From 7cfe6d7efe5415eaa5db7e1a9e3906e60dfc1d83 Mon Sep 17 00:00:00 2001 From: krahets Date: Wed, 17 May 2023 04:38:32 +0800 Subject: [PATCH] Add the section of number encoding and character encoding --- .../basic_data_types.md | 125 +++++++++++ .../character_encoding.assets/ascii_table.png | Bin 0 -> 197683 bytes .../unicode_hello_algo.png | Bin 0 -> 61664 bytes .../utf-8_hello_algo.png | Bin 0 -> 74825 bytes .../character_encoding.md | 65 ++++++ .../computer_memory_location.png | Bin .../classification_of_data_structure.md | 14 +- .../data_and_memory.assets/ieee_754_float.png | Bin 73710 -> 0 bytes .../chapter_data_structure/data_and_memory.md | 211 ------------------ .../1s_2s_complement.png | Bin 0 -> 64536 bytes .../number_encoding.assets/ieee_754_float.png | Bin 0 -> 73859 bytes .../chapter_data_structure/number_encoding.md | 145 ++++++++++++ docs/chapter_data_structure/summary.md | 11 +- mkdocs.yml | 10 +- 14 files changed, 360 insertions(+), 221 deletions(-) create mode 100644 docs/chapter_data_structure/basic_data_types.md create mode 100644 docs/chapter_data_structure/character_encoding.assets/ascii_table.png create mode 100644 docs/chapter_data_structure/character_encoding.assets/unicode_hello_algo.png create mode 100644 docs/chapter_data_structure/character_encoding.assets/utf-8_hello_algo.png create mode 100644 docs/chapter_data_structure/character_encoding.md rename docs/chapter_data_structure/{data_and_memory.assets => classification_of_data_structure.assets}/computer_memory_location.png (100%) delete mode 100644 docs/chapter_data_structure/data_and_memory.assets/ieee_754_float.png delete mode 100644 docs/chapter_data_structure/data_and_memory.md create mode 100644 docs/chapter_data_structure/number_encoding.assets/1s_2s_complement.png create mode 100644 docs/chapter_data_structure/number_encoding.assets/ieee_754_float.png create mode 100644 docs/chapter_data_structure/number_encoding.md diff --git a/docs/chapter_data_structure/basic_data_types.md b/docs/chapter_data_structure/basic_data_types.md new file mode 100644 index 00000000..5a1fc0e9 --- /dev/null +++ b/docs/chapter_data_structure/basic_data_types.md @@ -0,0 +1,125 @@ +# 基本数据类型 + +谈及计算机中的数据,我们会想到文本、图片、视频、语音、3D 模型等各种形式。尽管这些数据的组织形式各异,但它们都由各种基本数据类型构成。 + +**基本数据类型是 CPU 可以直接进行运算的类型,在算法中直接被使用**。它包括: + +- 整数类型 `byte` , `short` , `int` , `long` ; +- 浮点数类型 `float` , `double` ,用于表示小数; +- 字符类型 `char` ,用于表示各种语言的字母、标点符号、甚至表情符号等; +- 布尔类型 `bool` ,用于表示“是”与“否”判断; + +**所有基本数据类型都以二进制的形式存储在计算机中**。在计算机中,我们将 $1$ 个二进制位称为 $1$ 比特,并规定 $1$ 字节(byte)由 $8$ 比特(bits)组成。基本数据类型的取值范围取决于其占用的空间大小,例如: + +- 整数类型 `byte` 占用 $1$ byte = $8$ bits ,可以表示 $2^{8}$ 个不同的数字; +- 整数类型 `int` 占用 $4$ bytes = $32$ bits ,可以表示 $2^{32}$ 个数字; + +下表列举了各种基本数据类型的占用空间、取值范围和默认值。此表格无需硬背,大致理解即可,需要时可以通过查表来回忆。 + +
+ +| 类型 | 符号 | 占用空间 | 最小值 | 最大值 | 默认值 | +| ------ | -------- | ---------------- | ------------------------ | ----------------------- | -------------- | +| 整数 | `byte` | 1 byte | $-2^7$ ($-128$) | $2^7 - 1$ ($127$) | $0$ | +| | `short` | 2 bytes | $-2^{15}$ | $2^{15} - 1$ | $0$ | +| | `int` | 4 bytes | $-2^{31}$ | $2^{31} - 1$ | $0$ | +| | `long` | 8 bytes | $-2^{63}$ | $2^{63} - 1$ | $0$ | +| 浮点数 | `float` | 4 bytes | $1.175 \times 10^{-38}$ | $3.403 \times 10^{38}$ | $0.0 f$ | +| | `double` | 8 bytes | $2.225 \times 10^{-308}$ | $1.798 \times 10^{308}$ | $0.0$ | +| 字符 | `char` | 2 bytes / 1 byte | $0$ | $2^{16} - 1$ | $0$ | +| 布尔 | `bool` | 1 byte / 1 bit | $\text{false}$ | $\text{true}$ | $\text{false}$ | + +
+ +那么,基本数据类型与数据结构之间有什么联系与区别呢?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”**。如以下代码所示,我们可以使用不同基本数据类型来初始化数组。 + +=== "Java" + + ```java title="" + /* 使用多种「基本数据类型」来初始化「数组」 */ + int[] numbers = new int[5]; + float[] decimals = new float[5]; + char[] characters = new char[5]; + boolean[] booleans = new boolean[5]; + ``` + +=== "C++" + + ```cpp title="" + /* 使用多种「基本数据类型」来初始化「数组」 */ + int numbers[5]; + float decimals[5]; + char characters[5]; + bool booleans[5]; + ``` + +=== "Python" + + ```python title="" + # Python 的 list 可以自由存储各种基本数据类型和对象 + list = [0, 0.0, 'a', False] + ``` + +=== "Go" + + ```go title="" + // 使用多种「基本数据类型」来初始化「数组」 + var numbers = [5]int{} + var decimals = [5]float64{} + var characters = [5]byte{} + var booleans = [5]bool{} + ``` + +=== "JavaScript" + + ```javascript title="" + /* JavaScript 的数组可以自由存储各种基本数据类型和对象 */ + const array = [0, 0.0, 'a', false]; + ``` + +=== "TypeScript" + + ```typescript title="" + /* 使用多种「基本数据类型」来初始化「数组」 */ + const numbers: number[] = []; + const characters: string[] = []; + const booleans: boolean[] = []; + ``` + +=== "C" + + ```c title="" + /* 使用多种「基本数据类型」来初始化「数组」 */ + int numbers[10]; + float decimals[10]; + char characters[10]; + bool booleans[10]; + ``` + +=== "C#" + + ```csharp title="" + /* 使用多种「基本数据类型」来初始化「数组」 */ + int[] numbers = new int[5]; + float[] decimals = new float[5]; + char[] characters = new char[5]; + bool[] booleans = new bool[5]; + ``` + +=== "Swift" + + ```swift title="" + /* 使用多种「基本数据类型」来初始化「数组」 */ + let numbers = Array(repeating: Int(), count: 5) + let decimals = Array(repeating: Double(), count: 5) + let characters = Array(repeating: Character("a"), count: 5) + let booleans = Array(repeating: Bool(), count: 5) + ``` + +=== "Zig" + + ```zig title="" + + ``` diff --git a/docs/chapter_data_structure/character_encoding.assets/ascii_table.png b/docs/chapter_data_structure/character_encoding.assets/ascii_table.png new file mode 100644 index 0000000000000000000000000000000000000000..01dd642f3b6d2b7569a46a0f3e095169797d61ff GIT binary patch literal 197683 zcmZ_01ymeSx95un3lJ>0ySuvwcXw|fNU-4UZo%DMg1fsr0fIZh-CyzDJ8#y!c{jD7 zx>?m#)#vQ9<+uNpFhzMuM0gx{FfcGgsc&M+U|`U|ONeP$2w*2fC=Cwmz#Ww(MZl^i z@s5Bufo7Ug=5lgiG{9?EFlca8FvyQXfDH^B=RY6*_=q$(?*IE*8Jz0Bj{(jZVhINI zpX0&6w1NG{KSSW><1_!Yhs**0-)GE$_+LjugXKW}uh$UMz|mknV#1!l4$l6YmLnJ# zQvb&f_%B){T;RUJq{M_(UBUnT{h7lj{)_4@L;&+1ZZsa9OlA_5Jb+3VdIS~~{RIutsYbbJPxg%}HLd_I(sMsF{C+wZsa&JCF8q|@=!x5-o9qe<0wW)(d(ujBb6 z6*Wq=Te0`^_t&oXU)8dH|I;R1-2HCb5#aMUX7F|`_ic3ZqDa4SYaU zq^_)Y-fS^168axDF!;z4)nI>Y=@R(&=RWK0Fv>T<_k(xOet6SN-`~#P--aax@(P_Wm=pdOC$wn)4`&6gc7p!R-J0yB=)~vkz~}8K`^_?5isEsX zQjlA?!3hGMX#u#^*Nt50FQ-i+;7Ido0g@+wK_!nqz?a@LwH+RBN`**;fRy!v*2aew&A@!3MsFpbAk;k+(N5cGIP}A79$w((~x@9V_&pKw28c_%vt29j!39i}+jD z%UG9U04+!v2ZDb5F~;YGwEA39@M)&t^(%&<&|fQwQd-oRvej@3+HgLsWsP}kN%6yc z-W}L2hI|@x@O0;vnJhL=K7M?9H>ILm*I!EDgaxT&&U^~gVd!YGpSN=Q2;A7@VwshL z_|j0RDN=^>E&a)G)Pi~vJg-1jncyG4)Jx5084`YDhbjIp*n#hvs$OkccbB=inpaVX z+Juv5W_CEset$Xed2aGinKD`C(IW)466!lZg!6_y(Dzm*M`z7@a0vdQj-_TIpziMH zAefotvVgxKhQ0zF_a(_c-r(dphwISyG$#E)<;id4Gb`ZByOorlakiu%kLKh5yU} zEz1({w`Z+Q)~`wl80)Gy9!|E4k4tSD{5UgQqM8nyQB@5n)T{p%Zlwg%*pMXpv#hkp zK|o#90vE%ubEDvu;-0ZRUCn?Su_k)K{BiAU^W#!I9>pz^9eLfF5-LiZ-xnGBoS*!jo^wT)tMS~czwQWIhTGH;J`kcC zuA)+nX60VadW)V%=_o=Z1tLK2aH6irG|jT8RSEC8D=dJh1ml6LO;e&D2I|!)v?k3A zqaU(-3>{04qa$Egd}*H^_V%lsQz1;S<#IK?F327Y&qpLC?c}{mOejX?r*FZ${j(6% zSwnt7hObenRt++n^)93?{NabCQBR3)zr!&*Fs{1dE+Sf~G0Dyf%>w<&7s72Gxm#vj z?+^`scV?cFktClP>|@}-^o`F%vK4daKjw%x*OJ2PTD0N5TF1X|FXT9#1`2u`bYNMW z$$_$a4qwbjFpxSWbtDJpIs}~|vtmbdX*9Tz*#~EVCDheNZz}PoB`S8g$ zPDS1AWxwnF?U|lrif_?)g19uPXrB1U?#!GYUiEYBK6N2?v3~x}St+no{?eYn^}4Y& zh}`j|Y3C|evBu_?3h&95fl3N(-^(iAP8myUQ5mEe)Hp8(CA;2V&)b;upadnOw(Y>; 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+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 。 + +如下表所示,为各个编程语言的字符串默认编码方式。由于 UTF-16 和 UTF-32 属于等长的编码方法,因此编程语言可以直接计算字符串的长度,也可以快速地访问字符串中的任意字符。而如果使用 UTF-8 这种变长的编码方法,编程语言往往需要额外维护一个字符数组,才能实现高效的随机访问。 + +

diff --git a/docs/chapter_data_structure/data_and_memory.assets/computer_memory_location.png b/docs/chapter_data_structure/classification_of_data_structure.assets/computer_memory_location.png similarity index 100% rename from docs/chapter_data_structure/data_and_memory.assets/computer_memory_location.png rename to docs/chapter_data_structure/classification_of_data_structure.assets/computer_memory_location.png diff --git a/docs/chapter_data_structure/classification_of_data_structure.md b/docs/chapter_data_structure/classification_of_data_structure.md index 079cde1f..7f3f3dd0 100644 --- a/docs/chapter_data_structure/classification_of_data_structure.md +++ b/docs/chapter_data_structure/classification_of_data_structure.md @@ -15,11 +15,17 @@ ## 物理结构:连续与离散 -!!! note +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)。 - 如若阅读起来有困难,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。 +**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。 -**「物理结构」体现了数据在计算机内存中的存储方式**,可以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。 +**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。 + +![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。 + +**「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。 ![连续空间存储与离散空间存储](classification_of_data_structure.assets/classification_phisical_structure.png) @@ -32,4 +38,4 @@ !!! tip - 数组与链表是其他所有数据结构的“底层积木”,建议读者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。 diff --git a/docs/chapter_data_structure/data_and_memory.assets/ieee_754_float.png 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z%>^32>5*sA6yh)UAGPa1ehAy1>YXhu{{Y2~EVA&I1!HE~9{$I2DZ z9QFkR3+HpF3!rRR_zu3X&({v7F@_1$CT-69x^?-%Pzf*J+Z7nIuKh z5p9c7JQxZ0+f4p~-)-1-9r3 zB#Dw{M+4fpLFpo(w|u_Rv;*|_>O4u}6+mBN - -| 类别 | 符号 | 占用空间 | 取值范围 | 默认值 | -| ------ | ----------- | ----------------- | ---------------------------------------------- | -------------- | -| 整数 | byte | 1 byte | $-2^7$ ~ $2^7 - 1$ ( $-128$ ~ $127$ ) | $0$ | -| | short | 2 bytes | $-2^{15}$ ~ $2^{15} - 1$ | $0$ | -| | **int** | 4 bytes | $-2^{31}$ ~ $2^{31} - 1$ | $0$ | -| | long | 8 bytes | $-2^{63}$ ~ $2^{63} - 1$ | $0$ | -| 浮点数 | **float** | 4 bytes | $-3.4 \times 10^{38}$ ~ $3.4 \times 10^{38}$ | $0.0$ f | -| | double | 8 bytes | $-1.7 \times 10^{308}$ ~ $1.7 \times 10^{308}$ | $0.0$ | -| 字符 | **char** | 2 bytes / 1 byte | $0$ ~ $2^{16} - 1$ | $0$ | -| 布尔 | **bool** | 1 byte / 1 bit | $\text{true}$ 或 $\text{false}$ | $\text{false}$ | - - - -以上表格中,加粗项在算法题中最为常用。此表格无需硬背,大致理解即可,需要时可以通过查表来回忆。 - -### 整数表示方式 - -整数的取值范围取决于变量使用的内存长度,即字节(或比特)数。在计算机中,1 字节 (byte) = 8 比特 (bit),1 比特即 1 个二进制位。以 int 类型为例: - -1. 整数类型 int 占用 4 bytes = 32 bits ,可以表示 $2^{32}$ 个不同的数字; -2. 将最高位视为符号位,$0$ 代表正数,$1$ 代表负数,一共可表示 $2^{31}$ 个正数和 $2^{31}$ 个负数; -3. 当所有 bits 为 0 时代表数字 $0$ ,从零开始增大,可得最大正数为 $2^{31} - 1$ ; -4. 剩余 $2^{31}$ 个数字全部用来表示负数,因此最小负数为 $-2^{31}$ ;具体细节涉及“源码、反码、补码”的相关知识,有兴趣的同学可以查阅学习; - -其他整数类型 byte, short, long 的取值范围的计算方法与 int 类似,在此不再赘述。 - -### 浮点数表示方式 * - -!!! note - - 本书中,标题后的 * 符号代表选读章节。如果你觉得理解困难,建议先跳过,等学完必读章节后再单独攻克。 - -细心的你可能会发现:int 和 float 长度相同,都是 4 bytes,但为什么 float 的取值范围远大于 int ?按理说 float 需要表示小数,取值范围应该变小才对。 - -实际上,这是因为浮点数 float 采用了不同的表示方式。根据 IEEE 754 标准,32-bit 长度的 float 由以下部分构成: - -- 符号位 $\mathrm{S}$ :占 1 bit ; -- 指数位 $\mathrm{E}$ :占 8 bits ; -- 分数位 $\mathrm{N}$ :占 24 bits ,其中 23 位显式存储; - -设 32-bit 二进制数的第 $i$ 位为 $b_i$ ,则 float 值的计算方法定义为: - -$$ -\text { val } = (-1)^{b_{31}} \times 2^{\left(b_{30} b_{29} \ldots b_{23}\right)_2-127} \times\left(1 . b_{22} b_{21} \ldots b_0\right)_2 -$$ - -转化到十进制下的计算公式为 - -$$ -\text { val }=(-1)^{\mathrm{S}} \times 2^{\mathrm{E} -127} \times (1 + \mathrm{N}) -$$ - -其中各项的取值范围为 - -$$ -\begin{aligned} -\mathrm{S} \in & \{ 0, 1\} , \quad \mathrm{E} \in \{ 1, 2, \dots, 254 \} \newline -(1 + \mathrm{N}) = & (1 + \sum_{i=1}^{23} b_{23-i} 2^{-i}) \subset [1, 2 - 2^{-23}] -\end{aligned} -$$ - -![IEEE 754 标准下的 float 表示方式](data_and_memory.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 - -$$ -\text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875 -$$ - -现在我们可以回答最初的问题:**float 的表示方式包含指数位,导致其取值范围远大于 int** 。根据以上计算,float 可表示的最大正数为 $2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}$ ,切换符号位便可得到最小负数。 - -**尽管浮点数 float 扩展了取值范围,但其副作用是牺牲了精度**。整数类型 int 将全部 32 位用于表示数字,数字是均匀分布的;而由于指数位的存在,浮点数 float 的数值越大,相邻两个数字之间的差值就会趋向越大。 - -进一步地,指数位 $E = 0$ 和 $E = 255$ 具有特殊含义,**用于表示零、无穷大、$\mathrm{NaN}$ 等**。 - -
- -| 指数位 E | 分数位 $\mathrm{N} = 0$ | 分数位 $\mathrm{N} \ne 0$ | 计算公式 | -| ------------------ | ----------------------- | ---------------------------- | ------------------------------------------------------------ | -| $0$ | $\pm 0$ | 次正规数 | $(-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})$ | -| $1, 2, \dots, 254$ | 正规数 | 正规数 | $(-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})$ | -| $255$ | $\pm \infty$ | $\mathrm{NaN}$ | | - -
- -特别地,次正规数显著提升了浮点数的精度,这是因为: - -- 最小正正规数为 $2^{-126} \approx 1.18 \times 10^{-38}$ ; -- 最小正次正规数为 $2^{-126} \times 2^{-23} \approx 1.4 \times 10^{-45}$ ; - -双精度 double 也采用类似 float 的表示方法,此处不再详述。 - -### 基本数据类型与数据结构的关系 - -我们知道,**数据结构是在计算机中组织与存储数据的方式**,它的核心是“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」数据结构。数组的存储方式可以表示数字的相邻关系、顺序关系,但至于具体存储的是整数 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”。 - -=== "Java" - - ```java title="" - /* 使用多种「基本数据类型」来初始化「数组」 */ - int[] numbers = new int[5]; - float[] decimals = new float[5]; - char[] characters = new char[5]; - boolean[] booleans = new boolean[5]; - ``` - -=== "C++" - - ```cpp title="" - /* 使用多种「基本数据类型」来初始化「数组」 */ - int numbers[5]; - float decimals[5]; - char characters[5]; - bool booleans[5]; - ``` - -=== "Python" - - ```python title="" - # Python 的 list 可以自由存储各种基本数据类型和对象 - list = [0, 0.0, 'a', False] - ``` - -=== "Go" - - ```go title="" - // 使用多种「基本数据类型」来初始化「数组」 - var numbers = [5]int{} - var decimals = [5]float64{} - var characters = [5]byte{} - var booleans = [5]bool{} - ``` - -=== "JavaScript" - - ```javascript title="" - /* JavaScript 的数组可以自由存储各种基本数据类型和对象 */ - const array = [0, 0.0, 'a', false]; - ``` - -=== "TypeScript" - - ```typescript title="" - /* 使用多种「基本数据类型」来初始化「数组」 */ - const numbers: number[] = []; - const characters: string[] = []; - const booleans: boolean[] = []; - ``` - -=== "C" - - ```c title="" - /* 使用多种「基本数据类型」来初始化「数组」 */ - int numbers[10]; - float decimals[10]; - char characters[10]; - bool booleans[10]; - ``` - -=== "C#" - - ```csharp title="" - /* 使用多种「基本数据类型」来初始化「数组」 */ - int[] numbers = new int[5]; - float[] decimals = new float[5]; - char[] characters = new char[5]; - bool[] booleans = new bool[5]; - ``` - -=== "Swift" - - ```swift title="" - /* 使用多种「基本数据类型」来初始化「数组」 */ - let numbers = Array(repeating: Int(), count: 5) - let decimals = Array(repeating: Double(), count: 5) - let characters = Array(repeating: Character("a"), count: 5) - let booleans = Array(repeating: Bool(), count: 5) - ``` - -=== "Zig" - - ```zig title="" - - ``` - -## 计算机内存 - -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。 - -**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。 - -![内存条、内存空间、内存地址](data_and_memory.assets/computer_memory_location.png) - -**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**。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。我们需要根据剩余内存资源的实际情况来设计算法。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。 diff --git a/docs/chapter_data_structure/number_encoding.assets/1s_2s_complement.png b/docs/chapter_data_structure/number_encoding.assets/1s_2s_complement.png new file mode 100644 index 0000000000000000000000000000000000000000..ccc56863226e3b725f51deac83b403cfc2a2f554 GIT binary patch literal 64536 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zC#t!~_t9QMxqrS2I9$gf;k5@9x3rC6BW^wNtMK$;2d=Srvl#EUhey zE`JqD|7)da;05Lg=bvGmyOX#}zaC-q&Ho8SM5*)7zlM4*;dR(g=gNeAN#zQX?njpU z*ojJ3=sM9%%ZR5Yb2|U9ie42~jSJhXf7?SI3Y42)OIU)|Fw*8paeYGXohl>Hky>*` z6e>fJEh0awPBj>Rq#t>}VJhl4#agzt*eW7|MDt2W;aJgDv@s-eExMqIbD3zAX@!pZzxM{ab!D;18oRjV|a|ASnPJ4mcMAx=UdJ-bv$nTgaTW%Vm zWMvq|R5;&c)vEW4|9#LDabG2hx2bD}mv<5118n@y7AP&(nVo*I=tcTPCNX1Oovykaovj8yW>? whaV{X1%U-gT~etOoSws|<$vxK&?UgXVKWNlvk}QFIPgzVPEEE_+WhtZ0Y + +| 指数位 E | 分数位 $\mathrm{N} = 0$ | 分数位 $\mathrm{N} \ne 0$ | 计算公式 | +| ------------------ | ----------------------- | ------------------------- | ---------------------------------------------------------------------- | +| $0$ | $\pm 0$ | 次正规数 | $(-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})$ | +| $1, 2, \dots, 254$ | 正规数 | 正规数 | $(-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})$ | +| $255$ | $\pm \infty$ | $\mathrm{NaN}$ | | + + + +特别地,次正规数显著提升了浮点数的精度,这是因为: + +- 最小正正规数为 $2^{-126} \approx 1.18 \times 10^{-38}$ ; +- 最小正次正规数为 $2^{-126} \times 2^{-23} \approx 1.4 \times 10^{-45}$ ; + +双精度 `double` 也采用类似 `float` 的表示方法,此处不再详述。 diff --git a/docs/chapter_data_structure/summary.md b/docs/chapter_data_structure/summary.md index ca005bbc..5a29901d 100644 --- a/docs/chapter_data_structure/summary.md +++ b/docs/chapter_data_structure/summary.md @@ -1,7 +1,14 @@ # 小结 -- 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。 -- 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。 - 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。 +- 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。 +- 整数的原码的最高位是符号位,其余位是数字的值。数字的原码、反码和补码是可以相互转换的。 +- 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 位分数为构成。由于存在指数位,浮点数的取值范围远大于整数。 +- ASCII 码是最早出现的英文字符集,长度为 1 字节,共收录 127 个字符。GBK 字符集是常用的中文字符集,共收录两万多个汉字。 +- Unicode 致力于提供一个统一的字符集标准,其包含世界范围内的各种字符,从而解决由于字符编码不同导致的乱码问题。 +- UTF-8 是最受欢迎的 Unicode 编码方法,通用性非常好。它是一种变长的编码方法,具有很好的扩展性,有效提升了存储空间的使用效率。 +- UTF-16 和 UTF-32 是等长的编码方法。在编码中文时,UTF-16 比 UTF-8 的占用空间更小。Java, C# 等编程语言默认使用 UTF-16 编码。 diff --git a/mkdocs.yml b/mkdocs.yml index 56017ca4..dcb3143d 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -137,10 +137,12 @@ nav: - 2.2.   时间复杂度: chapter_computational_complexity/time_complexity.md - 2.3.   空间复杂度: chapter_computational_complexity/space_complexity.md - 2.4.   小结: chapter_computational_complexity/summary.md - - 3.     数据结构简介: - - 3.1.   数据与内存: chapter_data_structure/data_and_memory.md - - 3.2.   数据结构分类: chapter_data_structure/classification_of_data_structure.md - - 3.3.   小结: chapter_data_structure/summary.md + - 3.     数据结构与数据: + - 3.1.   数据结构分类: chapter_data_structure/classification_of_data_structure.md + - 3.2.   基本数据类型: chapter_data_structure/basic_data_types.md + - 3.3.   数字编码 *: chapter_data_structure/number_encoding.md + - 3.4.   字符编码 *: chapter_data_structure/character_encoding.md + - 3.5.   小结: chapter_data_structure/summary.md - 4.     数组与链表: - 4.1.   数组: chapter_array_and_linkedlist/array.md - 4.2.   链表: chapter_array_and_linkedlist/linked_list.md