xinyin025/hello-algo
1
0
Fork 0
mirror of https://github.com/krahets/hello-algo.git synced 2026-07-21 11:06:07 +00:00
Code Issues Packages Projects Releases Wiki Activity

deploy

Browse Source
This commit is contained in:
krahets
2026-04-02 03:08:50 +08:00
parent 09a136c9fa
commit aaf9f58eb3
157 changed files with 3002 additions and 2994 deletions
Download Patch File Download Diff File Expand all files Collapse all files
en/chapter_data_structure/basic_data_types/index.html
+25 -25
View File
@@ -6,7 +6,7 @@
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
@@ -576,7 +576,7 @@
<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
@@ -598,7 +598,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
</label>
@@ -1195,7 +1195,7 @@
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
@@ -1217,7 +1217,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
</label>
@@ -1323,7 +1323,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
@@ -1414,7 +1414,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
@@ -1436,7 +1436,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
@@ -1514,7 +1514,7 @@
<span class="md-ellipsis">
5.3 Double-Ended Queue
5.3 Deque
@@ -1605,7 +1605,7 @@
<span class="md-ellipsis">
Chapter 6. Hashing
Chapter 6. Hash Table
@@ -1627,7 +1627,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
@@ -1900,7 +1900,7 @@
<span class="md-ellipsis">
7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
@@ -2119,7 +2119,7 @@
<span class="md-ellipsis">
8.2 Building a Heap
8.2 Heap Construction Operation
@@ -2147,7 +2147,7 @@
<span class="md-ellipsis">
8.3 Top-K Problem
8.3 Top-k Problem
@@ -2505,7 +2505,7 @@
<span class="md-ellipsis">
10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
@@ -2533,7 +2533,7 @@
<span class="md-ellipsis">
10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
@@ -2589,7 +2589,7 @@
<span class="md-ellipsis">
10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
@@ -2738,7 +2738,7 @@
<span class="md-ellipsis">
11.1 Sorting Algorithms
11.1 Sorting Algorithm
@@ -3211,7 +3211,7 @@
<span class="md-ellipsis">
12.4 Hanoi Tower Problem
12.4 Hanota Problem
@@ -4123,7 +4123,7 @@
<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
@@ -4267,7 +4267,7 @@
<!-- 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>
<p>When we talk about data in computers, we think of various forms such as text, images, videos, audio, 3D models, and more. Although these data are organized in different ways, they are all composed of various basic data types.</p>
<p>When we talk about data stored in computers, we think of various forms such as text, images, videos, audio, 3D models, and more. Although these kinds of data are organized in different ways, they are all composed of various basic data types.</p>
<p><strong>Basic data types are types that the CPU can directly operate on</strong>, and they are directly used in algorithms, mainly including the following.</p>
<ul>
<li>Integer types <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>.</li>
@@ -4275,7 +4275,7 @@
<li>Character type <code>char</code>, used to represent letters, punctuation marks, and even emojis in various languages.</li>
<li>Boolean type <code>bool</code>, used to represent "yes" and "no" judgments.</li>
</ul>
<p><strong>Basic data types are stored in binary form in computers</strong>. One binary bit is <span class="arithmatex">\(1\)</span> bit. In most modern operating systems, <span class="arithmatex">\(1\)</span> byte consists of <span class="arithmatex">\(8\)</span> bits.</p>
<p><strong>Basic data types are stored in binary form in computers</strong>. A binary digit is one bit. In most modern operating systems, <span class="arithmatex">\(1\)</span> byte consists of <span class="arithmatex">\(8\)</span> bits.</p>
<p>The range of values for basic data types depends on the size of the space they occupy. Below is an example using Java.</p>
<ul>
<li>Integer type <code>byte</code> occupies <span class="arithmatex">\(1\)</span> byte = <span class="arithmatex">\(8\)</span> bits, and can represent <span class="arithmatex">\(2^{8}\)</span> numbers.</li>
@@ -4364,15 +4364,15 @@
</tbody>
</table>
</div>
<p>Please note that the above table is specific to Java's basic data types. Each programming language has its own data type definitions, and their space occupied, value ranges, and default values may vary.</p>
<p>Please note that Table 3-1 applies specifically to Java's basic data types. Each programming language has its own type definitions, and their space usage, value ranges, and default values may vary.</p>
<ul>
<li>In Python, the integer type <code>int</code> can be of any size, limited only by available memory; the floating-point type <code>float</code> is double-precision 64-bit; there is no <code>char</code> type, a single character is actually a string <code>str</code> of length 1.</li>
<li>C and C++ do not explicitly specify the size of basic data types, which varies by implementation and platform. The above table follows the LP64 <a href="https://en.cppreference.com/w/cpp/language/types#Properties">data model</a>, which is used in Unix 64-bit operating systems including Linux and macOS.</li>
<li>The size of character <code>char</code> is 1 byte in C and C++, and in most programming languages it depends on the specific character encoding method, as detailed in the "Character Encoding" section.</li>
<li>Even though representing a boolean value requires only 1 bit (<span class="arithmatex">\(0\)</span> or <span class="arithmatex">\(1\)</span>), it is usually stored as 1 byte in memory. This is because modern computer CPUs typically use 1 byte as the minimum addressable memory unit.</li>
</ul>
<p>So, what is the relationship between basic data types and data structures? We know that data structures are ways of organizing and storing data in computers. The subject of this statement is "structure", not "data".</p>
<p>If we want to represent "a row of numbers", we naturally think of using an array. This is because the linear structure of an array can represent the adjacency and order relationships of numbers, but the content stored—whether integer <code>int</code>, floating-point <code>float</code>, or character <code>char</code>—is unrelated to the "data structure".</p>
<p>So, what is the relationship between basic data types and data structures? We know that data structures are ways of organizing and storing data in computers. Here, the emphasis is on the "structure", not the "data".</p>
<p>If we want to represent "a row of numbers", we naturally think of using an array. This is because the linear structure of an array can represent the adjacency and order relationships of numbers, but whether the stored content is integer <code>int</code>, floating-point <code>float</code>, or character <code>char</code> is unrelated to the "data structure".</p>
<p>In other words, <strong>basic data types provide the "content type" of data, while data structures provide the "organization method" of data</strong>. For example, in the following code, we use the same data structure (array) to store and represent different basic data types, including <code>int</code>, <code>float</code>, <code>char</code>, <code>bool</code>, etc.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
<div class="tabbed-content">
en/chapter_data_structure/character_encoding/index.html
+25 -25
View File
@@ -6,7 +6,7 @@
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
@@ -576,7 +576,7 @@
<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
@@ -598,7 +598,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
</label>
@@ -1288,7 +1288,7 @@
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
@@ -1310,7 +1310,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
</label>
@@ -1416,7 +1416,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
@@ -1507,7 +1507,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
@@ -1529,7 +1529,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
@@ -1607,7 +1607,7 @@
<span class="md-ellipsis">
5.3 Double-Ended Queue
5.3 Deque
@@ -1698,7 +1698,7 @@
<span class="md-ellipsis">
Chapter 6. Hashing
Chapter 6. Hash Table
@@ -1720,7 +1720,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
@@ -1993,7 +1993,7 @@
<span class="md-ellipsis">
7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
@@ -2212,7 +2212,7 @@
<span class="md-ellipsis">
8.2 Building a Heap
8.2 Heap Construction Operation
@@ -2240,7 +2240,7 @@
<span class="md-ellipsis">
8.3 Top-K Problem
8.3 Top-k Problem
@@ -2598,7 +2598,7 @@
<span class="md-ellipsis">
10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
@@ -2626,7 +2626,7 @@
<span class="md-ellipsis">
10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
@@ -2682,7 +2682,7 @@
<span class="md-ellipsis">
10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
@@ -2831,7 +2831,7 @@
<span class="md-ellipsis">
11.1 Sorting Algorithms
11.1 Sorting Algorithm
@@ -3304,7 +3304,7 @@
<span class="md-ellipsis">
12.4 Hanoi Tower Problem
12.4 Hanota Problem
@@ -4216,7 +4216,7 @@
<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
@@ -4432,13 +4432,13 @@
<p>However, <strong>ASCII code can only represent English</strong>. With the globalization of computers, a character set called <u>EASCII</u> that can represent more languages emerged. It expands from the 7-bit basis of ASCII to 8 bits, and can represent 256 different characters.</p>
<p>Worldwide, a batch of EASCII character sets suitable for different regions have appeared successively. The first 128 characters of these character sets are unified as ASCII code, and the last 128 characters are defined differently to adapt to the needs 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>
<p>Later, people found that <strong>EASCII code still cannot meet the character quantity requirements of many languages</strong>. For example, there are nearly one hundred thousand Chinese characters, and several thousand are used daily. In 1980, the China National Standardization Administration released the <u>GB2312</u> character set, which included 6,763 Chinese characters, basically meeting the needs for computer processing of Chinese characters.</p>
<p>Later, people found that <strong>EASCII still could not provide enough characters for many languages</strong>. For example, there are nearly one hundred thousand Chinese characters, and several thousand are used in everyday life. In 1980, the China National Standardization Administration released the <u>GB2312</u> character set, which included 6,763 Chinese characters, basically meeting the needs of computer processing for Chinese.</p>
<p>However, GB2312 cannot handle some rare characters and traditional Chinese characters. The <u>GBK</u> character set is an extension based on GB2312, which includes a total of 21,886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented using one byte, and Chinese characters are represented using 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>
<p>With the vigorous development of computer technology, character sets and encoding standards flourished, which brought many problems. On the one hand, these character sets generally only define characters for specific languages and cannot work normally in multilingual environments. On the other hand, multiple character set standards exist for the same language, and if two computers use different encoding standards, garbled characters will appear during information transmission.</p>
<p>Researchers of that era thought: <strong>If a sufficiently complete character set is released that includes all languages and symbols in the world, wouldn't it be possible to solve cross-language environment and garbled character problems</strong>? Driven by this idea, a large and comprehensive character set, Unicode, was born.</p>
<p>Researchers of that era thought: <strong>If a sufficiently complete character set were released to include all languages and symbols in the world, wouldn't that solve problems in cross-language environments and eliminate garbled text</strong>? Driven by this idea, a large and comprehensive character set, Unicode, was born.</p>
<p><u>Unicode</u> is called "统一码" (Unified Code) in Chinese and can theoretically accommodate over one million characters. It is committed to including characters from around the world into a unified character set, providing a universal character set to handle and display various language texts, reducing garbled character problems caused by different encoding standards.</p>
<p>Since its release in 1991, Unicode has continuously expanded to include new languages and characters. As of September 2022, Unicode has included 149,186 characters, including characters, symbols, and even emojis from various languages. In the vast Unicode character set, commonly used characters occupy 2 bytes, and some rare characters occupy 3 bytes or even 4 bytes.</p>
<p>Since its release in 1991, Unicode has continuously expanded to include new languages and characters. As of September 2022, Unicode has included 149,186 characters, including characters, symbols, and even emojis from various languages. In practical storage and encoding schemes for this vast character set, commonly used characters often occupy 2 bytes, while some rare characters occupy 3 bytes or even 4 bytes.</p>
<p>Unicode is a universal character set that essentially assigns a number (called a "code point") to each character, <strong>but it does not specify how to store these character code points in computers</strong>. We can't help but ask: when Unicode code points of multiple lengths appear simultaneously in a text, how does the system parse the characters? For example, given an encoding with a length of 2 bytes, how does the system determine whether it is one 2-byte character or two 1-byte characters?</p>
<p>For the above problem, <strong>a straightforward solution is to store all characters as equal-length encodings</strong>. As shown in Figure 3-7, each character in "Hello" occupies 1 byte, and each character in "算法" (algorithm) occupies 2 bytes. We can encode all characters in "Hello 算法" as 2 bytes in length by padding the high bits with 0. In this way, the system can parse one character every 2 bytes and restore the content of this phrase.</p>
<p><img alt="Unicode encoding example" class="animation-figure" src="../character_encoding.assets/unicode_hello_algo.png" /></p>
@@ -4452,7 +4452,7 @@
<li>For 1-byte characters, set the highest bit to <span class="arithmatex">\(0\)</span>, and set the remaining 7 bits to the Unicode code point. It is worth noting that ASCII characters occupy the first 128 code points in the Unicode character set. That is to say, <strong>UTF-8 encoding is backward compatible with ASCII code</strong>. This means we can use UTF-8 to parse very old ASCII code text.</li>
<li>For characters with a length of <span class="arithmatex">\(n\)</span> bytes (where <span class="arithmatex">\(n &gt; 1\)</span>), set the highest <span class="arithmatex">\(n\)</span> bits of the first byte to <span class="arithmatex">\(1\)</span>, and set the <span class="arithmatex">\((n + 1)\)</span>-th bit to <span class="arithmatex">\(0\)</span>; starting from the second byte, set the highest 2 bits of each byte to <span class="arithmatex">\(10\)</span>; use all remaining bits to fill in the Unicode code point of the character.</li>
</ul>
<p>Figure 3-8 shows the UTF-8 encoding corresponding to "Hello算法". It can be observed that since the highest <span class="arithmatex">\(n\)</span> bits are all set to <span class="arithmatex">\(1\)</span>, the system can parse the length of the character as <span class="arithmatex">\(n\)</span> by reading the number of highest bits that are <span class="arithmatex">\(1\)</span>.</p>
<p>Figure 3-8 shows the UTF-8 encoding corresponding to "Hello算法". It can be observed that since the highest <span class="arithmatex">\(n\)</span> bits are all set to <span class="arithmatex">\(1\)</span>, the system can determine that the character length is <span class="arithmatex">\(n\)</span> by counting the leading <span class="arithmatex">\(1\)</span> bits.</p>
<p>But why set the highest 2 bits of all other bytes to <span class="arithmatex">\(10\)</span>? In fact, this <span class="arithmatex">\(10\)</span> can serve as a check symbol. Assuming 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 determine an anomaly.</p>
<p>The reason for using <span class="arithmatex">\(10\)</span> as a check symbol is that under UTF-8 encoding rules, it is impossible for a character's highest two bits to be <span class="arithmatex">\(10\)</span>. This conclusion can be proven by contradiction: assuming the highest two bits of a character are <span class="arithmatex">\(10\)</span>, it means the length of the character is <span class="arithmatex">\(1\)</span>, corresponding to ASCII code. However, the highest bit of ASCII code should be <span class="arithmatex">\(0\)</span>, which contradicts the assumption.</p>
<p><img alt="UTF-8 encoding example" class="animation-figure" src="../character_encoding.assets/utf-8_hello_algo.png" /></p>
@@ -4466,7 +4466,7 @@
<p>From the perspective of storage space occupation, using UTF-8 to represent English characters is very efficient because it only requires 1 byte; using UTF-16 encoding for some non-English characters (such as Chinese) will be more efficient because it only requires 2 bytes, while UTF-8 may require 3 bytes.</p>
<p>From a compatibility perspective, UTF-8 has the best universality, and many tools and libraries support UTF-8 first.</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>
<p>For most past programming languages, strings during program execution use fixed-length encodings such as UTF-16 or UTF-32. Under fixed-length encoding, we can treat strings as arrays for processing, and this approach has the following advantages.</p>
<p>For many programming languages in the past, strings during program execution used internal encodings such as UTF-16 or UTF-32. Under these representations, we can often treat strings like arrays during processing, and this approach has the following advantages.</p>
<ul>
<li><strong>Random access</strong>: UTF-16 encoded strings can be easily accessed randomly. UTF-8 is a variable-length encoding. To find the <span class="arithmatex">\(i\)</span>-th character, we need to traverse from the beginning of the string to the <span class="arithmatex">\(i\)</span>-th character, which requires <span class="arithmatex">\(O(n)\)</span> time.</li>
<li><strong>Character counting</strong>: Similar to random access, calculating the length of a UTF-16 encoded string is also an <span class="arithmatex">\(O(1)\)</span> operation. However, calculating the length of a UTF-8 encoded string requires traversing the entire string.</li>
en/chapter_data_structure/classification_of_data_structure/index.html
+24 -24
View File
@@ -6,7 +6,7 @@
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
@@ -576,7 +576,7 @@
<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
@@ -598,7 +598,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
</label>
@@ -1255,7 +1255,7 @@
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
@@ -1277,7 +1277,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
</label>
@@ -1383,7 +1383,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
@@ -1474,7 +1474,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
@@ -1496,7 +1496,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
@@ -1574,7 +1574,7 @@
<span class="md-ellipsis">
5.3 Double-Ended Queue
5.3 Deque
@@ -1665,7 +1665,7 @@
<span class="md-ellipsis">
Chapter 6. Hashing
Chapter 6. Hash Table
@@ -1687,7 +1687,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
@@ -1960,7 +1960,7 @@
<span class="md-ellipsis">
7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
@@ -2179,7 +2179,7 @@
<span class="md-ellipsis">
8.2 Building a Heap
8.2 Heap Construction Operation
@@ -2207,7 +2207,7 @@
<span class="md-ellipsis">
8.3 Top-K Problem
8.3 Top-k Problem
@@ -2565,7 +2565,7 @@
<span class="md-ellipsis">
10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
@@ -2593,7 +2593,7 @@
<span class="md-ellipsis">
10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
@@ -2649,7 +2649,7 @@
<span class="md-ellipsis">
10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
@@ -2798,7 +2798,7 @@
<span class="md-ellipsis">
11.1 Sorting Algorithms
11.1 Sorting Algorithm
@@ -3271,7 +3271,7 @@
<span class="md-ellipsis">
12.4 Hanoi Tower Problem
12.4 Hanota Problem
@@ -4183,7 +4183,7 @@
<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
@@ -4359,7 +4359,7 @@
<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 from two dimensions: "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>
<p><strong>Logical structure reveals the logical relationships between data elements</strong>. In arrays and linked lists, data is arranged in a certain order, embodying the linear relationship between data; while in trees, data is arranged hierarchically from top to bottom, showing the derived relationship between "ancestors" and "descendants"; graphs are composed of nodes and edges, reflecting complex network relationships.</p>
<p><strong>Logical structure reveals the logical relationships between data elements</strong>. In arrays and linked lists, data is arranged in a certain order, embodying linear relationships between elements; while in trees, data is arranged hierarchically from top to bottom, showing parent-descendant relationships; graphs are composed of nodes and edges, reflecting complex network relationships.</p>
<p>As shown in Figure 3-1, logical structures can be divided into two major categories: "linear" and "non-linear". Linear structures are more intuitive, indicating that data is linearly arranged in logical relationships; non-linear structures are the opposite, arranged non-linearly.</p>
<ul>
<li><strong>Linear data structures</strong>: Arrays, linked lists, stacks, queues, hash tables, where elements have a one-to-one sequential relationship.</li>
@@ -4381,10 +4381,10 @@
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>It is worth noting that comparing memory to an Excel spreadsheet is a simplified analogy. The actual working mechanism of memory is quite complex, involving concepts such as address space, memory management, cache mechanisms, virtual memory, and physical memory.</p>
<p>It should be noted that comparing memory to an Excel spreadsheet is only a simplified analogy. The actual workings of memory are much more complex, involving concepts such as address space, memory management, cache mechanisms, virtual memory, and physical memory.</p>
</div>
<p>Memory is a shared resource for all programs. When a block of memory is occupied by a program, it usually cannot be used by other programs at the same time. <strong>Therefore, in the design of data structures and algorithms, memory resources are an important consideration</strong>. For example, the peak memory occupied by an algorithm should not exceed the remaining free memory of the system; if there is a lack of contiguous large memory blocks, then the data structure chosen must be able to be stored in dispersed memory spaces.</p>
<p>As shown in Figure 3-3, <strong>physical structure reflects the way data is stored in computer memory</strong>, and can be divided into contiguous space storage (arrays) and dispersed space storage (linked lists). The two physical structures exhibit complementary characteristics in terms of time efficiency and space efficiency.</p>
<p>As shown in Figure 3-3, <strong>physical structure reflects the way data is stored in computer memory</strong>. It can be divided into contiguous-space storage (arrays) and dispersed-space storage (linked lists). At a low level, physical structure determines how data is accessed, updated, inserted, and deleted. These two physical structures exhibit complementary characteristics in terms of time efficiency and space efficiency.</p>
<p><img alt="Contiguous space storage and dispersed space storage" class="animation-figure" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></p>
<p align="center"> Figure 3-3 &nbsp; Contiguous space storage and dispersed space storage </p>
@@ -4393,7 +4393,7 @@
<li><strong>Can be implemented based on arrays</strong>: Stacks, queues, hash tables, trees, heaps, graphs, matrices, tensors (arrays with dimensions <span class="arithmatex">\(\geq 3\)</span>), etc.</li>
<li><strong>Can be implemented based on linked lists</strong>: Stacks, queues, hash tables, trees, heaps, graphs, etc.</li>
</ul>
<p>After initialization, linked lists can still adjust their length during program execution, so they are also called "dynamic data structures". After initialization, the length of arrays cannot be changed, so they are also called "static data structures". It is worth noting that arrays can achieve length changes by reallocating memory, thus possessing a certain degree of "dynamism".</p>
<p>After initialization, linked lists can still adjust their length during program execution, so they are also called "dynamic data structures". After initialization, the length of arrays cannot be changed, so they are also called "static data structures". It is worth noting that arrays can change length by reallocating memory, thus retaining a limited degree of flexibility.</p>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>If you find it difficult to understand physical structure, it is recommended to read the next chapter first, and then review this section.</p>
en/chapter_data_structure/index.html
+21 -21
View File
@@ -6,7 +6,7 @@
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
@@ -576,7 +576,7 @@
<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
@@ -598,7 +598,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
</label>
@@ -1185,7 +1185,7 @@
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
@@ -1207,7 +1207,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
</label>
@@ -1313,7 +1313,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
@@ -1404,7 +1404,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
@@ -1426,7 +1426,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
@@ -1504,7 +1504,7 @@
<span class="md-ellipsis">
5.3 Double-Ended Queue
5.3 Deque
@@ -1595,7 +1595,7 @@
<span class="md-ellipsis">
Chapter 6. Hashing
Chapter 6. Hash Table
@@ -1617,7 +1617,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
@@ -1890,7 +1890,7 @@
<span class="md-ellipsis">
7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
@@ -2109,7 +2109,7 @@
<span class="md-ellipsis">
8.2 Building a Heap
8.2 Heap Construction Operation
@@ -2137,7 +2137,7 @@
<span class="md-ellipsis">
8.3 Top-K Problem
8.3 Top-k Problem
@@ -2495,7 +2495,7 @@
<span class="md-ellipsis">
10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
@@ -2523,7 +2523,7 @@
<span class="md-ellipsis">
10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
@@ -2579,7 +2579,7 @@
<span class="md-ellipsis">
10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
@@ -2728,7 +2728,7 @@
<span class="md-ellipsis">
11.1 Sorting Algorithms
11.1 Sorting Algorithm
@@ -3201,7 +3201,7 @@
<span class="md-ellipsis">
12.4 Hanoi Tower Problem
12.4 Hanota Problem
@@ -4113,7 +4113,7 @@
<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
@@ -4279,7 +4279,7 @@
<p><img alt="Data structures" class="cover-image" src="../assets/covers/chapter_data_structure.jpg" /></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>Data structure is like a sturdy and diverse framework.</p>
<p>Data structures are like a sturdy and diverse framework.</p>
<p>It provides a blueprint for the orderly organization of data, upon which algorithms come to life.</p>
</div>
<h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
en/chapter_data_structure/number_encoding/index.html
+23 -23
View File
@@ -6,7 +6,7 @@
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
@@ -576,7 +576,7 @@
<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
@@ -598,7 +598,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
</label>
@@ -1255,7 +1255,7 @@
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
@@ -1277,7 +1277,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
</label>
@@ -1383,7 +1383,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
@@ -1474,7 +1474,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
@@ -1496,7 +1496,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
@@ -1574,7 +1574,7 @@
<span class="md-ellipsis">
5.3 Double-Ended Queue
5.3 Deque
@@ -1665,7 +1665,7 @@
<span class="md-ellipsis">
Chapter 6. Hashing
Chapter 6. Hash Table
@@ -1687,7 +1687,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
@@ -1960,7 +1960,7 @@
<span class="md-ellipsis">
7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
@@ -2179,7 +2179,7 @@
<span class="md-ellipsis">
8.2 Building a Heap
8.2 Heap Construction Operation
@@ -2207,7 +2207,7 @@
<span class="md-ellipsis">
8.3 Top-K Problem
8.3 Top-k Problem
@@ -2565,7 +2565,7 @@
<span class="md-ellipsis">
10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
@@ -2593,7 +2593,7 @@
<span class="md-ellipsis">
10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
@@ -2649,7 +2649,7 @@
<span class="md-ellipsis">
10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
@@ -2798,7 +2798,7 @@
<span class="md-ellipsis">
11.1 Sorting Algorithms
11.1 Sorting Algorithm
@@ -3271,7 +3271,7 @@
<span class="md-ellipsis">
12.4 Hanoi Tower Problem
12.4 Hanota Problem
@@ -4183,7 +4183,7 @@
<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
@@ -4362,7 +4362,7 @@
<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-sign-magnitude-1s-complement-and-2s-complement">3.3.1 &nbsp; Sign-Magnitude, 1's Complement, and 2's Complement<a class="headerlink" href="#331-sign-magnitude-1s-complement-and-2s-complement" title="Permanent link">&para;</a></h2>
<p>In the table from the previous section, we found that all integer types can represent one more negative number than positive numbers. For example, the <code>byte</code> range is <span class="arithmatex">\([-128, 127]\)</span>. This phenomenon is counterintuitive, and its underlying reason involves knowledge of sign-magnitude, 1's complement, and 2's complement.</p>
<p>In the table from the previous section, we found that all integer types can represent one more negative number than positive numbers. For example, the <code>byte</code> range is <span class="arithmatex">\([-128, 127]\)</span>. This phenomenon is counterintuitive, and its underlying cause lies in sign-magnitude, 1's complement, and 2's complement representations.</p>
<p>First, it should be noted that <strong>numbers are stored in computers in the form of "2's complement"</strong>. Before analyzing the reasons for this, let's first define these three concepts.</p>
<ul>
<li><strong>Sign-magnitude</strong>: We treat the highest bit of the binary representation of a number as the sign bit, where <span class="arithmatex">\(0\)</span> represents a positive number and <span class="arithmatex">\(1\)</span> represents a negative number, and the remaining bits represent the value of the number.</li>
@@ -4409,7 +4409,7 @@
\end{aligned}
\]</div>
<p>Adding <span class="arithmatex">\(1\)</span> to the 1's complement of negative zero produces a carry, but since the <code>byte</code> type has a length of only 8 bits, the <span class="arithmatex">\(1\)</span> that overflows to the 9<sup>th</sup> bit is discarded. That is to say, <strong>the 2's complement of negative zero is <span class="arithmatex">\(0000 \; 0000\)</span>, which is the same as the 2's complement of positive zero</strong>. This means that in 2's complement representation, there is only one zero, and the positive and negative zero ambiguity is thus resolved.</p>
<p>One last question remains: the range of the <code>byte</code> type is <span class="arithmatex">\([-128, 127]\)</span>, and how is the extra negative number <span class="arithmatex">\(-128\)</span> obtained? We notice that all integers in the interval <span class="arithmatex">\([-127, +127]\)</span> have corresponding sign-magnitude, 1's complement, and 2's complement, and sign-magnitude and 2's complement can be converted to each other.</p>
<p>One last question remains: the range of the <code>byte</code> type is <span class="arithmatex">\([-128, 127]\)</span>, so where does the extra negative number <span class="arithmatex">\(-128\)</span> come from? We notice that all integers in the interval <span class="arithmatex">\([-127, +127]\)</span> have corresponding sign-magnitude, 1's complement, and 2's complement, and sign-magnitude and 2's complement can be converted to each other.</p>
<p>However, <strong>the 2's complement <span class="arithmatex">\(1000 \; 0000\)</span> is an exception, and it does not have a corresponding sign-magnitude</strong>. According to the conversion method, we get that the sign-magnitude of this 2's complement is <span class="arithmatex">\(0000 \; 0000\)</span>. This is clearly contradictory because this sign-magnitude represents the number <span class="arithmatex">\(0\)</span>, and its 2's complement should be itself. The computer specifies that this special 2's complement <span class="arithmatex">\(1000 \; 0000\)</span> represents <span class="arithmatex">\(-128\)</span>. In fact, the result of calculating <span class="arithmatex">\((-1) + (-127)\)</span> in 2's complement is <span class="arithmatex">\(-128\)</span>.</p>
<div class="arithmatex">\[
\begin{aligned}
@@ -4423,7 +4423,7 @@
\]</div>
<p>You may have noticed that all the above calculations are addition operations. This hints at an important fact: <strong>the hardware circuits inside computers are mainly designed based on addition operations</strong>. This is because addition operations are simpler to implement in hardware compared to other operations (such as multiplication, division, and subtraction), easier to parallelize, and have faster operation speeds.</p>
<p>Please note that this does not mean that computers can only perform addition. <strong>By combining addition with some basic logical operations, computers can implement various other mathematical operations</strong>. For example, calculating the subtraction <span class="arithmatex">\(a - b\)</span> can be converted to calculating the addition <span class="arithmatex">\(a + (-b)\)</span>; calculating multiplication and division can be converted to calculating multiple additions or subtractions.</p>
<p>Now we can summarize the reasons why computers use 2's complement: based on 2's complement representation, computers can use the same circuits and operations to handle the addition of positive and negative numbers, without the need to design special hardware circuits to handle subtraction, and without the need to specially handle the ambiguity problem of positive and negative zero. This greatly simplifies hardware design and improves operational efficiency.</p>
<p>We can now summarize why computers use 2's complement: with 2's complement representation, computers can use the same circuits and operations to handle the addition of positive and negative numbers, without designing special hardware circuits for subtraction or separately handling the ambiguity of positive and negative zero. This greatly simplifies hardware design and improves efficiency.</p>
<p>The design of 2's complement is very ingenious. Due to space limitations, we will 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>
<p>Careful readers may have noticed: <code>int</code> and <code>float</code> have the same length, both are 4 bytes, but why does <code>float</code> have a much larger range than <code>int</code>? This is very counterintuitive because it stands to reason that <code>float</code> needs to represent decimals, so the range should be smaller.</p>
en/chapter_data_structure/summary/index.html
+26 -26
View File
@@ -6,7 +6,7 @@
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
@@ -576,7 +576,7 @@
<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
@@ -598,7 +598,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
</label>
@@ -1255,7 +1255,7 @@
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
@@ -1277,7 +1277,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
</label>
@@ -1383,7 +1383,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
@@ -1474,7 +1474,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
@@ -1496,7 +1496,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
@@ -1574,7 +1574,7 @@
<span class="md-ellipsis">
5.3 Double-Ended Queue
5.3 Deque
@@ -1665,7 +1665,7 @@
<span class="md-ellipsis">
Chapter 6. Hashing
Chapter 6. Hash Table
@@ -1687,7 +1687,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
@@ -1960,7 +1960,7 @@
<span class="md-ellipsis">
7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
@@ -2179,7 +2179,7 @@
<span class="md-ellipsis">
8.2 Building a Heap
8.2 Heap Construction Operation
@@ -2207,7 +2207,7 @@
<span class="md-ellipsis">
8.3 Top-K Problem
8.3 Top-k Problem
@@ -2565,7 +2565,7 @@
<span class="md-ellipsis">
10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
@@ -2593,7 +2593,7 @@
<span class="md-ellipsis">
10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
@@ -2649,7 +2649,7 @@
<span class="md-ellipsis">
10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
@@ -2798,7 +2798,7 @@
<span class="md-ellipsis">
11.1 Sorting Algorithms
11.1 Sorting Algorithm
@@ -3271,7 +3271,7 @@
<span class="md-ellipsis">
12.4 Hanoi Tower Problem
12.4 Hanota Problem
@@ -4183,7 +4183,7 @@
<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
@@ -4360,15 +4360,15 @@
<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 classified 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, and network structures. We typically classify data structures as 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>
<li>Common logical structures include linear, tree-like, and network structures. We typically classify data structures as 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>
<li>When a program runs, data is stored in computer memory. Each memory space has a corresponding memory address, and the program accesses data through these memory addresses.</li>
<li>Physical structures are primarily divided into contiguous space storage (arrays) and dispersed space storage (linked lists). All data structures are implemented using arrays, linked lists, or a combination of both.</li>
<li>Basic data types in computers include integers <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>, floating-point numbers <code>float</code>, <code>double</code>, characters <code>char</code>, and booleans <code>bool</code>. Their value ranges depend on the size of space they occupy and their representation method.</li>
<li>Sign-magnitude, 1's complement, and 2's complement are three methods for encoding numbers in computers, and they can be converted into each other. The most significant bit of sign-magnitude is the sign bit, and the remaining bits represent the value of the number.</li>
<li>Integers are stored in computers in 2's complement form. Under 2's complement representation, computers can treat the addition of positive and negative numbers uniformly, without needing to design special hardware circuits for subtraction, and there is no ambiguity of positive and negative zero.</li>
<li>The encoding of floating-point numbers consists of 1 sign bit, 8 exponent bits, and 23 fraction bits. Due to the exponent bits, the range of floating-point numbers is much larger than that of integers, at the cost of sacrificing precision.</li>
<li>ASCII is the earliest English character set, with a length of 1 byte, containing a total of 127 characters. GBK is a commonly used Chinese character set, containing over 20,000 Chinese characters. Unicode is committed to providing a complete character set standard, collecting characters from various languages around the world, thereby solving the garbled text problem caused by inconsistent character encoding methods.</li>
<li>UTF-8 is the most popular Unicode encoding method, with excellent universality. It is a variable-length encoding method with good scalability, effectively improving storage space efficiency. UTF-16 and UTF-32 are fixed-length encoding methods. When encoding Chinese characters, UTF-16 occupies less space than UTF-8. Programming languages such as Java and C# use UTF-16 encoding by default.</li>
<li>ASCII is the earliest English character set, with a length of 1 byte, containing a total of 128 characters. GBK is a commonly used Chinese character set, containing over 20,000 Chinese characters. Unicode is committed to providing a complete character set standard, collecting characters from various languages around the world, thereby solving the garbled text problem caused by inconsistent character encoding methods.</li>
<li>UTF-8 is the most popular Unicode encoding method and has excellent compatibility. It is a variable-length encoding method with good scalability, effectively improving storage space efficiency. UTF-16 and UTF-32 are common Unicode encoding methods. When encoding Chinese characters, UTF-16 occupies less space than UTF-8. Programming languages such as Java and C# use UTF-16 encoding by default.</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>: Why do hash tables contain both linear and non-linear data structures?</p>
@@ -4379,7 +4379,7 @@
<p><strong>Q</strong>: Is there ambiguity in referring to array-based data structures as "static data structures"? Stacks can also perform "dynamic" operations such as push and pop.</p>
<p>Stacks can indeed implement dynamic data operations, but the data structure is still "static" (fixed length). Although array-based data structures can dynamically add or remove elements, their capacity is fixed. If the data volume exceeds the pre-allocated size, a new larger array needs to be created, and the contents of the old array must be copied to the new array.</p>
<p><strong>Q</strong>: When constructing a stack (queue), its size is not specified. Why are they "static data structures"?</p>
<p>In high-level programming languages, we do not need to manually specify the initial capacity of a stack (queue); this work is automatically completed within the class. For example, the initial capacity of Java's <code>ArrayList</code> is typically 10. Additionally, the expansion operation is also automatically implemented. See the subsequent "List" section for details.</p>
<p>In high-level programming languages, we do not need to manually specify the initial capacity of a stack (queue); the class handles this automatically. For example, the initial capacity of Java's <code>ArrayList</code> is typically 10. Additionally, the expansion operation is also automatically implemented. See the subsequent "List" section for details.</p>
<p><strong>Q</strong>: The method of converting sign-magnitude to 2's complement is "first negate then add 1". So converting 2's complement to sign-magnitude should be the inverse operation "first subtract 1 then negate". However, 2's complement can also be converted to sign-magnitude through "first negate then add 1". Why is this?</p>
<p>This is because the mutual conversion between sign-magnitude and 2's complement is actually the process of computing the "complement". Let us first define the complement: assuming <span class="arithmatex">\(a + b = c\)</span>, then we say that <span class="arithmatex">\(a\)</span> is the complement of <span class="arithmatex">\(b\)</span> to <span class="arithmatex">\(c\)</span>, and conversely, <span class="arithmatex">\(b\)</span> is the complement of <span class="arithmatex">\(a\)</span> to <span class="arithmatex">\(c\)</span>.</p>
<p>Given an <span class="arithmatex">\(n = 4\)</span> bit binary number <span class="arithmatex">\(0010\)</span>, if we treat this number as sign-magnitude (ignoring the sign bit), then its 2's complement can be obtained through "first negate then add 1":</p>
@@ -4445,7 +4445,7 @@ aria-label="Footer"
<a
href="../../chapter_array_and_linkedlist/"
class="md-footer__link md-footer__link--next"
aria-label="Next: Chapter 4. &amp;nbsp; Array and Linked List"
aria-label="Next: Chapter 4. &amp;nbsp; Arrays and Linked Lists"
rel="next"
>
<div class="md-footer__title">
@@ -4453,7 +4453,7 @@ aria-label="Footer"
Next
</span>
<div class="md-ellipsis">
Chapter 4. &nbsp; Array and Linked List
Chapter 4. &nbsp; Arrays and Linked Lists
</div>
</div>
<div class="md-footer__button md-icon">