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en/docs/chapter_data_structure/character_encoding.md

@@ -8,7 +8,7 @@ In the computer system, all data is stored in binary form, and characters (repre
 
 ## 3.4.1   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).
+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 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" }
 
@@ -36,7 +36,7 @@ Since its release in 1991, Unicode has continually expanded to include new langu
 
 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.
+A straightforward solution to this problem is to store all characters as equal-length encodings. As shown in 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" }
 
@@ -53,7 +53,7 @@ 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 $1$.
+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.
 

en/docs/chapter_data_structure/classification_of_data_structure.md

@@ -10,7 +10,7 @@ Common data structures include arrays, linked lists, stacks, queues, hash tables
 
 **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.
+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 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.
@@ -27,9 +27,9 @@ Non-linear data structures can be further divided into tree structures and netwo
 
 ## 3.1.2   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.
+**During the execution of an algorithm, the data being processed is stored in memory**. 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.
+**The system accesses the data at the target location by means of a memory address**. As shown in 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" }
 
@@ -41,7 +41,7 @@ Non-linear data structures can be further divided into tree structures and netwo
 
 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.
+As illustrated in 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" }
 

en/docs/chapter_data_structure/number_encoding.md

@@ -143,7 +143,7 @@ Now we can answer the initial question: **The representation of `float` includes
 
 **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 $\mathrm{E} = 0$ and $\mathrm{E} = 255$ have special meanings, **used to represent zero, infinity, $\mathrm{NaN}$, etc.**
+As shown in Table 3-2, exponent bits $\mathrm{E} = 0$ and $\mathrm{E} = 255$ have special meanings, **used to represent zero, infinity, $\mathrm{NaN}$, etc.**
 
 <p align="center"> Table 3-2 &nbsp; Meaning of exponent bits </p>