build

en/docs/chapter_graph/graph.md

@@ -14,7 +14,7 @@ G & = \{ V, E \} \newline
 \end{aligned}
 $$
 
-If we view vertices as nodes and edges as references (pointers) connecting the nodes, we can see graphs as a data structure extended from linked lists. As shown in Figure 9-1, **compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex**.
+If we view vertices as nodes and edges as references (pointers) connecting them, we can regard a graph as an extension of the linked list data structure. As shown in Figure 9-1, **compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex**.
 
 ![Relationships among linked lists, trees, and graphs](graph.assets/linkedlist_tree_graph.png){ class="animation-figure" }
 
@@ -24,8 +24,8 @@ If we view vertices as nodes and edges as references (pointers) connecting the n
 
 Graphs can be divided into <u>undirected graphs</u> and <u>directed graphs</u> based on whether edges have direction, as shown in Figure 9-2.
 
-- In undirected graphs, edges represent a "bidirectional" connection between two vertices, such as the "friend relationship" on WeChat or QQ.
-- In directed graphs, edges have directionality, meaning edges $A \rightarrow B$ and $A \leftarrow B$ are independent of each other, such as the "follow" and "be followed" relationships on Weibo or TikTok.
+- In undirected graphs, edges represent a "bidirectional" connection between two vertices, such as friendships on WeChat or QQ.
+- In directed graphs, edges have directionality, meaning edges $A \rightarrow B$ and $A \leftarrow B$ are independent of each other, such as following and follower relationships on Weibo or TikTok.
 
 ![Directed and undirected graphs](graph.assets/directed_graph.png){ class="animation-figure" }
 
@@ -40,7 +40,7 @@ Graphs can be divided into <u>connected graphs</u> and <u>disconnected graphs</u
 
 <p align="center"> Figure 9-3 &nbsp; Connected and disconnected graphs </p>
 
-We can also add a "weight" variable to edges, resulting in <u>weighted graphs</u> as shown in Figure 9-4. For example, in mobile games like "Honor of Kings", the system calculates the "intimacy" between players based on their shared game time, and such intimacy networks can be represented using weighted graphs.
+We can also add a "weight" variable to edges, resulting in <u>weighted graphs</u> as shown in Figure 9-4. For example, in mobile games like "Honor of Kings", the system calculates the "intimacy" between players based on how long they have played together, and such intimacy networks can be represented using weighted graphs.
 
 ![Weighted and unweighted graphs](graph.assets/weighted_graph.png){ class="animation-figure" }
 
@@ -50,7 +50,7 @@ Graph data structures include the following commonly used terms.
 
 - <u>Adjacency</u>: When two vertices are connected by an edge, these two vertices are said to be "adjacent". In Figure 9-4, the adjacent vertices of vertex 1 are vertices 2, 3, and 5.
 - <u>Path</u>: The sequence of edges from vertex A to vertex B is called a "path" from A to B. In Figure 9-4, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4.
-- <u>Degree</u>: The number of edges a vertex has. For directed graphs, <u>in-degree</u> indicates how many edges point to the vertex, and <u>out-degree</u> indicates how many edges point out from the vertex.
+- <u>Degree</u>: The number of edges a vertex has. For directed graphs, <u>in-degree</u> indicates how many edges point to the vertex, and <u>out-degree</u> indicates how many edges leave the vertex.
 
 ## 9.1.2 &nbsp; Representation of Graphs
 
@@ -70,7 +70,7 @@ Adjacency matrices have the following properties.
 
 - In simple graphs, vertices cannot connect to themselves, so the elements on the main diagonal of the adjacency matrix are meaningless.
 - For undirected graphs, edges in both directions are equivalent, so the adjacency matrix is symmetric about the main diagonal.
-- Replacing the elements of the adjacency matrix from $1$ and $0$ to weights allows representation of weighted graphs.
+- Replacing the $1$ and $0$ entries in the adjacency matrix with weights allows it to represent weighted graphs.
 
 When using adjacency matrices to represent graphs, we can directly access matrix elements to obtain edges, resulting in highly efficient addition, deletion, lookup, and modification operations, all with a time complexity of $O(1)$. However, the space complexity of the matrix is $O(n^2)$, which consumes significant memory.
 
@@ -82,9 +82,9 @@ An <u>adjacency list</u> uses $n$ linked lists to represent a graph, with linked
 
 <p align="center"> Figure 9-6 &nbsp; Adjacency list representation of a graph </p>
 
-Adjacency lists only store edges that actually exist, and the total number of edges is typically much less than $n^2$, making them more space-efficient. However, finding edges in an adjacency list requires traversing the linked list, so its time efficiency is inferior to that of adjacency matrices.
+Adjacency lists only store edges that actually exist, and the total number of edges is typically much less than $n^2$, making them more space-efficient. However, finding edges in an adjacency list requires traversing the linked list, so it is less time-efficient than an adjacency matrix.
 
-Observing Figure 9-6, **the structure of adjacency lists is very similar to "chaining" in hash tables, so we can adopt similar methods to optimize efficiency**. For example, when linked lists are long, they can be converted to AVL trees or red-black trees, thereby optimizing time efficiency from $O(n)$ to $O(\log n)$; linked lists can also be converted to hash tables, thereby reducing time complexity to $O(1)$.
+As shown in Figure 9-6, **the structure of adjacency lists is very similar to separate chaining in hash tables, so we can use similar methods to improve efficiency**. For example, when a linked list becomes long, it can be converted into an AVL tree or red-black tree, improving the time complexity from $O(n)$ to $O(\log n)$; it can also be converted into a hash table, reducing the time complexity to $O(1)$.
 
 ## 9.1.3 &nbsp; Common Applications of Graphs
 

en/docs/chapter_graph/graph_operations.md

@@ -4,7 +4,7 @@ comments: true
 
 # 9.2   Basic Operations on Graphs
 
-Basic operations on graphs can be divided into operations on "edges" and operations on "vertices". Under the two representation methods of "adjacency matrix" and "adjacency list", the implementation methods differ.
+Basic operations on graphs can be divided into operations on "edges" and operations on "vertices". Their implementations differ depending on whether the graph is represented as an "adjacency matrix" or an "adjacency list".
 
 ## 9.2.1   Implementation Based on Adjacency Matrix
 
@@ -13,21 +13,21 @@ Given an undirected graph with $n$ vertices, the various operations are implemen
 - **Adding or removing an edge**: Directly modify the specified edge in the adjacency matrix, using $O(1)$ time. Since it is an undirected graph, both directions of the edge need to be updated simultaneously.
 - **Adding a vertex**: Add a row and a column at the end of the adjacency matrix and fill them all with $0$s, using $O(n)$ time.
 - **Removing a vertex**: Delete a row and a column in the adjacency matrix. The worst case occurs when removing the first row and column, requiring $(n-1)^2$ elements to be "moved up and to the left", thus using $O(n^2)$ time.
-- **Initialization**: Pass in $n$ vertices, initialize a vertex list `vertices` of length $n$, using $O(n)$ time; initialize an adjacency matrix `adjMat` of size $n \times n$, using $O(n^2)$ time.
+- **Initialization**: Given $n$ vertices, initialize a vertex list `vertices` of length $n$, using $O(n)$ time; initialize an adjacency matrix `adjMat` of size $n \times n$, using $O(n^2)$ time.
 
-=== "Initialize adjacency matrix"
+=== "<1>"
     ![Initialization, adding and removing edges, adding and removing vertices in adjacency matrix](graph_operations.assets/adjacency_matrix_step1_initialization.png){ class="animation-figure" }
 
-=== "Add an edge"
+=== "<2>"
     ![adjacency_matrix_add_edge](graph_operations.assets/adjacency_matrix_step2_add_edge.png){ class="animation-figure" }
 
-=== "Remove an edge"
+=== "<3>"
     ![adjacency_matrix_remove_edge](graph_operations.assets/adjacency_matrix_step3_remove_edge.png){ class="animation-figure" }
 
-=== "Add a vertex"
+=== "<4>"
     ![adjacency_matrix_add_vertex](graph_operations.assets/adjacency_matrix_step4_add_vertex.png){ class="animation-figure" }
 
-=== "Remove a vertex"
+=== "<5>"
     ![adjacency_matrix_remove_vertex](graph_operations.assets/adjacency_matrix_step5_remove_vertex.png){ class="animation-figure" }
 
 <p align="center"> Figure 9-7 &nbsp; Initialization, adding and removing edges, adding and removing vertices in adjacency matrix </p>
@@ -1212,33 +1212,33 @@ Given an undirected graph with a total of $n$ vertices and $m$ edges, the variou
 
 - **Adding an edge**: Add the edge at the end of the corresponding vertex's linked list, using $O(1)$ time. Since it is an undirected graph, edges in both directions need to be added simultaneously.
 - **Removing an edge**: Find and remove the specified edge in the corresponding vertex's linked list, using $O(m)$ time. In an undirected graph, edges in both directions need to be removed simultaneously.
-- **Adding a vertex**: Add a linked list in the adjacency list and set the new vertex as the head node of the list, using $O(1)$ time.
+- **Adding a vertex**: Add a linked list to the adjacency list, with the new vertex as the head node, using $O(1)$ time.
 - **Removing a vertex**: Traverse the entire adjacency list and remove all edges containing the specified vertex, using $O(n + m)$ time.
 - **Initialization**: Create $n$ vertices and $2m$ edges in the adjacency list, using $O(n + m)$ time.
 
-=== "Initialize adjacency list"
+=== "<1>"
     ![Initialization, adding and removing edges, adding and removing vertices in adjacency list](graph_operations.assets/adjacency_list_step1_initialization.png){ class="animation-figure" }
 
-=== "Add an edge"
+=== "<2>"
     ![adjacency_list_add_edge](graph_operations.assets/adjacency_list_step2_add_edge.png){ class="animation-figure" }
 
-=== "Remove an edge"
+=== "<3>"
     ![adjacency_list_remove_edge](graph_operations.assets/adjacency_list_step3_remove_edge.png){ class="animation-figure" }
 
-=== "Add a vertex"
+=== "<4>"
     ![adjacency_list_add_vertex](graph_operations.assets/adjacency_list_step4_add_vertex.png){ class="animation-figure" }
 
-=== "Remove a vertex"
+=== "<5>"
     ![adjacency_list_remove_vertex](graph_operations.assets/adjacency_list_step5_remove_vertex.png){ class="animation-figure" }
 
 <p align="center"> Figure 9-8 &nbsp; Initialization, adding and removing edges, adding and removing vertices in adjacency list </p>
 
-The following is the adjacency list code implementation. Compared to Figure 9-8, the actual code has the following differences.
+The following code shows the adjacency list implementation. Compared with Figure 9-8, the actual code differs in the following ways.
 
 - For convenience in adding and removing vertices, and to simplify the code, we use lists (dynamic arrays) instead of linked lists.
 - A hash table is used to store the adjacency list, where `key` is the vertex instance and `value` is the list (linked list) of adjacent vertices for that vertex.
 
-Additionally, we use the `Vertex` class to represent vertices in the adjacency list. The reason for this is: if we used list indices to distinguish different vertices as with adjacency matrices, then to delete the vertex at index $i$, we would need to traverse the entire adjacency list and decrement all indices greater than $i$ by $1$, which is very inefficient. However, if each vertex is a unique `Vertex` instance, deleting a vertex does not require modifying other vertices.
+Additionally, we use the `Vertex` class to represent vertices in the adjacency list for the following reason: if we used list indices to distinguish different vertices, as with adjacency matrices, then to delete the vertex at index $i$, we would need to traverse the entire adjacency list and decrement all indices greater than $i$ by $1$, which is very inefficient. However, if each vertex is a unique `Vertex` instance, deleting one vertex does not require modifying the others.
 
 === "Python"
 
@@ -2361,7 +2361,7 @@ Additionally, we use the `Vertex` class to represent vertices in the adjacency l
 
 ## 9.2.3 &nbsp; Efficiency Comparison
 
-Assuming the graph has $n$ vertices and $m$ edges, Table 9-2 compares the time efficiency and space efficiency of adjacency matrices and adjacency lists. Note that the adjacency list (linked list) corresponds to the implementation in this text, while the adjacency list (hash table) refers specifically to the implementation where all linked lists are replaced with hash tables.
+Assuming the graph has $n$ vertices and $m$ edges, Table 9-2 compares the time efficiency and space efficiency of adjacency matrices and adjacency lists. Note that the adjacency list (linked list) corresponds to the implementation used in this section, while the adjacency list (hash table) refers specifically to the implementation where all linked lists are replaced with hash tables.
 
 <p align="center"> Table 9-2 &nbsp; Comparison of adjacency matrix and adjacency list </p>
 

en/docs/chapter_graph/graph_traversal.md

@@ -10,7 +10,7 @@ Both graphs and trees require the application of search algorithms to implement
 
 ## 9.3.1   Breadth-First Search
 
-**Breadth-first search is a near-to-far traversal method that, starting from a certain node, always prioritizes visiting the nearest vertices and expands outward layer by layer**. As shown in Figure 9-9, starting from the top-left vertex, first traverse all adjacent vertices of that vertex, then traverse all adjacent vertices of the next vertex, and so on, until all vertices have been visited.
+**Breadth-first search proceeds from near to far: starting from a given node, it always visits the nearest vertices first and expands outward layer by layer**. As shown in Figure 9-9, starting from the top-left vertex, first traverse all adjacent vertices of that vertex, then traverse all adjacent vertices of the next vertex, and so on, until all vertices have been visited.
 
 ![Breadth-first search of a graph](graph_traversal.assets/graph_bfs.png){ class="animation-figure" }
 
@@ -28,7 +28,7 @@ To prevent revisiting vertices, we use a hash set `visited` to record which node
 
 !!! tip
 
-    A hash set can be viewed as a hash table that stores only `key` without storing `value`. It can perform addition, deletion, lookup, and modification operations on `key` in $O(1)$ time complexity. Based on the uniqueness of `key`, hash sets are typically used for data deduplication and similar scenarios.
+    A hash set can be viewed as a hash table that stores only `key` without storing `value`. It supports insertion, deletion, lookup, and update operations on `key` in $O(1)$ time. Based on the uniqueness of `key`, hash sets are typically used for data deduplication and similar scenarios.
 
 === "Python"
 
@@ -933,9 +933,9 @@ This "go as far as possible then return" algorithm paradigm is typically impleme
 The algorithm flow of depth-first search is shown in Figure 9-12.
 
 - **Straight dashed lines represent downward recursion**, indicating that a new recursive method has been initiated to visit a new vertex.
-- **Curved dashed lines represent upward backtracking**, indicating that this recursive method has returned to the position where it was initiated.
+- **Curved dashed lines represent upward backtracking**, indicating that this recursive call has returned to the point where it was made.
 
-To deepen understanding, it is recommended to combine Figure 9-12 with the code to mentally simulate (or draw out) the entire DFS process, including when each recursive method is initiated and when it returns.
+To deepen understanding, it is recommended to combine Figure 9-12 with the code to mentally simulate (or draw out) the entire DFS process, including when each recursive call begins and when it returns.
 
 === "<1>"
     ![Steps of depth-first search of a graph](graph_traversal.assets/graph_dfs_step1.png){ class="animation-figure" }
@@ -974,7 +974,7 @@ To deepen understanding, it is recommended to combine Figure 9-12 with the code
 
 !!! question "Is the depth-first traversal sequence unique?"
 
-    Similar to breadth-first search, the order of depth-first traversal sequences is also not unique. Given a certain vertex, exploring in any direction first is valid, meaning the order of adjacent vertices can be arbitrarily shuffled, all being depth-first search.
+    Similar to breadth-first search, depth-first traversal sequences are also not unique. Given a vertex, any exploration direction may be chosen first; that is, the order of adjacent vertices can be arbitrarily rearranged and still constitute depth-first search.
 
     Taking tree traversal as an example, "root $\rightarrow$ left $\rightarrow$ right", "left $\rightarrow$ root $\rightarrow$ right", and "left $\rightarrow$ right $\rightarrow$ root" correspond to pre-order, in-order, and post-order traversals, respectively. They represent three different traversal priorities, yet all three belong to depth-first search.
 

en/docs/chapter_graph/summary.md

@@ -7,16 +7,16 @@ comments: true
 ### 1.   Key Review
 
 - Graphs consist of vertices and edges and can be represented as a set of vertices and a set of edges.
-- Compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex.
-- Directed graphs have edges with directionality, connected graphs have all vertices reachable from any vertex, and weighted graphs have edges that each contain a weight variable.
+- Compared with the linear relationships modeled by linked lists and the divide-and-conquer relationships modeled by trees, the network relationships modeled by graphs offer much greater flexibility and are therefore more complex.
+- In directed graphs, edges have direction; in connected graphs, every vertex is reachable from any other vertex; and in weighted graphs, each edge carries a weight.
 - Adjacency matrices use matrices to represent graphs, where each row (column) represents a vertex, and matrix elements represent edges, using $1$ or $0$ to indicate whether two vertices have an edge or not. Adjacency matrices are highly efficient for addition, deletion, lookup, and modification operations, but consume significant space.
-- Adjacency lists use multiple linked lists to represent graphs, where the $i$-th linked list corresponds to vertex $i$ and stores all adjacent vertices of that vertex. Adjacency lists are more space-efficient than adjacency matrices, but have lower time efficiency because they require traversing linked lists to find edges.
+- Adjacency lists use multiple linked lists to represent a graph: the $i$-th linked list corresponds to vertex $i$ and stores all vertices adjacent to it. Compared with adjacency matrices, adjacency lists use less space, but edge lookups are less efficient because the linked list must be traversed.
 - When linked lists in adjacency lists become too long, they can be converted to red-black trees or hash tables, thereby improving lookup efficiency.
 - From an algorithmic perspective, adjacency matrices embody "trading space for time", while adjacency lists embody "trading time for space".
 - Graphs can be used to model various real-world systems, such as social networks and subway lines.
 - Trees are a special case of graphs, and tree traversal is a special case of graph traversal.
-- Breadth-first search of graphs is a near-to-far, layer-by-layer expansion search method, typically implemented using a queue.
-- Depth-first search of graphs is a search method that prioritizes going as far as possible and backtracks when no path remains, commonly implemented using recursion.
+- Breadth-first search in graphs explores from near to far, expanding layer by layer, and is typically implemented with a queue.
+- Depth-first search in graphs follows a path as deep as possible and backtracks when it can go no farther, and is commonly implemented with recursion.
 
 ### 2.   Q & A
 
@@ -28,8 +28,8 @@ In this text, a path is viewed as a sequence of edges, not a sequence of vertice
 
 **Q**: In a disconnected graph, will there be unreachable vertices?
 
-In a disconnected graph, starting from a certain vertex, at least one vertex cannot be reached. Traversing a disconnected graph requires setting multiple starting points to traverse all connected components of the graph.
+In a disconnected graph, if you start from one vertex, at least one other vertex will be unreachable. To traverse a disconnected graph, you need multiple starting points so that all connected components are covered.
 
-**Q**: In an adjacency list, is there a requirement for the order of "all vertices connected to that vertex"?
+**Q**: In an adjacency list, is there any required ordering for the vertices adjacent to a given vertex?
 
-It can be in any order. However, in practical applications, it may be necessary to sort according to specified rules, such as the order in which vertices were added, or the order of vertex values, which helps quickly find vertices "with certain extreme values".
+They can appear in any order. In practice, however, they may need to be sorted according to specific rules, such as the order in which vertices were added or the order of vertex values, which helps when quickly finding a vertex with some extreme value.