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en/chapter_data_structure/basic_data_types/index.html
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<title>3.2 Fundamental Data Types - Hello Algo</title>
<title>3.2 Fundamental data types - Hello Algo</title>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
</div>
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
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<label class="md-nav__title" for="__nav_2">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
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<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
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<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash Table
Chapter 6. Hash table
</label>
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<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
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<!-- Page content -->
<h1 id="32-basic-data-types">3.2 &nbsp; Basic Data Types<a class="headerlink" href="#32-basic-data-types" title="Permanent link">&para;</a></h1>
<h1 id="32-basic-data-types">3.2 &nbsp; Basic data types<a class="headerlink" href="#32-basic-data-types" title="Permanent link">&para;</a></h1>
<p>When discussing data in computers, various forms like text, images, videos, voice and 3D models comes to mind. Despite their different organizational forms, they are all composed of various basic data types.</p>
<p><strong>Basic data types are those that the CPU can directly operate on</strong> and are directly used in algorithms, mainly including the following.</p>
<ul>
@@ -2012,7 +2018,7 @@
<li>The integer type <code>int</code> occupies 4 bytes = 32 bits and can represent <span class="arithmatex">\(2^{32}\)</span> numbers.</li>
</ul>
<p>The following table lists the space occupied, value range, and default values of various basic data types in Java. While memorizing this table isn't necessary, having a general understanding of it and referencing it when required is recommended.</p>
<p align="center"> Table 3-1 &nbsp; Space Occupied and Value Range of Basic Data Types </p>
<p align="center"> Table 3-1 &nbsp; Space occupied and value range of basic data types </p>
<div class="center-table">
<table>
@@ -2229,7 +2235,7 @@ aria-label="Footer"
<a
href="../classification_of_data_structure/"
class="md-footer__link md-footer__link--prev"
aria-label="Previous: 3.1 Classification of Data Structures"
aria-label="Previous: 3.1 Classification of data structures"
rel="prev"
>
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Previous
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3.1 Classification of Data Structures
3.1 Classification of data structures
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<a
href="../number_encoding/"
class="md-footer__link md-footer__link--next"
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aria-label="Next: 3.3 Number encoding *"
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3.3 Number Encoding *
3.3 Number encoding *
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<nav class="md-footer__inner md-grid" aria-label="Footer" >
<a href="../classification_of_data_structure/" class="md-footer__link md-footer__link--prev" aria-label="Previous: 3.1 Classification of Data Structures">
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Previous
</span>
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3.1 Classification of Data Structures
3.1 Classification of data structures
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en/chapter_data_structure/character_encoding/index.html
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<title>3.4 Character Encoding * - Hello Algo</title>
<title>3.4 Character encoding * - Hello Algo</title>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
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<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
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<label class="md-nav__title" for="__nav_3">
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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<label class="md-nav__title" for="__nav_4">
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Chapter 3. Data Structures
Chapter 3. Data structures
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@@ -838,7 +844,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<li class="md-nav__item">
<a href="#341-ascii-character-set" class="md-nav__link">
<span class="md-ellipsis">
3.4.1 &nbsp; ASCII Character Set
3.4.1 &nbsp; ASCII character set
</span>
</a>
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<li class="md-nav__item">
<a href="#342-gbk-character-set" class="md-nav__link">
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3.4.2 &nbsp; GBK Character Set
3.4.2 &nbsp; GBK character set
</span>
</a>
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<li class="md-nav__item">
<a href="#343-unicode-character-set" class="md-nav__link">
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3.4.3 &nbsp; Unicode Character Set
3.4.3 &nbsp; Unicode character set
</span>
</a>
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<li class="md-nav__item">
<a href="#344-utf-8-encoding" class="md-nav__link">
<span class="md-ellipsis">
3.4.4 &nbsp; UTF-8 Encoding
3.4.4 &nbsp; UTF-8 encoding
</span>
</a>
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<li class="md-nav__item">
<a href="#345-character-encoding-in-programming-languages" class="md-nav__link">
<span class="md-ellipsis">
3.4.5 &nbsp; Character Encoding in Programming Languages
3.4.5 &nbsp; Character encoding in programming languages
</span>
</a>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
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<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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3.4.1 &nbsp; ASCII character set
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3.4.2 &nbsp; GBK character set
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3.4.3 &nbsp; Unicode Character Set
3.4.3 &nbsp; Unicode character set
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3.4.4 &nbsp; UTF-8 Encoding
3.4.4 &nbsp; UTF-8 encoding
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3.4.5 &nbsp; Character Encoding in Programming Languages
3.4.5 &nbsp; Character encoding in programming languages
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<!-- Page content -->
<h1 id="34-character-encoding">3.4 &nbsp; Character Encoding *<a class="headerlink" href="#34-character-encoding" title="Permanent link">&para;</a></h1>
<h1 id="34-character-encoding">3.4 &nbsp; Character encoding *<a class="headerlink" href="#34-character-encoding" title="Permanent link">&para;</a></h1>
<p>In the computer system, all data is stored in binary form, and characters (represented by char) are no exception. To represent characters, we need to develop a "character set" that defines a one-to-one mapping between each character and binary numbers. With the character set, computers can convert binary numbers to characters by looking up the table.</p>
<h2 id="341-ascii-character-set">3.4.1 &nbsp; ASCII Character Set<a class="headerlink" href="#341-ascii-character-set" title="Permanent link">&para;</a></h2>
<h2 id="341-ascii-character-set">3.4.1 &nbsp; ASCII character set<a class="headerlink" href="#341-ascii-character-set" title="Permanent link">&para;</a></h2>
<p>The "ASCII code" is one of the earliest character sets, officially known as the American Standard Code for Information Interchange. It uses 7 binary digits (the lower 7 bits of a byte) to represent a character, allowing for a maximum of 128 different characters. As shown in the Figure 3-6 , ASCII includes uppercase and lowercase English letters, numbers 0 ~ 9, various punctuation marks, and certain control characters (such as newline and tab).</p>
<p><a class="glightbox" href="../character_encoding.assets/ascii_table.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="ASCII Code" class="animation-figure" src="../character_encoding.assets/ascii_table.png" /></a></p>
<p align="center"> Figure 3-6 &nbsp; ASCII Code </p>
<p><a class="glightbox" href="../character_encoding.assets/ascii_table.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="ASCII code" class="animation-figure" src="../character_encoding.assets/ascii_table.png" /></a></p>
<p align="center"> Figure 3-6 &nbsp; ASCII code </p>
<p>However, <strong>ASCII can only represent English characters</strong>. With the globalization of computers, a character set called "EASCII" was developed to represent more languages. It expands from the 7-bit structure of ASCII to 8 bits, enabling the representation of 256 characters.</p>
<p>Globally, various region-specific EASCII character sets have been introduced. The first 128 characters of these sets are consistent with the ASCII, while the remaining 128 characters are defined differently to accommodate the requirements of different languages.</p>
<h2 id="342-gbk-character-set">3.4.2 &nbsp; GBK Character Set<a class="headerlink" href="#342-gbk-character-set" title="Permanent link">&para;</a></h2>
<h2 id="342-gbk-character-set">3.4.2 &nbsp; GBK character set<a class="headerlink" href="#342-gbk-character-set" title="Permanent link">&para;</a></h2>
<p>Later, it was found that <strong>EASCII still could not meet the character requirements of many languages</strong>. For instance, there are nearly a hundred thousand Chinese characters, with several thousand used regularly. In 1980, the Standardization Administration of China released the "GB2312" character set, which included 6763 Chinese characters, essentially fulfilling the computer processing needs for the Chinese language.</p>
<p>However, GB2312 could not handle some rare and traditional characters. The "GBK" character set expands GB2312 and includes 21886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented with one byte, while Chinese characters use two bytes.</p>
<h2 id="343-unicode-character-set">3.4.3 &nbsp; Unicode Character Set<a class="headerlink" href="#343-unicode-character-set" title="Permanent link">&para;</a></h2>
<h2 id="343-unicode-character-set">3.4.3 &nbsp; Unicode character set<a class="headerlink" href="#343-unicode-character-set" title="Permanent link">&para;</a></h2>
<p>With the rapid evolution of computer technology and a plethora of character sets and encoding standards, numerous problems arose. On the one hand, these character sets generally only defined characters for specific languages and could not function properly in multilingual environments. On the other hand, the existence of multiple character set standards for the same language caused garbled text when information was exchanged between computers using different encoding standards.</p>
<p>Researchers of that era thought: <strong>What if a comprehensive character set encompassing all global languages and symbols was developed? Wouldn't this resolve the issues associated with cross-linguistic environments and garbled text?</strong> Inspired by this idea, the extensive character set, Unicode, was born.</p>
<p>"Unicode" is referred to as "统一码" (Unified Code) in Chinese, theoretically capable of accommodating over a million characters. It aims to incorporate characters from all over the world into a single set, providing a universal character set for processing and displaying various languages and reducing the issues of garbled text due to different encoding standards.</p>
<p>Since its release in 1991, Unicode has continually expanded to include new languages and characters. As of September 2022, Unicode contains 149,186 characters, including characters, symbols, and even emojis from various languages. In the vast Unicode character set, commonly used characters occupy 2 bytes, while some rare characters may occupy 3 or even 4 bytes.</p>
<p>Unicode is a universal character set that assigns a number (called a "code point") to each character, <strong>but it does not specify how these character code points should be stored in a computer system</strong>. One might ask: How does a system interpret Unicode code points of varying lengths within a text? For example, given a 2-byte code, how does the system determine if it represents a single 2-byte character or two 1-byte characters?</p>
<p>A straightforward solution to this problem is to store all characters as equal-length encodings. As shown in the Figure 3-7 , each character in "Hello" occupies 1 byte, while each character in "算法" (algorithm) occupies 2 bytes. We could encode all characters in "Hello 算法" as 2 bytes by padding the higher bits with zeros. This method would enable the system to interpret a character every 2 bytes, recovering the content of the phrase.</p>
<p><a class="glightbox" href="../character_encoding.assets/unicode_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Unicode Encoding Example" class="animation-figure" src="../character_encoding.assets/unicode_hello_algo.png" /></a></p>
<p align="center"> Figure 3-7 &nbsp; Unicode Encoding Example </p>
<p><a class="glightbox" href="../character_encoding.assets/unicode_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Unicode encoding example" class="animation-figure" src="../character_encoding.assets/unicode_hello_algo.png" /></a></p>
<p align="center"> Figure 3-7 &nbsp; Unicode encoding example </p>
<p>However, as ASCII has shown us, encoding English only requires 1 byte. Using the above approach would double the space occupied by English text compared to ASCII encoding, which is a waste of memory space. Therefore, a more efficient Unicode encoding method is needed.</p>
<h2 id="344-utf-8-encoding">3.4.4 &nbsp; UTF-8 Encoding<a class="headerlink" href="#344-utf-8-encoding" title="Permanent link">&para;</a></h2>
<h2 id="344-utf-8-encoding">3.4.4 &nbsp; UTF-8 encoding<a class="headerlink" href="#344-utf-8-encoding" title="Permanent link">&para;</a></h2>
<p>Currently, UTF-8 has become the most widely used Unicode encoding method internationally. <strong>It is a variable-length encoding</strong>, using 1 to 4 bytes to represent a character, depending on the complexity of the character. ASCII characters need only 1 byte, Latin and Greek letters require 2 bytes, commonly used Chinese characters need 3 bytes, and some other rare characters need 4 bytes.</p>
<p>The encoding rules for UTF-8 are not complex and can be divided into two cases:</p>
<ul>
@@ -2157,22 +2163,22 @@
<p>The Figure 3-8 shows the UTF-8 encoding for "Hello算法". It can be observed that since the highest <span class="arithmatex">\(n\)</span> bits are set to <span class="arithmatex">\(1\)</span>, the system can determine the length of the character as <span class="arithmatex">\(n\)</span> by counting the number of highest bits set to <span class="arithmatex">\(1\)</span>.</p>
<p>But why set the highest 2 bits of the remaining bytes to <span class="arithmatex">\(10\)</span>? Actually, this <span class="arithmatex">\(10\)</span> serves as a kind of checksum. If the system starts parsing text from an incorrect byte, the <span class="arithmatex">\(10\)</span> at the beginning of the byte can help the system quickly detect anomalies.</p>
<p>The reason for using <span class="arithmatex">\(10\)</span> as a checksum is that, under UTF-8 encoding rules, it's impossible for the highest two bits of a character to be <span class="arithmatex">\(10\)</span>. This can be proven by contradiction: If the highest two bits of a character are <span class="arithmatex">\(10\)</span>, it indicates that the character's length is <span class="arithmatex">\(1\)</span>, corresponding to ASCII. However, the highest bit of an ASCII character should be <span class="arithmatex">\(0\)</span>, which contradicts the assumption.</p>
<p><a class="glightbox" href="../character_encoding.assets/utf-8_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="UTF-8 Encoding Example" class="animation-figure" src="../character_encoding.assets/utf-8_hello_algo.png" /></a></p>
<p align="center"> Figure 3-8 &nbsp; UTF-8 Encoding Example </p>
<p><a class="glightbox" href="../character_encoding.assets/utf-8_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="UTF-8 encoding example" class="animation-figure" src="../character_encoding.assets/utf-8_hello_algo.png" /></a></p>
<p align="center"> Figure 3-8 &nbsp; UTF-8 encoding example </p>
<p>Apart from UTF-8, other common encoding methods include:</p>
<ul>
<li><strong>UTF-16 Encoding</strong>: Uses 2 or 4 bytes to represent a character. All ASCII characters and commonly used non-English characters are represented with 2 bytes; a few characters require 4 bytes. For 2-byte characters, the UTF-16 encoding equals the Unicode code point.</li>
<li><strong>UTF-32 Encoding</strong>: Every character uses 4 bytes. This means UTF-32 occupies more space than UTF-8 and UTF-16, especially for texts with a high proportion of ASCII characters.</li>
<li><strong>UTF-16 encoding</strong>: Uses 2 or 4 bytes to represent a character. All ASCII characters and commonly used non-English characters are represented with 2 bytes; a few characters require 4 bytes. For 2-byte characters, the UTF-16 encoding equals the Unicode code point.</li>
<li><strong>UTF-32 encoding</strong>: Every character uses 4 bytes. This means UTF-32 occupies more space than UTF-8 and UTF-16, especially for texts with a high proportion of ASCII characters.</li>
</ul>
<p>From the perspective of storage space, using UTF-8 to represent English characters is very efficient because it only requires 1 byte; using UTF-16 to encode some non-English characters (such as Chinese) can be more efficient because it only requires 2 bytes, while UTF-8 might need 3 bytes.</p>
<p>From a compatibility perspective, UTF-8 is the most versatile, with many tools and libraries supporting UTF-8 as a priority.</p>
<h2 id="345-character-encoding-in-programming-languages">3.4.5 &nbsp; Character Encoding in Programming Languages<a class="headerlink" href="#345-character-encoding-in-programming-languages" title="Permanent link">&para;</a></h2>
<h2 id="345-character-encoding-in-programming-languages">3.4.5 &nbsp; Character encoding in programming languages<a class="headerlink" href="#345-character-encoding-in-programming-languages" title="Permanent link">&para;</a></h2>
<p>Historically, many programming languages utilized fixed-length encodings such as UTF-16 or UTF-32 for processing strings during program execution. This allows strings to be handled as arrays, offering several advantages:</p>
<ul>
<li><strong>Random Access</strong>: Strings encoded in UTF-16 can be accessed randomly with ease. For UTF-8, which is a variable-length encoding, locating the <span class="arithmatex">\(i^{th}\)</span> character requires traversing the string from the start to the <span class="arithmatex">\(i^{th}\)</span> position, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
<li><strong>Character Counting</strong>: Similar to random access, counting the number of characters in a UTF-16 encoded string is an <span class="arithmatex">\(O(1)\)</span> operation. However, counting characters in a UTF-8 encoded string requires traversing the entire string.</li>
<li><strong>String Operations</strong>: Many string operations like splitting, concatenating, inserting, and deleting are easier on UTF-16 encoded strings. These operations generally require additional computation on UTF-8 encoded strings to ensure the validity of the UTF-8 encoding.</li>
<li><strong>Random access</strong>: Strings encoded in UTF-16 can be accessed randomly with ease. For UTF-8, which is a variable-length encoding, locating the <span class="arithmatex">\(i^{th}\)</span> character requires traversing the string from the start to the <span class="arithmatex">\(i^{th}\)</span> position, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
<li><strong>Character counting</strong>: Similar to random access, counting the number of characters in a UTF-16 encoded string is an <span class="arithmatex">\(O(1)\)</span> operation. However, counting characters in a UTF-8 encoded string requires traversing the entire string.</li>
<li><strong>String operations</strong>: Many string operations like splitting, concatenating, inserting, and deleting are easier on UTF-16 encoded strings. These operations generally require additional computation on UTF-8 encoded strings to ensure the validity of the UTF-8 encoding.</li>
</ul>
<p>The design of character encoding schemes in programming languages is an interesting topic involving various factors:</p>
<ul>
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<title>3.1 Classification of Data Structures - Hello Algo</title>
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3.1 Classification of Data Structures
3.1 Classification of data structures
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繁體中文
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.1.1 &nbsp; Logical Structure: Linear and Non-Linear
3.1.1 &nbsp; Logical structure: linear and non-linear
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3.1.2 &nbsp; Physical Structure: Contiguous and Dispersed
3.1.2 &nbsp; Physical structure: contiguous and dispersed
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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<a href="#311-logical-structure-linear-and-non-linear" class="md-nav__link">
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3.1.1 &nbsp; Logical Structure: Linear and Non-Linear
3.1.1 &nbsp; Logical structure: linear and non-linear
</span>
</a>
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<a href="#312-physical-structure-contiguous-and-dispersed" class="md-nav__link">
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3.1.2 &nbsp; Physical Structure: Contiguous and Dispersed
3.1.2 &nbsp; Physical structure: contiguous and dispersed
</span>
</a>
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<!-- Page content -->
<h1 id="31-classification-of-data-structures">3.1 &nbsp; Classification of Data Structures<a class="headerlink" href="#31-classification-of-data-structures" title="Permanent link">&para;</a></h1>
<h1 id="31-classification-of-data-structures">3.1 &nbsp; Classification of data structures<a class="headerlink" href="#31-classification-of-data-structures" title="Permanent link">&para;</a></h1>
<p>Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be classified into "logical structure" and "physical structure".</p>
<h2 id="311-logical-structure-linear-and-non-linear">3.1.1 &nbsp; Logical Structure: Linear and Non-Linear<a class="headerlink" href="#311-logical-structure-linear-and-non-linear" title="Permanent link">&para;</a></h2>
<h2 id="311-logical-structure-linear-and-non-linear">3.1.1 &nbsp; Logical structure: linear and non-linear<a class="headerlink" href="#311-logical-structure-linear-and-non-linear" title="Permanent link">&para;</a></h2>
<p><strong>The logical structures reveal the logical relationships between data elements</strong>. In arrays and linked lists, data are arranged in a specific sequence, demonstrating the linear relationship between data; while in trees, data are arranged hierarchically from the top down, showing the derived relationship between "ancestors" and "descendants"; and graphs are composed of nodes and edges, reflecting the intricate network relationship.</p>
<p>As shown in the Figure 3-1 , logical structures can be divided into two major categories: "linear" and "non-linear". Linear structures are more intuitive, indicating data is arranged linearly in logical relationships; non-linear structures, conversely, are arranged non-linearly.</p>
<ul>
<li><strong>Linear Data Structures</strong>: Arrays, Linked Lists, Stacks, Queues, Hash Tables.</li>
<li><strong>Non-Linear Data Structures</strong>: Trees, Heaps, Graphs, Hash Tables.</li>
<li><strong>Linear data structures</strong>: Arrays, Linked Lists, Stacks, Queues, Hash Tables.</li>
<li><strong>Non-linear data structures</strong>: Trees, Heaps, Graphs, Hash Tables.</li>
</ul>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_logic_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linear and Non-Linear Data Structures" class="animation-figure" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></a></p>
<p align="center"> Figure 3-1 &nbsp; Linear and Non-Linear Data Structures </p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_logic_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linear and non-linear data structures" class="animation-figure" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></a></p>
<p align="center"> Figure 3-1 &nbsp; Linear and non-linear data structures </p>
<p>Non-linear data structures can be further divided into tree structures and network structures.</p>
<ul>
<li><strong>Linear Structures</strong>: Arrays, linked lists, queues, stacks, and hash tables, where elements have a one-to-one sequential relationship.</li>
<li><strong>Tree Structures</strong>: Trees, Heaps, Hash Tables, where elements have a one-to-many relationship.</li>
<li><strong>Network Structures</strong>: Graphs, where elements have a many-to-many relationships.</li>
<li><strong>Linear structures</strong>: Arrays, linked lists, queues, stacks, and hash tables, where elements have a one-to-one sequential relationship.</li>
<li><strong>Tree structures</strong>: Trees, Heaps, Hash Tables, where elements have a one-to-many relationship.</li>
<li><strong>Network structures</strong>: Graphs, where elements have a many-to-many relationships.</li>
</ul>
<h2 id="312-physical-structure-contiguous-and-dispersed">3.1.2 &nbsp; Physical Structure: Contiguous and Dispersed<a class="headerlink" href="#312-physical-structure-contiguous-and-dispersed" title="Permanent link">&para;</a></h2>
<h2 id="312-physical-structure-contiguous-and-dispersed">3.1.2 &nbsp; Physical structure: contiguous and dispersed<a class="headerlink" href="#312-physical-structure-contiguous-and-dispersed" title="Permanent link">&para;</a></h2>
<p><strong>During the execution of an algorithm, the data being processed is stored in memory</strong>. The Figure 3-2 shows a computer memory stick where each black square is a physical memory space. We can think of memory as a vast Excel spreadsheet, with each cell capable of storing a certain amount of data.</p>
<p><strong>The system accesses the data at the target location by means of a memory address</strong>. As shown in the Figure 3-2 , the computer assigns a unique identifier to each cell in the table according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access the data stored in memory.</p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/computer_memory_location.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Memory Stick, Memory Spaces, Memory Addresses" class="animation-figure" src="../classification_of_data_structure.assets/computer_memory_location.png" /></a></p>
<p align="center"> Figure 3-2 &nbsp; Memory Stick, Memory Spaces, Memory Addresses </p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/computer_memory_location.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Memory stick, memory spaces, memory addresses" class="animation-figure" src="../classification_of_data_structure.assets/computer_memory_location.png" /></a></p>
<p align="center"> Figure 3-2 &nbsp; Memory stick, memory spaces, memory addresses </p>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
@@ -2100,8 +2106,8 @@
</div>
<p>Memory is a shared resource for all programs. When a block of memory is occupied by one program, it cannot be simultaneously used by other programs. <strong>Therefore, considering memory resources is crucial in designing data structures and algorithms</strong>. For instance, the algorithm's peak memory usage should not exceed the remaining free memory of the system; if there is a lack of contiguous memory blocks, then the data structure chosen must be able to be stored in non-contiguous memory blocks.</p>
<p>As illustrated in the Figure 3-3 , <strong>the physical structure reflects the way data is stored in computer memory</strong> and it can be divided into contiguous space storage (arrays) and non-contiguous space storage (linked lists). The two types of physical structures exhibit complementary characteristics in terms of time efficiency and space efficiency.</p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_phisical_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Contiguous Space Storage and Dispersed Space Storage" class="animation-figure" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></a></p>
<p align="center"> Figure 3-3 &nbsp; Contiguous Space Storage and Dispersed Space Storage </p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_phisical_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Contiguous space storage and dispersed space storage" class="animation-figure" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></a></p>
<p align="center"> Figure 3-3 &nbsp; Contiguous space storage and dispersed space storage </p>
<p><strong>It is worth noting that all data structures are implemented based on arrays, linked lists, or a combination of both</strong>. For example, stacks and queues can be implemented using either arrays or linked lists; while implementations of hash tables may involve both arrays and linked lists.
- <strong>Array-based implementations</strong>: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, Matrices, Tensors (arrays with dimensions <span class="arithmatex">\(\geq 3\)</span>).
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<title>Chapter 3.   Data Structures - Hello Algo</title>
<title>Chapter 3.   Data structures - Hello Algo</title>
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Chapter 3. &nbsp; Data Structures
Chapter 3. &nbsp; Data structures
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中文
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0.1 About This Book
0.1 About this book
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<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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<!-- Page content -->
<h1 id="chapter-3-data-structures">Chapter 3. &nbsp; Data Structures<a class="headerlink" href="#chapter-3-data-structures" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_data_structure.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Data Structures" class="cover-image" src="../assets/covers/chapter_data_structure.jpg" /></a></p>
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<p><a class="glightbox" href="../assets/covers/chapter_data_structure.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Data structures" class="cover-image" src="../assets/covers/chapter_data_structure.jpg" /></a></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>Data structures serve as a robust and diverse framework.</p>
@@ -2012,10 +2018,10 @@
</div>
<h2 id="chapter-contents">Chapter Contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/classification_of_data_structure/">3.1 &nbsp; Classification of Data Structures</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/basic_data_types/">3.2 &nbsp; Fundamental Data Types</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/number_encoding/">3.3 &nbsp; Number Encoding *</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/character_encoding/">3.4 &nbsp; Character Encoding *</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/classification_of_data_structure/">3.1 &nbsp; Classification of data structures</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/basic_data_types/">3.2 &nbsp; Fundamental data types</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/number_encoding/">3.3 &nbsp; Number encoding *</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/character_encoding/">3.4 &nbsp; Character encoding *</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/summary/">3.5 &nbsp; Summary</a></li>
</ul>
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<title>3.3 Number Encoding * - Hello Algo</title>
<title>3.3 Number encoding * - Hello Algo</title>
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<div class="md-header__topic" data-md-component="header-topic">
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3.3 Number Encoding *
3.3 Number encoding *
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0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
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<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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3.3 Number Encoding *
3.3 Number encoding *
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3.3.1 &nbsp; Integer Encoding
3.3.1 &nbsp; Integer encoding
</span>
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<a href="#332-floating-point-number-encoding" class="md-nav__link">
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3.3.2 &nbsp; Floating-Point Number Encoding
3.3.2 &nbsp; Floating-point number encoding
</span>
</a>
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3.4 Character Encoding *
3.4 Character encoding *
</span>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
</label>
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6.1 Hash Table
6.1 Hash table
</span>
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6.2 Hash Collision
6.2 Hash collision
</span>
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6.3 Hash Algorithm
6.3 Hash algorithm
</span>
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3.3.1 &nbsp; Integer Encoding
3.3.1 &nbsp; Integer encoding
</span>
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<a href="#332-floating-point-number-encoding" class="md-nav__link">
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3.3.2 &nbsp; Floating-Point Number Encoding
3.3.2 &nbsp; Floating-point number encoding
</span>
</a>
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<!-- Page content -->
<h1 id="33-number-encoding">3.3 &nbsp; Number Encoding *<a class="headerlink" href="#33-number-encoding" title="Permanent link">&para;</a></h1>
<h1 id="33-number-encoding">3.3 &nbsp; Number encoding *<a class="headerlink" href="#33-number-encoding" title="Permanent link">&para;</a></h1>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>In this book, chapters marked with an asterisk '*' are optional readings. If you are short on time or find them challenging, you may skip these initially and return to them after completing the essential chapters.</p>
</div>
<h2 id="331-integer-encoding">3.3.1 &nbsp; Integer Encoding<a class="headerlink" href="#331-integer-encoding" title="Permanent link">&para;</a></h2>
<h2 id="331-integer-encoding">3.3.1 &nbsp; Integer encoding<a class="headerlink" href="#331-integer-encoding" title="Permanent link">&para;</a></h2>
<p>In the table from the previous section, we observed that all integer types can represent one more negative number than positive numbers, such as the <code>byte</code> range of <span class="arithmatex">\([-128, 127]\)</span>. This phenomenon seems counterintuitive, and its underlying reason involves knowledge of sign-magnitude, one's complement, and two's complement encoding.</p>
<p>Firstly, it's important to note that <strong>numbers are stored in computers using the two's complement form</strong>. Before analyzing why this is the case, let's define these three encoding methods:</p>
<ul>
@@ -2084,8 +2090,8 @@
<li><strong>Two's complement</strong>: The two's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by adding <span class="arithmatex">\(1\)</span> to their one's complement.</li>
</ul>
<p>The following diagram illustrates the conversions among sign-magnitude, one's complement, and two's complement:</p>
<p><a class="glightbox" href="../number_encoding.assets/1s_2s_complement.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Conversions between Sign-Magnitude, One's Complement, and Two's Complement" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></a></p>
<p align="center"> Figure 3-4 &nbsp; Conversions between Sign-Magnitude, One's Complement, and Two's Complement </p>
<p><a class="glightbox" href="../number_encoding.assets/1s_2s_complement.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Conversions between sign-magnitude, one's complement, and two's complement" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></a></p>
<p align="center"> Figure 3-4 &nbsp; Conversions between sign-magnitude, one's complement, and two's complement </p>
<p>Although sign-magnitude is the most intuitive, it has limitations. For one, <strong>negative numbers in sign-magnitude cannot be directly used in calculations</strong>. For example, in sign-magnitude, calculating <span class="arithmatex">\(1 + (-2)\)</span> results in <span class="arithmatex">\(-3\)</span>, which is incorrect.</p>
<div class="arithmatex">\[
@@ -2139,7 +2145,7 @@
<p>It's important to note that this doesn't mean computers can only perform addition. <strong>By combining addition with basic logical operations, computers can execute a variety of other mathematical operations</strong>. For example, the subtraction <span class="arithmatex">\(a - b\)</span> can be translated into <span class="arithmatex">\(a + (-b)\)</span>; multiplication and division can be translated into multiple additions or subtractions.</p>
<p>We can now summarize the reason for using two's complement in computers: with two's complement representation, computers can use the same circuits and operations to handle both positive and negative number addition, eliminating the need for special hardware circuits for subtraction and avoiding the ambiguity of positive and negative zero. This greatly simplifies hardware design and enhances computational efficiency.</p>
<p>The design of two's complement is quite ingenious, and due to space constraints, we'll stop here. Interested readers are encouraged to explore further.</p>
<h2 id="332-floating-point-number-encoding">3.3.2 &nbsp; Floating-Point Number Encoding<a class="headerlink" href="#332-floating-point-number-encoding" title="Permanent link">&para;</a></h2>
<h2 id="332-floating-point-number-encoding">3.3.2 &nbsp; Floating-point number encoding<a class="headerlink" href="#332-floating-point-number-encoding" title="Permanent link">&para;</a></h2>
<p>You might have noticed something intriguing: despite having the same length of 4 bytes, why does a <code>float</code> have a much larger range of values compared to an <code>int</code>? This seems counterintuitive, as one would expect the range to shrink for <code>float</code> since it needs to represent fractions.</p>
<p>In fact, <strong>this is due to the different representation method used by floating-point numbers (<code>float</code>)</strong>. Let's consider a 32-bit binary number as:</p>
<div class="arithmatex">\[
@@ -2166,8 +2172,8 @@ b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
(1 + \mathrm{N}) = &amp; (1 + \sum_{i=1}^{23} b_{23-i} \times 2^{-i}) \subset [1, 2 - 2^{-23}]
\end{aligned}
\]</div>
<p><a class="glightbox" href="../number_encoding.assets/ieee_754_float.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example Calculation of a float in IEEE 754 Standard" class="animation-figure" src="../number_encoding.assets/ieee_754_float.png" /></a></p>
<p align="center"> Figure 3-5 &nbsp; Example Calculation of a float in IEEE 754 Standard </p>
<p><a class="glightbox" href="../number_encoding.assets/ieee_754_float.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example calculation of a float in IEEE 754 standard" class="animation-figure" src="../number_encoding.assets/ieee_754_float.png" /></a></p>
<p align="center"> Figure 3-5 &nbsp; Example calculation of a float in IEEE 754 standard </p>
<p>Observing the diagram, given an example data <span class="arithmatex">\(\mathrm{S} = 0\)</span>, <span class="arithmatex">\(\mathrm{E} = 124\)</span>, <span class="arithmatex">\(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)</span>, we have:</p>
<div class="arithmatex">\[
@@ -2176,7 +2182,7 @@ b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
<p>Now we can answer the initial question: <strong>The representation of <code>float</code> includes an exponent bit, leading to a much larger range than <code>int</code></strong>. Based on the above calculation, the maximum positive number representable by <code>float</code> is approximately <span class="arithmatex">\(2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}\)</span>, and the minimum negative number is obtained by switching the sign bit.</p>
<p><strong>However, the trade-off for <code>float</code>'s expanded range is a sacrifice in precision</strong>. The integer type <code>int</code> uses all 32 bits to represent the number, with values evenly distributed; but due to the exponent bit, the larger the value of a <code>float</code>, the greater the difference between adjacent numbers.</p>
<p>As shown in the Table 3-2 , exponent bits <span class="arithmatex">\(E = 0\)</span> and <span class="arithmatex">\(E = 255\)</span> have special meanings, <strong>used to represent zero, infinity, <span class="arithmatex">\(\mathrm{NaN}\)</span>, etc.</strong></p>
<p align="center"> Table 3-2 &nbsp; Meaning of Exponent Bits </p>
<p align="center"> Table 3-2 &nbsp; Meaning of exponent bits </p>
<div class="center-table">
<table>
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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1. &nbsp; Key Review
1. &nbsp; Key review
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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1. &nbsp; Key Review
1. &nbsp; Key review
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<!-- Page content -->
<h1 id="35-summary">3.5 &nbsp; Summary<a class="headerlink" href="#35-summary" title="Permanent link">&para;</a></h1>
<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
<h3 id="1-key-review">1. &nbsp; Key review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
<ul>
<li>Data structures can be categorized from two perspectives: logical structure and physical structure. Logical structure describes the logical relationships between data elements, while physical structure describes how data is stored in computer memory.</li>
<li>Common logical structures include linear, tree-like, and network structures. We generally classify data structures into linear (arrays, linked lists, stacks, queues) and non-linear (trees, graphs, heaps) based on their logical structure. The implementation of hash tables may involve both linear and non-linear data structures.</li>
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3.4 Character Encoding *
3.4 Character encoding *
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