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# 3.2   Basic Data Types
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.
**Basic data types are those that the CPU can directly operate on** and are directly used in algorithms, mainly including the following.
- Integer types: `byte`, `short`, `int`, `long`.
- Floating-point types: `float`, `double`, used to represent decimals.
- Character type: `char`, used to represent letters, punctuation, and even emojis in various languages.
- Boolean type: `bool`, used to represent "yes" or "no" decisions.
**Basic data types are stored in computers in binary form**. One binary digit is 1 bit. In most modern operating systems, 1 byte consists of 8 bits.
The range of values for basic data types depends on the size of the space they occupy. Below, we take Java as an example.
- The integer type `byte` occupies 1 byte = 8 bits and can represent $2^8$ numbers.
- The integer type `int` occupies 4 bytes = 32 bits and can represent $2^{32}$ numbers.
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 align="center"> Table 3-1 &nbsp; Space Occupied and Value Range of Basic Data Types </p>
<div class="center-table" markdown>
| Type | Symbol | Space Occupied | Minimum Value | Maximum Value | Default Value |
| ------- | -------- | -------------- | ------------------------ | ----------------------- | -------------- |
| Integer | `byte` | 1 byte | $-2^7$ ($-128$) | $2^7 - 1$ ($127$) | 0 |
| | `short` | 2 bytes | $-2^{15}$ | $2^{15} - 1$ | 0 |
| | `int` | 4 bytes | $-2^{31}$ | $2^{31} - 1$ | 0 |
| | `long` | 8 bytes | $-2^{63}$ | $2^{63} - 1$ | 0 |
| Float | `float` | 4 bytes | $1.175 \times 10^{-38}$ | $3.403 \times 10^{38}$ | $0.0\text{f}$ |
| | `double` | 8 bytes | $2.225 \times 10^{-308}$ | $1.798 \times 10^{308}$ | 0.0 |
| Char | `char` | 2 bytes | 0 | $2^{16} - 1$ | 0 |
| Boolean | `bool` | 1 byte | $\text{false}$ | $\text{true}$ | $\text{false}$ |
</div>
Please note that the above table is specific to Java's basic data types. Every programming language has its own data type definitions, which might differ in space occupied, value ranges, and default values.
- In Python, the integer type `int` can be of any size, limited only by available memory; the floating-point `float` is double precision 64-bit; there is no `char` type, as a single character is actually a string `str` of length 1.
- C and C++ do not specify the size of basic data types, it varies with implementation and platform. The above table follows the LP64 [data model](https://en.cppreference.com/w/cpp/language/types#Properties), used for Unix 64-bit operating systems including Linux and macOS.
- The size of `char` in C and C++ is 1 byte, while in most programming languages, it depends on the specific character encoding method, as detailed in the "Character Encoding" chapter.
- Even though representing a boolean only requires 1 bit (0 or 1), it is usually stored in memory as 1 byte. This is because modern computer CPUs typically use 1 byte as the smallest addressable memory unit.
So, what is the connection between basic data types and data structures? We know that data structures are ways to organize and store data in computers. The focus here is on "structure" rather than "data".
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 the ordering of the numbers, but whether the stored content is an integer `int`, a decimal `float`, or a character `char`, is irrelevant to the "data structure".
In other words, **basic data types provide the "content type" of data, while data structures provide the "way of organizing" data**. For example, in the following code, we use the same data structure (array) to store and represent different basic data types, including `int`, `float`, `char`, `bool`, etc.
=== "Python"
```python title=""
# Using various basic data types to initialize arrays
numbers: list[int] = [0] * 5
decimals: list[float] = [0.0] * 5
# Python's characters are actually strings of length 1
characters: list[str] = ['0'] * 5
bools: list[bool] = [False] * 5
# Python's lists can freely store various basic data types and object references
data = [0, 0.0, 'a', False, ListNode(0)]
```
=== "C++"
```cpp title=""
// Using various basic data types to initialize arrays
int numbers[5];
float decimals[5];
char characters[5];
bool bools[5];
```
=== "Java"
```java title=""
// Using various basic data types to initialize arrays
int[] numbers = new int[5];
float[] decimals = new float[5];
char[] characters = new char[5];
boolean[] bools = new boolean[5];
```
=== "C#"
```csharp title=""
// Using various basic data types to initialize arrays
int[] numbers = new int[5];
float[] decimals = new float[5];
char[] characters = new char[5];
bool[] bools = new bool[5];
```
=== "Go"
```go title=""
// Using various basic data types to initialize arrays
var numbers = [5]int{}
var decimals = [5]float64{}
var characters = [5]byte{}
var bools = [5]bool{}
```
=== "Swift"
```swift title=""
// Using various basic data types to initialize arrays
let numbers = Array(repeating: 0, count: 5)
let decimals = Array(repeating: 0.0, count: 5)
let characters: [Character] = Array(repeating: "a", count: 5)
let bools = Array(repeating: false, count: 5)
```
=== "JS"
```javascript title=""
// JavaScript's arrays can freely store various basic data types and objects
const array = [0, 0.0, 'a', false];
```
=== "TS"
```typescript title=""
// Using various basic data types to initialize arrays
const numbers: number[] = [];
const characters: string[] = [];
const bools: boolean[] = [];
```
=== "Dart"
```dart title=""
// Using various basic data types to initialize arrays
List<int> numbers = List.filled(5, 0);
List<double> decimals = List.filled(5, 0.0);
List<String> characters = List.filled(5, 'a');
List<bool> bools = List.filled(5, false);
```
=== "Rust"
```rust title=""
// Using various basic data types to initialize arrays
let numbers: Vec<i32> = vec![0; 5];
let decimals: Vec<f32> = vec![0.0, 5];
let characters: Vec<char> = vec!['0'; 5];
let bools: Vec<bool> = vec![false; 5];
```
=== "C"
```c title=""
// Using various basic data types to initialize arrays
int numbers[10];
float decimals[10];
char characters[10];
bool bools[10];
```
=== "Zig"
```zig title=""
// Using various basic data types to initialize arrays
var numbers: [5]i32 = undefined;
var decimals: [5]f32 = undefined;
var characters: [5]u8 = undefined;
var bools: [5]bool = undefined;
```

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# 3.4 &nbsp; Character Encoding *
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.
## 3.4.1 &nbsp; ASCII Character Set
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).
![ASCII Code](character_encoding.assets/ascii_table.png){ class="animation-figure" }
<p align="center"> Figure 3-6 &nbsp; ASCII Code </p>
However, **ASCII can only represent English characters**. 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.
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.
## 3.4.2 &nbsp; GBK Character Set
Later, it was found that **EASCII still could not meet the character requirements of many languages**. 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.
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.
## 3.4.3 &nbsp; Unicode Character Set
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.
Researchers of that era thought: **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?** Inspired by this idea, the extensive character set, Unicode, was born.
"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.
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.
Unicode is a universal character set that assigns a number (called a "code point") to each character, **but it does not specify how these character code points should be stored in a computer system**. 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?
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.
![Unicode Encoding Example](character_encoding.assets/unicode_hello_algo.png){ class="animation-figure" }
<p align="center"> Figure 3-7 &nbsp; Unicode Encoding Example </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.
## 3.4.4 &nbsp; UTF-8 Encoding
Currently, UTF-8 has become the most widely used Unicode encoding method internationally. **It is a variable-length encoding**, 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.
The encoding rules for UTF-8 are not complex and can be divided into two cases:
- For 1-byte characters, set the highest bit to $0$, and the remaining 7 bits to the Unicode code point. Notably, ASCII characters occupy the first 128 code points in the Unicode set. This means that **UTF-8 encoding is backward compatible with ASCII**. This implies that UTF-8 can be used to parse ancient ASCII text.
- For characters of length $n$ bytes (where $n > 1$), set the highest $n$ bits of the first byte to $1$, and the $(n + 1)^{\text{th}}$ bit to $0$; starting from the second byte, set the highest 2 bits of each byte to $10$; the rest of the bits are used to fill the Unicode code point.
The Figure 3-8 shows the UTF-8 encoding for "Hello算法". It can be observed that since the highest $n$ bits are set to $1$, the system can determine the length of the character as $n$ by counting the number of highest bits set to $1$.
But why set the highest 2 bits of the remaining bytes to $10$? Actually, this $10$ serves as a kind of checksum. If the system starts parsing text from an incorrect byte, the $10$ at the beginning of the byte can help the system quickly detect anomalies.
The reason for using $10$ as a checksum is that, under UTF-8 encoding rules, it's impossible for the highest two bits of a character to be $10$. This can be proven by contradiction: If the highest two bits of a character are $10$, it indicates that the character's length is $1$, corresponding to ASCII. However, the highest bit of an ASCII character should be $0$, which contradicts the assumption.
![UTF-8 Encoding Example](character_encoding.assets/utf-8_hello_algo.png){ class="animation-figure" }
<p align="center"> Figure 3-8 &nbsp; UTF-8 Encoding Example </p>
Apart from UTF-8, other common encoding methods include:
- **UTF-16 Encoding**: 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.
- **UTF-32 Encoding**: 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.
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.
From a compatibility perspective, UTF-8 is the most versatile, with many tools and libraries supporting UTF-8 as a priority.
## 3.4.5 &nbsp; Character Encoding in Programming Languages
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:
- **Random Access**: Strings encoded in UTF-16 can be accessed randomly with ease. For UTF-8, which is a variable-length encoding, locating the $i^{th}$ character requires traversing the string from the start to the $i^{th}$ position, taking $O(n)$ time.
- **Character Counting**: Similar to random access, counting the number of characters in a UTF-16 encoded string is an $O(1)$ operation. However, counting characters in a UTF-8 encoded string requires traversing the entire string.
- **String Operations**: 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.
The design of character encoding schemes in programming languages is an interesting topic involving various factors:
- Java’s `String` type uses UTF-16 encoding, with each character occupying 2 bytes. This was based on the initial belief that 16 bits were sufficient to represent all possible characters and proven incorrect later. As the Unicode standard expanded beyond 16 bits, characters in Java may now be represented by a pair of 16-bit values, known as “surrogate pairs.”
- JavaScript and TypeScript use UTF-16 encoding for similar reasons as Java. When JavaScript was first introduced by Netscape in 1995, Unicode was still in its early stages, and 16-bit encoding was sufficient to represent all Unicode characters.
- C# uses UTF-16 encoding, largely because the .NET platform, designed by Microsoft, and many Microsoft technologies, including the Windows operating system, extensively use UTF-16 encoding.
Due to the underestimation of character counts, these languages had to use "surrogate pairs" to represent Unicode characters exceeding 16 bits. This approach has its drawbacks: strings containing surrogate pairs may have characters occupying 2 or 4 bytes, losing the advantage of fixed-length encoding. Additionally, handling surrogate pairs adds complexity and debugging difficulty to programming.
Addressing these challenges, some languages have adopted alternative encoding strategies:
- Python’s `str` type uses Unicode encoding with a flexible representation where the storage length of characters depends on the largest Unicode code point in the string. If all characters are ASCII, each character occupies 1 byte, 2 bytes for characters within the Basic Multilingual Plane (BMP), and 4 bytes for characters beyond the BMP.
- Go’s `string` type internally uses UTF-8 encoding. Go also provides the `rune` type for representing individual Unicode code points.
- Rust’s `str` and `String` types use UTF-8 encoding internally. Rust also offers the `char` type for individual Unicode code points.
It’s important to note that the above discussion pertains to how strings are stored in programming languages, **which is different from how strings are stored in files or transmitted over networks**. For file storage or network transmission, strings are usually encoded in UTF-8 format for optimal compatibility and space efficiency.

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# 3.1 &nbsp; Classification of Data Structures
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".
## 3.1.1 &nbsp; Logical Structure: Linear and Non-Linear
**The logical structures reveal the logical relationships between data elements**. 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.
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.
- **Linear Data Structures**: Arrays, Linked Lists, Stacks, Queues, Hash Tables.
- **Non-Linear Data Structures**: Trees, Heaps, Graphs, Hash Tables.
![Linear and Non-Linear Data Structures](classification_of_data_structure.assets/classification_logic_structure.png){ class="animation-figure" }
<p align="center"> Figure 3-1 &nbsp; Linear and Non-Linear Data Structures </p>
Non-linear data structures can be further divided into tree structures and network structures.
- **Linear Structures**: Arrays, linked lists, queues, stacks, and hash tables, where elements have a one-to-one sequential relationship.
- **Tree Structures**: Trees, Heaps, Hash Tables, where elements have a one-to-many relationship.
- **Network Structures**: Graphs, where elements have a many-to-many relationships.
## 3.1.2 &nbsp; Physical Structure: Contiguous and Dispersed
**During the execution of an algorithm, the data being processed is stored in memory**. 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.
**The system accesses the data at the target location by means of a memory address**. 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.
![Memory Stick, Memory Spaces, Memory Addresses](classification_of_data_structure.assets/computer_memory_location.png){ class="animation-figure" }
<p align="center"> Figure 3-2 &nbsp; Memory Stick, Memory Spaces, Memory Addresses </p>
!!! tip
It's worth noting that comparing memory to an Excel spreadsheet is a simplified analogy. The actual working mechanism of memory is more complex, involving concepts like address space, memory management, cache mechanisms, virtual memory, and physical memory.
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. **Therefore, considering memory resources is crucial in designing data structures and algorithms**. 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.
As illustrated in the Figure 3-3 , **the physical structure reflects the way data is stored in computer memory** 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.
![Contiguous Space Storage and Dispersed Space Storage](classification_of_data_structure.assets/classification_phisical_structure.png){ class="animation-figure" }
<p align="center"> Figure 3-3 &nbsp; Contiguous Space Storage and Dispersed Space Storage </p>
**It is worth noting that all data structures are implemented based on arrays, linked lists, or a combination of both**. 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.
- **Array-based implementations**: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, Matrices, Tensors (arrays with dimensions $\geq 3$).
- **Linked-list-based implementations**: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, etc.
Data structures implemented based on arrays are also called “Static Data Structures,” meaning their length cannot be changed after initialization. Conversely, those based on linked lists are called “Dynamic Data Structures,” which can still adjust their size during program execution.
!!! tip
If you find it challenging to comprehend the physical structure, it is recommended that you read the next chapter, "Arrays and Linked Lists," and revisit this section later.

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# Chapter 3. &nbsp; Data Structures
<div class="center-table" markdown>
![Data Structures](../assets/covers/chapter_data_structure.jpg){ class="cover-image" }
</div>
!!! abstract
Data structures serve as a robust and diverse framework.
They offer a blueprint for the orderly organization of data, upon which algorithms come to life.
## Chapter Contents
- [3.1 &nbsp; Classification of Data Structures](https://www.hello-algo.com/en/chapter_data_structure/classification_of_data_structure/)
- [3.2 &nbsp; Fundamental Data Types](https://www.hello-algo.com/en/chapter_data_structure/basic_data_types/)
- [3.3 &nbsp; Number Encoding *](https://www.hello-algo.com/en/chapter_data_structure/number_encoding/)
- [3.4 &nbsp; Character Encoding *](https://www.hello-algo.com/en/chapter_data_structure/character_encoding/)
- [3.5 &nbsp; Summary](https://www.hello-algo.com/en/chapter_data_structure/summary/)

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# 3.3 &nbsp; Number Encoding *
!!! note
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.
## 3.3.1 &nbsp; Integer Encoding
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 `byte` range of $[-128, 127]$. This phenomenon seems counterintuitive, and its underlying reason involves knowledge of sign-magnitude, one's complement, and two's complement encoding.
Firstly, it's important to note that **numbers are stored in computers using the two's complement form**. Before analyzing why this is the case, let's define these three encoding methods:
- **Sign-magnitude**: The highest bit of a binary representation of a number is considered the sign bit, where $0$ represents a positive number and $1$ represents a negative number. The remaining bits represent the value of the number.
- **One's complement**: The one's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by inverting all bits except the sign bit.
- **Two's complement**: The two's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by adding $1$ to their one's complement.
The following diagram illustrates the conversions among sign-magnitude, one's complement, and two's complement:
![Conversions between Sign-Magnitude, One's Complement, and Two's Complement](number_encoding.assets/1s_2s_complement.png){ class="animation-figure" }
<p align="center"> Figure 3-4 &nbsp; Conversions between Sign-Magnitude, One's Complement, and Two's Complement </p>
Although sign-magnitude is the most intuitive, it has limitations. For one, **negative numbers in sign-magnitude cannot be directly used in calculations**. For example, in sign-magnitude, calculating $1 + (-2)$ results in $-3$, which is incorrect.
$$
\begin{aligned}
& 1 + (-2) \newline
& \rightarrow 0000 \; 0001 + 1000 \; 0010 \newline
& = 1000 \; 0011 \newline
& \rightarrow -3
\end{aligned}
$$
To address this, computers introduced the **one's complement**. If we convert to one's complement and calculate $1 + (-2)$, then convert the result back to sign-magnitude, we get the correct result of $-1$.
$$
\begin{aligned}
& 1 + (-2) \newline
& \rightarrow 0000 \; 0001 \; \text{(Sign-magnitude)} + 1000 \; 0010 \; \text{(Sign-magnitude)} \newline
& = 0000 \; 0001 \; \text{(One's complement)} + 1111 \; 1101 \; \text{(One's complement)} \newline
& = 1111 \; 1110 \; \text{(One's complement)} \newline
& = 1000 \; 0001 \; \text{(Sign-magnitude)} \newline
& \rightarrow -1
\end{aligned}
$$
Additionally, **there are two representations of zero in sign-magnitude**: $+0$ and $-0$. This means two different binary encodings for zero, which could lead to ambiguity. For example, in conditional checks, not differentiating between positive and negative zero might result in incorrect outcomes. Addressing this ambiguity would require additional checks, potentially reducing computational efficiency.
$$
\begin{aligned}
+0 & \rightarrow 0000 \; 0000 \newline
-0 & \rightarrow 1000 \; 0000
\end{aligned}
$$
Like sign-magnitude, one's complement also suffers from the positive and negative zero ambiguity. Therefore, computers further introduced the **two's complement**. Let's observe the conversion process for negative zero in sign-magnitude, one's complement, and two's complement:
$$
\begin{aligned}
-0 \rightarrow \; & 1000 \; 0000 \; \text{(Sign-magnitude)} \newline
= \; & 1111 \; 1111 \; \text{(One's complement)} \newline
= 1 \; & 0000 \; 0000 \; \text{(Two's complement)} \newline
\end{aligned}
$$
Adding $1$ to the one's complement of negative zero produces a carry, but with `byte` length being only 8 bits, the carried-over $1$ to the 9th bit is discarded. Therefore, **the two's complement of negative zero is $0000 \; 0000$**, the same as positive zero, thus resolving the ambiguity.
One last puzzle is the $[-128, 127]$ range for `byte`, with an additional negative number, $-128$. We observe that for the interval $[-127, +127]$, all integers have corresponding sign-magnitude, one's complement, and two's complement, allowing for mutual conversion between them.
However, **the two's complement $1000 \; 0000$ is an exception without a corresponding sign-magnitude**. According to the conversion method, its sign-magnitude would be $0000 \; 0000$, indicating zero. This presents a contradiction because its two's complement should represent itself. Computers designate this special two's complement $1000 \; 0000$ as representing $-128$. In fact, the calculation of $(-1) + (-127)$ in two's complement results in $-128$.
$$
\begin{aligned}
& (-127) + (-1) \newline
& \rightarrow 1111 \; 1111 \; \text{(Sign-magnitude)} + 1000 \; 0001 \; \text{(Sign-magnitude)} \newline
& = 1000 \; 0000 \; \text{(One's complement)} + 1111 \; 1110 \; \text{(One's complement)} \newline
& = 1000 \; 0001 \; \text{(Two's complement)} + 1111 \; 1111 \; \text{(Two's complement)} \newline
& = 1000 \; 0000 \; \text{(Two's complement)} \newline
& \rightarrow -128
\end{aligned}
$$
As you might have noticed, all these calculations are additions, hinting at an important fact: **computers' internal hardware circuits are primarily designed around addition operations**. This is because addition is simpler to implement in hardware compared to other operations like multiplication, division, and subtraction, allowing for easier parallelization and faster computation.
It's important to note that this doesn't mean computers can only perform addition. **By combining addition with basic logical operations, computers can execute a variety of other mathematical operations**. For example, the subtraction $a - b$ can be translated into $a + (-b)$; multiplication and division can be translated into multiple additions or subtractions.
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.
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.
## 3.3.2 &nbsp; Floating-Point Number Encoding
You might have noticed something intriguing: despite having the same length of 4 bytes, why does a `float` have a much larger range of values compared to an `int`? This seems counterintuitive, as one would expect the range to shrink for `float` since it needs to represent fractions.
In fact, **this is due to the different representation method used by floating-point numbers (`float`)**. Let's consider a 32-bit binary number as:
$$
b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
$$
According to the IEEE 754 standard, a 32-bit `float` consists of the following three parts:
- Sign bit $\mathrm{S}$: Occupies 1 bit, corresponding to $b_{31}$.
- Exponent bit $\mathrm{E}$: Occupies 8 bits, corresponding to $b_{30} b_{29} \ldots b_{23}$.
- Fraction bit $\mathrm{N}$: Occupies 23 bits, corresponding to $b_{22} b_{21} \ldots b_0$.
The value of a binary `float` number is calculated as:
$$
\text{val} = (-1)^{b_{31}} \times 2^{\left(b_{30} b_{29} \ldots b_{23}\right)_2 - 127} \times \left(1 . b_{22} b_{21} \ldots b_0\right)_2
$$
Converted to a decimal formula, this becomes:
$$
\text{val} = (-1)^{\mathrm{S}} \times 2^{\mathrm{E} - 127} \times (1 + \mathrm{N})
$$
The range of each component is:
$$
\begin{aligned}
\mathrm{S} \in & \{ 0, 1\}, \quad \mathrm{E} \in \{ 1, 2, \dots, 254 \} \newline
(1 + \mathrm{N}) = & (1 + \sum_{i=1}^{23} b_{23-i} \times 2^{-i}) \subset [1, 2 - 2^{-23}]
\end{aligned}
$$
![Example Calculation of a float in IEEE 754 Standard](number_encoding.assets/ieee_754_float.png){ class="animation-figure" }
<p align="center"> Figure 3-5 &nbsp; Example Calculation of a float in IEEE 754 Standard </p>
Observing the diagram, given an example data $\mathrm{S} = 0$, $\mathrm{E} = 124$, $\mathrm{N} = 2^{-2} + 2^{-3} = 0.375$, we have:
$$
\text{val} = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875
$$
Now we can answer the initial question: **The representation of `float` includes an exponent bit, leading to a much larger range than `int`**. Based on the above calculation, the maximum positive number representable by `float` is approximately $2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}$, and the minimum negative number is obtained by switching the sign bit.
**However, the trade-off for `float`'s expanded range is a sacrifice in precision**. The integer type `int` uses all 32 bits to represent the number, with values evenly distributed; but due to the exponent bit, the larger the value of a `float`, the greater the difference between adjacent numbers.
As shown in the Table 3-2 , exponent bits $E = 0$ and $E = 255$ have special meanings, **used to represent zero, infinity, $\mathrm{NaN}$, etc.**
<p align="center"> Table 3-2 &nbsp; Meaning of Exponent Bits </p>
<div class="center-table" markdown>
| Exponent Bit E | Fraction Bit $\mathrm{N} = 0$ | Fraction Bit $\mathrm{N} \ne 0$ | Calculation Formula |
| ------------------ | ----------------------------- | ------------------------------- | ---------------------------------------------------------------------- |
| $0$ | $\pm 0$ | Subnormal Numbers | $(-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})$ |
| $1, 2, \dots, 254$ | Normal Numbers | Normal Numbers | $(-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})$ |
| $255$ | $\pm \infty$ | $\mathrm{NaN}$ | |
</div>
It's worth noting that subnormal numbers significantly improve the precision of floating-point numbers. The smallest positive normal number is $2^{-126}$, and the smallest positive subnormal number is $2^{-126} \times 2^{-23}$.
Double-precision `double` also uses a similar representation method to `float`, which is not elaborated here for brevity.

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# 3.5 &nbsp; Summary
### 1. &nbsp; Key Review
- 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.
- 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.
- 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 addresses.
- 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.
- Basic data types in computers include integers (`byte`, `short`, `int`, `long`), floating-point numbers (`float`, `double`), characters (`char`), and booleans (`boolean`). Their range depends on the size of the space occupied and the representation method.
- Original code, complement code, and two's complement code are three methods of encoding numbers in computers, and they can be converted into each other. The highest bit of the original code of an integer is the sign bit, and the remaining bits represent the value of the number.
- Integers are stored in computers in the form of two's complement. In this representation, the computer can treat the addition of positive and negative numbers uniformly, without the need for special hardware circuits for subtraction, and there is no ambiguity of positive and negative zero.
- The encoding of floating-point numbers consists of 1 sign bit, 8 exponent bits, and 23 fraction bits. Due to the presence of the exponent bit, the range of floating-point numbers is much greater than that of integers, but at the cost of sacrificing precision.
- ASCII is the earliest English character set, 1 byte in length, and includes 127 characters. The GBK character set is a commonly used Chinese character set, including more than 20,000 Chinese characters. Unicode strives to provide a complete character set standard, including characters from various languages worldwide, thus solving the problem of garbled characters caused by inconsistent character encoding methods.
- UTF-8 is the most popular Unicode encoding method, with excellent universality. It is a variable-length encoding method with good scalability and effectively improves the efficiency of space usage. UTF-16 and UTF-32 are fixed-length encoding methods. When encoding Chinese characters, UTF-16 occupies less space than UTF-8. Programming languages like Java and C# use UTF-16 encoding by default.
### 2. &nbsp; Q & A
**Q**: Why does a hash table contain both linear and non-linear data structures?
The underlying structure of a hash table is an array. To resolve hash collisions, we may use "chaining": each bucket in the array points to a linked list, which, when exceeding a certain threshold, might be transformed into a tree (usually a red-black tree).
From a storage perspective, the foundation of a hash table is an array, where each bucket slot might contain a value, a linked list, or a tree. Therefore, hash tables may contain both linear data structures (arrays, linked lists) and non-linear data structures (trees).
**Q**: Is the length of the `char` type 1 byte?
The length of the `char` type is determined by the encoding method used by the programming language. For example, Java, JavaScript, TypeScript, and C# all use UTF-16 encoding (to save Unicode code points), so the length of the char type is 2 bytes.
**Q**: Is there ambiguity in calling data structures based on arrays "static data structures"? Because operations like push and pop on stacks are "dynamic".
While stacks indeed allow for dynamic data operations, the data structure itself remains "static" (with unchangeable length). Even though data structures based on arrays 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 copied into it.
**Q**: When building stacks (queues) without specifying their size, why are they considered "static data structures"?
In high-level programming languages, we don't need to manually specify the initial capacity of stacks (queues); this task is automatically handled internally by the class. For example, the initial capacity of Java's ArrayList is usually 10. Furthermore, the expansion operation is also implemented automatically. See the subsequent "List" chapter for details.