)} + 1111 \space 1110 \space \text{(反码)} \newline & = 1000 \space 0001 \space \text{(补码)} + 1111 \space 1111 \space \text{(补码)} \newline & = 1000 \space 0000 \space \text{(补码)} \newline -& = -128 +& \rightarrow -128 \end{aligned} $$ -你可能已经发现,上述的所有计算都是加法运算。这暗示着一个重要事实:**计算机内部的硬件电路主要是基于加法运算设计的**。这是因为加法运算相对于其他运算(比如乘法、除法和减法)来说,硬件实现起来更简单,更容易进行并行化处理,从而提高运算速度。 +你可能已经发现,上述的所有计算都是加法运算。这暗示着一个重要事实:**计算机内部的硬件电路主要是基于加法运算设计的**。这是因为加法运算相对于其他运算(比如乘法、除法和减法)来说,硬件实现起来更简单,更容易进行并行化处理,运算速度更快。 -然而,这并不意味着计算机只能做加法。**通过将加法与一些基本逻辑运算结合,计算机能够实现各种其他的数学运算**。例如,计算减法 $a - b$ 可以转换为计算加法 $a + (-b)$ ;计算乘法和除法可以转换为计算多次加法或减法。 +请注意,这并不意味着计算机只能做加法。**通过将加法与一些基本逻辑运算结合,计算机能够实现各种其他的数学运算**。例如,计算减法 $a - b$ 可以转换为计算加法 $a + (-b)$ ;计算乘法和除法可以转换为计算多次加法或减法。 -现在,我们可以总结出计算机使用补码的原因:基于补码表示,计算机可以用同样的电路和操作来处理正数和负数的加法,不需要设计特殊的硬件电路来处理减法,并且无需特别处理正负零的歧义问题。这大大简化了硬件设计,并提高了运算效率。 +现在我们可以总结出计算机使用补码的原因:基于补码表示,计算机可以用同样的电路和操作来处理正数和负数的加法,不需要设计特殊的硬件电路来处理减法,并且无需特别处理正负零的歧义问题。这大大简化了硬件设计,提高了运算效率。 -补码的设计非常精妙,由于篇幅关系我们先介绍到这里。建议有兴趣的读者进一步深度了解。 +补码的设计非常精妙,因篇幅关系我们就先介绍到这里,建议有兴趣的读者进一步深度了解。 ## 浮点数编码 @@ -123,7 +125,7 @@ $$ ![IEEE 754 标准下的 float 表示方式](number_encoding.assets/ieee_754_float.png) -以上图为例,$\mathrm{S} = 0$ , $\mathrm{E} = 124$ ,$\mathrm{N} = 2^{-2} + 2^{-3} = 0.375$ ,易得 +给定一个示例数据 $\mathrm{S} = 0$ , $\mathrm{E} = 124$ ,$\mathrm{N} = 2^{-2} + 2^{-3} = 0.375$ ,则有: $$ \text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875 diff --git a/docs/chapter_data_structure/summary.md b/docs/chapter_data_structure/summary.md index 74cdc549e..2f17c078b 100644 --- a/docs/chapter_data_structure/summary.md +++ b/docs/chapter_data_structure/summary.md @@ -1,15 +1,10 @@ # 小结 -**数据结构分类** - - 数据结构可以从逻辑结构和物理结构两个角度进行分类。逻辑结构描述了数据元素之间的逻辑关系,而物理结构描述了数据在计算机内存中的存储方式。 - 常见的逻辑结构包括线性、树状和网状等。通常我们根据逻辑结构将数据结构分为线性(数组、链表、栈、队列)和非线性(树、图、堆)两种。哈希表的实现可能同时包含线性和非线性结构。 - 当程序运行时,数据被存储在计算机内存中。每个内存空间都拥有对应的内存地址,程序通过这些内存地址访问数据。 - 物理结构主要分为连续空间存储(数组)和离散空间存储(链表)。所有数据结构都是由数组、链表或两者的组合实现的。 - -**数据类型与编码** - -- 计算机中的基本数据类型包括整数 byte, short, int, long 、浮点数 float, double 、字符 char 和布尔 boolean 。它们的取值范围取决于占用空间大小和表示方式。 +- 计算机中的基本数据类型包括整数 `byte` , `short` , `int` , `long` 、浮点数 `float` , `double` 、字符 `char` 和布尔 `boolean` 。它们的取值范围取决于占用空间大小和表示方式。 - 原码、反码和补码是在计算机中编码数字的三种方法,它们之间是可以相互转换的。整数的原码的最高位是符号位,其余位是数字的值。 - 整数在计算机中是以补码的形式存储的。在补码表示下,计算机可以对正数和负数的加法一视同仁,不需要为减法操作单独设计特殊的硬件电路,并且不存在正负零歧义的问题。 - 浮点数的编码由 1 位符号位、8 位指数位和 23 位分数位构成。由于存在指数位,浮点数的取值范围远大于整数,代价是牺牲了精度。 @@ -20,8 +15,8 @@ !!! 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“拉链法”(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。 -!!! question "char 类型的长度是 1 byte 吗?" +!!! question "`char` 类型的长度是 1 byte 吗?" - char 类型的长度由编程语言采用的编码方法决定。例如,Java, JS, TS, C# 都采用 UTF-16 编码(保存 Unicode 码点),因此 char 类型的长度为 2 bytes 。 + `char` 类型的长度由编程语言采用的编码方法决定。例如,Java, JS, TS, C# 都采用 UTF-16 编码(保存 Unicode 码点),因此 char 类型的长度为 2 bytes 。 diff --git a/docs/chapter_dynamic_programming/unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step2.png b/docs/chapter_dynamic_programming/unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step2.png index 85fc94a01870e41b5e32063196962dcbb8d6d1bb..42a766f4c15f78b9c1e88f517a3f03d9e5ad9796 100644 GIT binary patch delta 37328 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