Merge branch 'main' into dev

codes/go/chapter_divide_and_conquer/binary_search_recur.go

@@ -18,7 +18,7 @@ func dfs(nums []int, target, i, j int) int {
 		// 递归子问题 f(m+1, j)
 		return dfs(nums, target, m+1, j)
 	} else if nums[m] > target {
-		// 小于则递归左半数组
+		// 大于则递归左半数组
 		// 递归子问题 f(i, m-1)
 		return dfs(nums, target, i, m-1)
 	} else {

codes/go/chapter_stack_and_queue/array_deque.go

@@ -72,6 +72,9 @@ func (q *arrayDeque) pushLast(num int) {
 /* 队首出队 */
 func (q *arrayDeque) popFirst() any {
 	num := q.peekFirst()
+	if num == nil {
+		return nil
+	}
 	// 队首指针向后移动一位
 	q.front = q.index(q.front + 1)
 	q.queSize--
@@ -81,6 +84,9 @@ func (q *arrayDeque) popFirst() any {
 /* 队尾出队 */
 func (q *arrayDeque) popLast() any {
 	num := q.peekLast()
+	if num == nil {
+		return nil
+	}
 	q.queSize--
 	return num
 }

codes/go/chapter_stack_and_queue/array_queue.go

@@ -49,6 +49,10 @@ func (q *arrayQueue) push(num int) {
 /* 出队 */
 func (q *arrayQueue) pop() any {
 	num := q.peek()
+	if num == nil {
+		return nil
+	}
+
 	// 队首指针向后移动一位，若越过尾部，则返回到数组头部
 	q.front = (q.front + 1) % q.queCapacity
 	q.queSize--

codes/go/chapter_stack_and_queue/queue_test.go

@@ -46,9 +46,13 @@ func TestQueue(t *testing.T) {
 }
 
 func TestArrayQueue(t *testing.T) {
+
 	// 初始化队列，使用队列的通用接口
 	capacity := 10
 	queue := newArrayQueue(capacity)
+	if queue.pop() != nil {
+		t.Errorf("want:%v,got:%v", nil, queue.pop())
+	}
 
 	// 元素入队
 	queue.push(1)

codes/rust/chapter_backtracking/permutations_i.rs

@@ -23,7 +23,7 @@ fn backtrack(mut state: Vec<i32>, choices: &[i32], selected: &mut [bool], res: &
             backtrack(state.clone(), choices, selected, res);
             // 回退：撤销选择，恢复到之前的状态
             selected[i] = false;
-            state.remove(state.len() - 1);
+            state.pop();
         }
     }
 }

codes/rust/chapter_backtracking/permutations_ii.rs

@@ -27,7 +27,7 @@ fn backtrack(mut state: Vec<i32>, choices: &[i32], selected: &mut [bool], res: &
             backtrack(state.clone(), choices, selected, res);
             // 回退：撤销选择，恢复到之前的状态
             selected[i] = false;
-            state.remove(state.len() - 1);
+            state.pop();
         }
     }
 }

codes/rust/chapter_computational_complexity/time_complexity.rs

@@ -113,7 +113,7 @@ fn linear_log_recur(n: i32) -> i32 {
         return 1;
     }
     let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);
-    for _ in 0..n as i32 {
+    for _ in 0..n {
         count += 1;
     }
     return count;

docs/chapter_computational_complexity/time_complexity.md

@@ -30,7 +30,7 @@
         a = a + 1;  // 1 ns
         a = a * 2;  // 10 ns
         // 循环 n 次
-        for (int i = 0; i < n; i++) {  // 1 ns ，每轮都要执行 i++
+        for (int i = 0; i < n; i++) {  // 1 ns
             cout << 0 << endl;         // 5 ns
         }
     }
@@ -45,7 +45,7 @@
         a = a + 1;  // 1 ns
         a = a * 2;  // 10 ns
         // 循环 n 次
-        for (int i = 0; i < n; i++) {  // 1 ns ，每轮都要执行 i++
+        for (int i = 0; i < n; i++) {  // 1 ns
             System.out.println(0);     // 5 ns
         }
     }
@@ -60,7 +60,7 @@
         a = a + 1;  // 1 ns
         a = a * 2;  // 10 ns
         // 循环 n 次
-        for (int i = 0; i < n; i++) {  // 1 ns ，每轮都要执行 i++
+        for (int i = 0; i < n; i++) {  // 1 ns
             Console.WriteLine(0);      // 5 ns
         }
     }
@@ -105,7 +105,7 @@
         a = a + 1; // 1 ns
         a = a * 2; // 10 ns
         // 循环 n 次
-        for(let i = 0; i < n; i++) { // 1 ns ，每轮都要执行 i++
+        for(let i = 0; i < n; i++) { // 1 ns
             console.log(0); // 5 ns
         }
     }
@@ -120,7 +120,7 @@
         a = a + 1; // 1 ns
         a = a * 2; // 10 ns
         // 循环 n 次
-        for(let i = 0; i < n; i++) { // 1 ns ，每轮都要执行 i++
+        for(let i = 0; i < n; i++) { // 1 ns
             console.log(0); // 5 ns
         }
     }
@@ -135,7 +135,7 @@
       a = a + 1; // 1 ns
       a = a * 2; // 10 ns
       // 循环 n 次
-      for (int i = 0; i < n; i++) { // 1 ns ，每轮都要执行 i++
+      for (int i = 0; i < n; i++) { // 1 ns
         print(0); // 5 ns
       }
     }
@@ -150,7 +150,7 @@
         a = a + 1;          // 1 ns
         a = a * 2;          // 10 ns
         // 循环 n 次
-        for _ in 0..n {     // 1 ns ，每轮都要执行 i++
+        for _ in 0..n {     // 1 ns
             println!("{}", 0);  // 5 ns
         }
     }
@@ -165,7 +165,7 @@
         a = a + 1;  // 1 ns
         a = a * 2;  // 10 ns
         // 循环 n 次
-        for (int i = 0; i < n; i++) {   // 1 ns ，每轮都要执行 i++
+        for (int i = 0; i < n; i++) {   // 1 ns
             printf("%d", 0);            // 5 ns
         }
     }
@@ -180,7 +180,7 @@
         a = a + 1 // 1 ns
         a = a * 2 // 10 ns
         // 循环 n 次
-        for (i in 0..<n) {  // 1 ns ，每轮都要执行 i++
+        for (i in 0..<n) {  // 1 ns
             println(0)      // 5 ns
         }
     }

en/docs/chapter_data_structure/summary.md

@@ -2,32 +2,65 @@
 
 ### 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.
+- Data structures can be categorized from two perspectives: logical structure and physical structure. Logical structure describes the logical relationships between data, while physical structure describes how data is stored in 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.
+- Frequently used logical structures include linear structures, trees, and networks. We usually divide 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.
+- When a program is running, data is stored in memory. Each memory space has a corresponding 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.
+- Physical structures can be divided into continuous space storage (arrays) and discrete 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.
+- The basic data types in computers include integers (`byte`, `short`, `int`, `long`), floating-point numbers (`float`, `double`), characters (`char`), and booleans (`bool`). The value range of a data type depends on its size and representation.
-- 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.
+- Sign-magnitude, 1's complement, 2's complement are three methods of encoding integers in computers, and they can be converted into each other. The most significant bit of the sign-magnitude 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.
+- Integers are encoded by 2's complement in computers. The benefits of this representation include (i) the computer can unify the addition of positive and negative integers, (ii) no need to design special hardware circuits for subtraction, and (iii) 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.
+- The encoding of floating-point numbers consists of 1 sign bit, 8 exponent bits, and 23 fraction bits. Due to the exponent bit, the range of floating-point numbers is much greater than that of integers, but at the cost of 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.
+- ASCII is the earliest English character set, with 1 byte in length and a total of 127 characters. GBK is a popular Chinese character set, which includes more than 20,000 Chinese characters. Unicode aims to provide a complete character set standard that includes characters from various languages in the world, thus solving the garbled character problem 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.
+- UTF-8 is the most popular and general Unicode encoding method. It is a variable-length encoding method with good scalability and space efficiency. UTF-16 and UTF-32 are fixed-length encoding methods. When encoding Chinese characters, UTF-16 takes up less space than UTF-8. Programming languages like Java and C# use UTF-16 encoding by default.
 
 ### 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).
+The underlying structure of a hash table is an array. To resolve hash collisions, we may use "chaining" (discussed in a later section, "Hash collision"): each bucket in the array points to a linked list, which may transform into a tree (usually a red-black tree) when its length is larger than a certain threshold.
-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).
+From a storage perspective, the underlying structure of a hash table is an array, where each bucket 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.
+The length of the `char` type is determined by the encoding method of 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".
+**Q**: Is there any ambiguity when we refer to array-based data structures as "static data structures"? The stack can also perform "dynamic" operations such as popping and pushing.
 
-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.
+The stack can implement dynamic data operations, but the data structure is still "static" (the length is fixed). Although array-based data structures can dynamically add or remove elements, their capacity is fixed. If the stack size exceeds the pre-allocated size, then the old array will be copied into a newly created and larger array.
 
-**Q**: When building stacks (queues) without specifying their size, why are they considered "static data structures"?
+**Q**: When building a stack (queue), its size is not specified, so why are they "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.
+In high-level programming languages, we do not need to manually specify the initial capacity of stacks (queues); this task is automatically completed within the class. For example, the initial capacity of Java's `ArrayList` is usually 10. Furthermore, the expansion operation is also completed automatically. See the subsequent "List" chapter for details.
+
+**Q**：The method of converting the sign-magnitude to the 2's complement is "first negate and then add 1", so converting the 2's complement to the sign-magnitude should be its inverse operation "first subtract 1 and then negate".
+However, the 2's complement can also be converted to the sign-magnitude through "first negate and then add 1", why is this?
+
+**A**：This is because the mutual conversion between the sign-magnitude and the 2's complement is equivalent to computing the "complement". We first define the complement: assuming $a + b = c$, then we say that $a$ is the complement of $b$ to $c$, and vice versa, $b$ is the complement of $a$ to $c$.
+
+Given a binary number $0010$ with length $n = 4$, if this number is the sign-magnitude (ignoring the sign bit), then its 2's complement can be obtained by "first negating and then adding 1":
+
+$$
+0010 \rightarrow 1101 \rightarrow 1110
+$$
+
+Observe that the sum of the sign-magnitude and the 2's complement is $0010 + 1110 = 10000$, i.e., the 2's complement $1110$ is the "complement" of the sign-magnitude $0010$ to $10000$. **This means that the above "first negate and then add 1" is equivalent to computing the complement to $10000$**.
+
+So, what is the "complement" of $1110$ to $10000$? We can still compute it by "negating first and then adding 1":
+
+$$
+1110 \rightarrow 0001 \rightarrow 0010
+$$
+
+In other words, the sign-magnitude and the 2's complement are each other's "complement" to $10000$, so "sign-magnitude to 2's complement" and "2's complement to sign-magnitude" can be implemented with the same operation (first negate and then add 1).
+
+Of course, we can also use the inverse operation of "first negate and then add 1" to find the sign-magnitude of the 2's complement $1110$, that is, "first subtract 1 and then negate":
+
+$$
+1110 \rightarrow 1101 \rightarrow 0010
+$$
+
+To sum up, "first negate and then add 1" and "first subtract 1 and then negate" are both computing the complement to $10000$, and they are equivalent.
+
+Essentially, the "negate" operation is actually to find the complement to $1111$ (because `sign-magnitude + 1's complement = 1111` always holds); and the 1's complement plus 1 is equal to the 2's complement to $10000$.
+
+We take $n = 4$ as an example in the above, and it can be generalized to any binary number with any number of digits.

en/docs/chapter_graph/graph.md

@@ -14,23 +14,23 @@ If vertices are viewed as nodes and edges as references (pointers) connecting th
 
 ![Relationship between linked lists, trees, and graphs](graph.assets/linkedlist_tree_graph.png)
 
-## Common types of graphs
+## Common types and terminologies of graphs
 
-Based on whether edges have direction, graphs can be divided into <u>undirected graphs</u> and <u>directed graphs</u>, as shown in the figure below.
+Graphs can be divided into  <u>undirected graphs</u> and <u>directed graphs</u> depending on whether edges have direction, as shown in the figure below.
 
-- In undirected graphs, edges represent a "bidirectional" connection between two vertices, for example, the "friendship" in WeChat or QQ.
+- In undirected graphs, edges represent a "bidirectional" connection between two vertices, for example, the "friends" in Facebook.
-- In directed graphs, edges have directionality, that is, the edges $A \rightarrow B$ and $A \leftarrow B$ are independent of each other, for example, the "follow" and "be followed" relationship on Weibo or TikTok.
+- In directed graphs, edges have directionality, that is, the edges $A \rightarrow B$ and $A \leftarrow B$ are independent of each other. For example, the "follow" and "followed" relationship on Instagram or TikTok.
 
 ![Directed and undirected graphs](graph.assets/directed_graph.png)
 
-Based on whether all vertices are connected, graphs can be divided into <u>connected graphs</u> and <u>disconnected graphs</u>, as shown in the figure below.
+Depending on whether all vertices are connected, graphs can be divided into <u>connected graphs</u> and <u>disconnected graphs</u>, as shown in the figure below.
 
-- For connected graphs, it is possible to reach any other vertex starting from a certain vertex.
+- For connected graphs, it is possible to reach any other vertex starting from an arbitrary vertex.
-- For disconnected graphs, there is at least one vertex that cannot be reached from a certain starting vertex.
+- For disconnected graphs, there is at least one vertex that cannot be reached from an arbitrary starting vertex.
 
 ![Connected and disconnected graphs](graph.assets/connected_graph.png)
 
-We can also add a weight variable to edges, resulting in <u>weighted graphs</u> as shown in the figure below. For example, in mobile games like "Honor of Kings", the system calculates the "closeness" between players based on shared gaming time, and this closeness network can be represented with a weighted graph.
+We can also add a weight variable to edges, resulting in <u>weighted graphs</u> as shown in the figure below. For example, in Instagram, the system sorts your follower and following list by the level of interaction between you and other users (likes, views, comments, etc.). Such an interaction network can be represented by a weighted graph.
 
 ![Weighted and unweighted graphs](graph.assets/weighted_graph.png)
 
@@ -42,7 +42,7 @@ Graph data structures include the following commonly used terms.
 
 ## Representation of graphs
 
-Common representations of graphs include "adjacency matrices" and "adjacency lists". The following examples use undirected graphs.
+Common representations of graphs include "adjacency matrix" and "adjacency list". The following examples use undirected graphs.
 
 ### Adjacency matrix
 
@@ -55,10 +55,10 @@ As shown in the figure below, let the adjacency matrix be $M$, and the list of v
 Adjacency matrices have the following characteristics.
 
 - A vertex cannot be connected to itself, so the elements on the main diagonal of the adjacency matrix are meaningless.
-- For undirected graphs, edges in both directions are equivalent, thus the adjacency matrix is symmetric about the main diagonal.
+- For undirected graphs, edges in both directions are equivalent, thus the adjacency matrix is symmetric with regard to the main diagonal.
-- By replacing the elements of the adjacency matrix from $1$ and $0$ to weights, it can represent weighted graphs.
+- By replacing the elements of the adjacency matrix from $1$ and $0$ to weights, we can represent weighted graphs.
 
-When representing graphs with adjacency matrices, it is possible to directly access matrix elements to obtain edges, thus operations of addition, deletion, lookup, and modification are very efficient, all with a time complexity of $O(1)$. However, the space complexity of the matrix is $O(n^2)$, which consumes more memory.
+When representing graphs with adjacency matrices, it is possible to directly access matrix elements to obtain edges, resulting in efficient operations of addition, deletion, lookup, and modification, all with a time complexity of $O(1)$. However, the space complexity of the matrix is $O(n^2)$, which consumes more memory.
 
 ### Adjacency list
 
@@ -78,6 +78,6 @@ As shown in the table below, many real-world systems can be modeled with graphs,
 
 |                 | Vertices         | Edges                                         | Graph Computing Problem          |
 | --------------- | ---------------- | --------------------------------------------- | -------------------------------- |
-| Social Networks | Users            | Friendships                                   | Potential Friend Recommendations |
+| Social Networks | Users            | Follow / Followed                                   | Potential Following Recommendations |
 | Subway Lines    | Stations         | Connectivity Between Stations                 | Shortest Route Recommendations   |
 | Solar System    | Celestial Bodies | Gravitational Forces Between Celestial Bodies | Planetary Orbit Calculations     |

en/docs/chapter_graph/index.md

@@ -4,6 +4,6 @@
 
 !!! abstract
 
-    In the journey of life, we are like individual nodes, connected by countless invisible edges.
+    In the journey of life, each of us is a node, connected by countless invisible edges.
     
-    Each encounter and parting leaves a distinctive imprint on this vast network graph.
+    Each encounter and parting leaves a unique imprint on this vast graph of life.

en/docs/chapter_hashing/hash_map.md

@@ -1,30 +1,30 @@
 # Hash table
 
-A <u>hash table</u> achieves efficient element querying by establishing a mapping between keys and values. Specifically, when we input a `key` into the hash table, we can retrieve the corresponding `value` in $O(1)$ time.
+A <u>hash table</u>, also known as a <u>hash map</u>, is a data structure that establishes a mapping between keys and values, enabling efficient element retrieval. Specifically, when we input a `key` into the hash table, we can retrive the corresponding `value` in $O(1)$ time complexity.
 
-As shown in the figure below, given $n$ students, each with two pieces of data: "name" and "student number". If we want to implement a query feature that returns the corresponding name when given a student number, we can use the hash table shown in the figure below.
+As shown in the figure below, given $n$ students, each student has two data fields: "Name" and "Student ID". If we want to implement a query function that takes a student ID as input and returns the corresponding name, we can use the hash table shown in the figure below.
 
 ![Abstract representation of a hash table](hash_map.assets/hash_table_lookup.png)
 
-Apart from hash tables, arrays and linked lists can also be used to implement querying functions. Their efficiency is compared in the table below.
+In addition to hash tables, arrays and linked lists can also be used to implement query functionality, but the time complexity is different. Their efficiency is compared in the table below:
 
-- **Adding elements**: Simply add the element to the end of the array (or linked list), using $O(1)$ time.
+- **Inserting elements**: Simply append the element to the tail of the array (or linked list). The time complexity of this operation is $O(1)$.
-- **Querying elements**: Since the array (or linked list) is unordered, it requires traversing all the elements, using $O(n)$ time.
+- **Searching for elements**: As the array (or linked list) is unsorted, searching for an element requires traversing through all of the elements. The time complexity of this operation is $O(n)$.
-- **Deleting elements**: First, locate the element, then delete it from the array (or linked list), using $O(n)$ time.
+- **Deleting elements**: To remove an element, we first need to locate it. Then, we delete it from the array (or linked list). The time complexity of this operation is $O(n)$.
 
-<p align="center"> Table <id> &nbsp; Comparison of element query efficiency </p>
+<p align="center"> Table <id> &nbsp; Comparison of time efficiency for common operations </p>
 
 |                | Array  | Linked List | Hash Table |
 | -------------- | ------ | ----------- | ---------- |
-| Find Element   | $O(n)$ | $O(n)$      | $O(1)$     |
+| Search Elements   | $O(n)$ | $O(n)$      | $O(1)$     |
-| Add Element    | $O(1)$ | $O(1)$      | $O(1)$     |
+| Insert Elements    | $O(1)$ | $O(1)$      | $O(1)$     |
-| Delete Element | $O(n)$ | $O(n)$      | $O(1)$     |
+| Delete Elements | $O(n)$ | $O(n)$      | $O(1)$     |
 
-Observations reveal that **the time complexity for adding, deleting, and querying in a hash table is $O(1)$**, which is highly efficient.
+It can be seen that **the time complexity for operations (insertion, deletion, searching, and modification) in a hash table is $O(1)$**, which is highly efficient.
 
 ## Common operations of hash table
 
-Common operations of a hash table include initialization, querying, adding key-value pairs, and deleting key-value pairs, etc. Example code is as follows:
+Common operations of a hash table include: initialization, querying, adding key-value pairs, and deleting key-value pairs. Here is an example code:
 
 === "Python"
 
@@ -283,7 +283,7 @@ Common operations of a hash table include initialization, querying, adding key-v
 
     https://pythontutor.com/render.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B7%BB%E5%8A%A0%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E6%B7%BB%E5%8A%A0%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%9F%A5%E8%AF%A2%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%90%91%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E8%BE%93%E5%85%A5%E9%94%AE%20key%20%EF%BC%8C%E5%BE%97%E5%88%B0%E5%80%BC%20value%0A%20%20%20%20name%20%3D%20hmap%5B15937%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E5%88%A0%E9%99%A4%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap.pop%2810583%29&cumulative=false&curInstr=2&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false
 
-There are three common ways to traverse a hash table: traversing key-value pairs, keys, and values. Example code is as follows:
+There are three common ways to traverse a hash table: traversing key-value pairs, traversing keys, and traversing values. Here is an example code:
 
 === "Python"
 
@@ -484,24 +484,24 @@ There are three common ways to traverse a hash table: traversing key-value pairs
 
     https://pythontutor.com/render.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B7%BB%E5%8A%A0%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E6%B7%BB%E5%8A%A0%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E9%94%AE%E5%80%BC%E5%AF%B9%20key-%3Evalue%0A%20%20%20%20for%20key,%20value%20in%20hmap.items%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key,%20%22-%3E%22,%20value%29%0A%20%20%20%20%23%20%E5%8D%95%E7%8B%AC%E9%81%8D%E5%8E%86%E9%94%AE%20key%0A%20%20%20%20for%20key%20in%20hmap.keys%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key%29%0A%20%20%20%20%23%20%E5%8D%95%E7%8B%AC%E9%81%8D%E5%8E%86%E5%80%BC%20value%0A%20%20%20%20for%20value%20in%20hmap.values%28%29%3A%0A%20%20%20%20%20%20%20%20print%28value%29&cumulative=false&curInstr=8&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false
 
-## Simple implementation of hash table
+## Simple implementation of a hash table
 
-First, let's consider the simplest case: **implementing a hash table using just an array**. In the hash table, each empty slot in the array is called a <u>bucket</u>, and each bucket can store one key-value pair. Therefore, the query operation involves finding the bucket corresponding to the `key` and retrieving the `value` from it.
+First, let's consider the simplest case: **implementing a hash table using only one array**. In the hash table, each empty slot in the array is called a <u>bucket</u>, and each bucket can store a key-value pair. Therefore, the query operation involves finding the bucket corresponding to the `key` and retrieving the `value` from it.
 
-So, how do we locate the appropriate bucket based on the `key`? This is achieved through a <u>hash function</u>. The role of the hash function is to map a larger input space to a smaller output space. In a hash table, the input space is all possible keys, and the output space is all buckets (array indices). In other words, input a `key`, **and we can use the hash function to determine the storage location of the corresponding key-value pair in the array**.
+So, how do we locate the corresponding bucket based on the `key`? This is achieved through a <u>hash function</u>. The role of the hash function is to map a larger input space to a smaller output space. In a hash table, the input space consists of all the keys, and the output space consists of all the buckets (array indices). In other words, given a `key`, **we can use the hash function to determine the storage location of the corresponding key-value pair in the array**.
 
-The calculation process of the hash function for a given `key` is divided into the following two steps:
+When given a `key`, the calculation process of the hash function consists of the following two steps:
 
-1. Calculate the hash value using a certain hash algorithm `hash()`.
+1. Calculate the hash value by using a certain hash algorithm `hash()`.
-2. Take the modulus of the hash value with the number of buckets (array length) `capacity` to obtain the array index `index`.
+2. Take the modulus of the hash value with the bucket count (array length) `capacity` to obtain the array `index` corresponding to that key.
 
 ```shell
 index = hash(key) % capacity
 ```
 
-Afterward, we can use `index` to access the corresponding bucket in the hash table and thereby retrieve the `value`.
+Afterward, we can use the `index` to access the corresponding bucket in the hash table and thereby retrieve the `value`.
 
-Assuming array length `capacity = 100` and hash algorithm `hash(key) = key`, the hash function is `key % 100`. The figure below uses `key` as the student number and `value` as the name to demonstrate the working principle of the hash function.
+Let's assume that the array length is `capacity = 100`, and the hash algorithm is defined as `hash(key) = key`. Therefore, the hash function can be expressed as `key % 100`. The following figure illustrates the working principle of the hash function using `key` as student ID and `value` as name.
 
 ![Working principle of hash function](hash_map.assets/hash_function.png)
 
@@ -513,25 +513,25 @@ The following code implements a simple hash table. Here, we encapsulate `key` an
 
 ## Hash collision and resizing
 
-Fundamentally, the role of the hash function is to map the entire input space of all keys to the output space of all array indices. However, the input space is often much larger than the output space. Therefore, **theoretically, there must be situations where "multiple inputs correspond to the same output"**.
+Essentially, the role of the hash function is to map the entire input space of all keys to the output space of all array indices. However, the input space is often much larger than the output space. Therefore, **theoretically, there will always be cases where "multiple inputs correspond to the same output"**.
 
-For the hash function in the above example, if the last two digits of the input `key` are the same, the output of the hash function will also be the same. For example, when querying for students with student numbers 12836 and 20336, we find:
+In the example above, with the given hash function, when the last two digits of the input `key` are the same, the hash function produces the same output. For instance, when querying two students with student IDs 12836 and 20336, we find:
 
 ```shell
 12836 % 100 = 36
 20336 % 100 = 36
 ```
 
-As shown in the figure below, both student numbers point to the same name, which is obviously incorrect. This situation where multiple inputs correspond to the same output is known as <u>hash collision</u>.
+As shown in the figure below, both student IDs point to the same name, which is obviously incorrect. This situation where multiple inputs correspond to the same output is called <u>hash collision</u>.
 
 ![Example of hash collision](hash_map.assets/hash_collision.png)
 
-It is easy to understand that the larger the capacity $n$ of the hash table, the lower the probability of multiple keys being allocated to the same bucket, and the fewer the collisions. Therefore, **expanding the capacity of the hash table can reduce hash collisions**.
+It is easy to understand that as the capacity $n$ of the hash table increases, the probability of multiple keys being assigned to the same bucket decreases, resulting in fewer collisions. Therefore, **we can reduce hash collisions by resizing the hash table**.
 
-As shown in the figure below, before expansion, key-value pairs `(136, A)` and `(236, D)` collided; after expansion, the collision is resolved.
+As shown in the figure below, before resizing, the key-value pairs `(136, A)` and `(236, D)` collide. However, after resizing, the collision is resolved.
 
-![Hash table expansion](hash_map.assets/hash_table_reshash.png)
+![Hash table resizing](hash_map.assets/hash_table_reshash.png)
 
-Similar to array expansion, resizing a hash table requires migrating all key-value pairs from the original hash table to the new one, which is time-consuming. Furthermore, since the capacity `capacity` of the hash table changes, we need to recalculate the storage positions of all key-value pairs using the hash function, which adds to the computational overhead of the resizing process. Therefore, programming languages often reserve a sufficiently large capacity for the hash table to prevent frequent resizing.
+Similar to array expansion, resizing a hash table requires migrating all key-value pairs from the original hash table to the new one, which is time-consuming. Furthermore, since the `capacity` of the hash table changes, we need to recalculate the storage positions of all key-value pairs using the hash function, further increasing the computational overhead of the resizing process. Therefore, programming languages often allocate a sufficiently large capacity for the hash table to prevent frequent resizing.
 
-The <u>load factor</u> is an important concept for hash tables. It is defined as the ratio of the number of elements in the hash table to the number of buckets. It is used to measure the severity of hash collisions and **is often used as a trigger for resizing the hash table**. For example, in Java, when the load factor exceeds $0.75$, the system will resize the hash table to twice its original size.
+The <u>load factor</u> is an important concept in hash tables. It is defined as the ratio of the number of elements in the hash table to the number of buckets. It is used to measure the severity of hash collisions and **often serves as a trigger for hash table resizing**. For example, in Java, when the load factor exceeds $0.75$, the system will resize the hash table to twice its original size.

en/docs/chapter_heap/summary.md

@@ -2,16 +2,16 @@
 
 ### Key review
 
-- A heap is a complete binary tree, which can be divided into a max heap and a min heap based on its property. The top element of a max (min) heap is the largest (smallest).
+- A heap is a complete binary tree that can be categorized as either a max heap or a min heap based on its building property, where the top element of a max heap is the largest and the top element of a min heap is the smallest.
 - A priority queue is defined as a queue with dequeue priority, usually implemented using a heap.
 - Common operations of a heap and their corresponding time complexities include: element insertion into the heap $O(\log n)$, removing the top element from the heap $O(\log n)$, and accessing the top element of the heap $O(1)$.
 - A complete binary tree is well-suited to be represented by an array, thus heaps are commonly stored using arrays.
 - Heapify operations are used to maintain the properties of the heap and are used in both heap insertion and removal operations.
-- The time complexity of inserting $n$ elements into a heap and building the heap can be optimized to $O(n)$, which is highly efficient.
+- The time complexity of building a heap given an input of $n$ elements can be optimized to $O(n)$, which is highly efficient.
 - Top-k is a classic algorithm problem that can be efficiently solved using the heap data structure, with a time complexity of $O(n \log k)$.
 
 ### Q & A
 
 **Q**: Is the "heap" in data structures the same concept as the "heap" in memory management?
 
-The two are not the same concept, even though they are both referred to as "heap". The heap in computer system memory is part of dynamic memory allocation, where the program can use it to store data during execution. The program can request a certain amount of heap memory to store complex structures like objects and arrays. When these data are no longer needed, the program needs to release this memory to prevent memory leaks. Compared to stack memory, the management and usage of heap memory need to be more cautious, as improper use may lead to memory leaks and dangling pointers.
+The two are not the same concept, even though they are both referred to as "heap". The heap in computer system memory is part of dynamic memory allocation, where the program can use it to store data during execution. The program can request a certain amount of heap memory to store complex structures like objects and arrays. When the allocated data is no longer needed, the program needs to release this memory to prevent memory leaks. Compared to stack memory, the management and usage of heap memory demands more caution, as improper use may lead to memory leaks and dangling pointers.