Merge a158c8bbf2 into 4db5c19011

* translation: graph summary updates

* resolve grammar issues

en/docs/chapter_computational_complexity/performance_evaluation.md

@@ -2,10 +2,10 @@
 
 In algorithm design, we pursue the following two objectives in sequence.
 
-1. **Finding a Solution to the Problem**: The algorithm should reliably find the correct solution within the stipulated range of inputs.
+1. **Finding a Solution to the Problem**: The algorithm should reliably find the correct solution within the specified range of inputs.
 2. **Seeking the Optimal Solution**: For the same problem, multiple solutions might exist, and we aim to find the most efficient algorithm possible.
 
-In other words, under the premise of being able to solve the problem, algorithm efficiency has become the main criterion for evaluating the merits of an algorithm, which includes the following two dimensions.
+In other words, under the premise of being able to solve the problem, algorithm efficiency has become the main criterion for evaluating the quality of an algorithm, which includes the following two dimensions.
 
 - **Time efficiency**: The speed at which an algorithm runs.
 - **Space efficiency**: The size of the memory space occupied by an algorithm.
@@ -16,11 +16,11 @@ There are mainly two methods of efficiency assessment: actual testing and theore
 
 ## Actual testing
 
-Suppose we have algorithms `A` and `B`, both capable of solving the same problem, and we need to compare their efficiencies. The most direct method is to use a computer to run these two algorithms and monitor and record their runtime and memory usage. This assessment method reflects the actual situation but has significant limitations.
+Suppose we have algorithms `A` and `B`, both capable of solving the same problem, and we need to compare their efficiencies. The most direct method is to use a computer to run these two algorithms, monitor and record their runtime and memory usage. This assessment method reflects the actual situation, but it has significant limitations.
 
-On one hand, **it's difficult to eliminate interference from the testing environment**. Hardware configurations can affect algorithm performance. For example, algorithm `A` might run faster than `B` on one computer, but the opposite result may occur on another computer with different configurations. This means we would need to test on a variety of machines to calculate average efficiency, which is impractical.
+On one hand, **it's difficult to eliminate interference from the testing environment**. Hardware configurations can affect algorithm performance. For example, an algorithm with a high degree of parallelism is better suited for running on multi-core CPUs, while an algorithm that involves intensive memory operations performs better with high-performance memory. The test results of an algorithm may vary across different machines. This means we would need to test the algorithm on various machines to calculate average efficiency, which is impractical.
 
-On the other hand, **conducting a full test is very resource-intensive**. As the volume of input data changes, the efficiency of the algorithms may vary. For example, with smaller data volumes, algorithm `A` might run faster than `B`, but the opposite might be true with larger data volumes. Therefore, to draw convincing conclusions, we need to test a wide range of input data sizes, which requires significant computational resources.
+On the other hand, **conducting a full test is very resource-intensive**. As the volume of input data changes, the efficiency of the algorithms also changes. For example, with smaller data volumes, algorithm `A` might run faster than `B`, but with larger data volumes, the test results may be the opposite. Therefore, to draw convincing conclusions, we need to test a wide range of input data sizes, which requires excessive computational resources.
 
 ## Theoretical estimation
 
@@ -30,19 +30,20 @@ Complexity analysis reflects the relationship between the time and space resourc
 
 - "Time and space resources" correspond to <u>time complexity</u> and <u>space complexity</u>, respectively.
 - "As the size of input data increases" means that complexity reflects the relationship between algorithm efficiency and the volume of input data.
-- "The trend of growth in time and space" indicates that complexity analysis focuses not on the specific values of runtime or space occupied but on the "rate" at which time or space grows.
+- "The trend of growth in time and space" indicates that complexity analysis focuses not on the specific values of runtime or space occupied, but on the "rate" at which time or space increases.
 
 **Complexity analysis overcomes the disadvantages of actual testing methods**, reflected in the following aspects:
 
+- It does not require actually running the code, making it more environmentally friendly and energy efficient.
 - It is independent of the testing environment and applicable to all operating platforms.
 - It can reflect algorithm efficiency under different data volumes, especially in the performance of algorithms with large data volumes.
 
 !!! tip
 
-    If you're still confused about the concept of complexity, don't worry. We will introduce it in detail in subsequent chapters.
+    If you're still confused about the concept of complexity, don't worry. We will cover it in detail in subsequent chapters.
 
-Complexity analysis provides us with a "ruler" to measure the time and space resources needed to execute an algorithm and compare the efficiency between different algorithms.
+Complexity analysis provides us with a "ruler" to evaluate the efficiency of an algorithm, enabling us to measure the time and space resources required to execute it and compare the efficiency of different algorithms.
 
-Complexity is a mathematical concept and may be abstract and challenging for beginners. From this perspective, complexity analysis might not be the best content to introduce first. However, when discussing the characteristics of a particular data structure or algorithm, it's hard to avoid analyzing its speed and space usage.
+Complexity is a mathematical concept that might be abstract and challenging for beginners. From this perspective, complexity analysis might not be the most suitable topic to introduce first. However, when discussing the characteristics of a particular data structure or algorithm, it's hard to avoid analyzing its speed and space usage.
 
-In summary, it's recommended that you establish a preliminary understanding of complexity analysis before diving deep into data structures and algorithms, **so that you can carry out simple complexity analyses of algorithms**.
+In summary, it is recommended to develop a basic understanding of complexity analysis before diving deep into data structures and algorithms, **so that you can perform complexity analysis on simple algorithms**.

en/docs/chapter_graph/summary.md

@@ -2,30 +2,30 @@
 
 ### Key review
 
-- A graph consists of vertices and edges and can be represented as a set comprising a group of vertices and a group 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.
-- The edges of a directed graph have directionality, any vertex in a connected graph is reachable, and each edge in a weighted graph contains a weight variable.
-- Adjacency matrices use matrices to represent graphs, with each row (column) representing a vertex and matrix elements representing edges, using $1$ or $0$ to indicate the presence or absence of an edge between two vertices. Adjacency matrices are highly efficient for add, delete, find, and modify operations, but they consume more space.
-- Adjacency lists use multiple linked lists to represent graphs, with the $i^{th}$ list corresponding to vertex $i$, containing all its adjacent vertices. Adjacency lists save more space compared to adjacency matrices, but since it is necessary to traverse the list to find edges, their time efficiency is lower.
-- When the linked lists in the adjacency list are too long, they can be converted into red-black trees or hash tables to improve query efficiency.
-- From the perspective of algorithmic thinking, adjacency matrices embody the principle of "space for time," while adjacency lists embody "time for space."
-- Graphs can be used to model various real systems, such as social networks, subway routes, etc.
+- A graph is made up of vertices and edges. It can be described as a set of vertices and a set of edges.
+- Compared to linear relationships (like linked lists) and hierarchical relationships (like trees), network relationships (graphs) offer greater flexibility, making them more complex.
+- In a directed graph, edges have directions. In a connected graph, any vertex can be reached from any other vertex. In a weighted graph, each edge has an associated weight variable.
+- An adjacency matrix is a way to represent a graph using matrix (2D array). The rows and columns represent the vertices. The matrix element value indicates whether there is an edge between two vertices, using $1$ for an edge or $0$ for no edge. Adjacency matrices are highly efficient for operations like adding, deleting, or checking edges, but they require more space.
+- An adjacency list is another common way to represent a graph using a collection of linked lists. Each vertex in the graph has a list that contains all its adjacent vertices. The $i^{th}$ list represents vertex $i$. Adjacency lists use less space compared to adjacency matrices. However, since it requires traversing the list to find edges, the time efficiency is lower.
+- When the linked lists in an adjacency list are long enough, they can be converted into red-black trees or hash tables to improve lookup efficiency.
+- From the perspective of algorithmic design, an adjacency matrix reflects the concept of "trading space for time", whereas an adjacency list reflects "trading time for space".
+- Graphs can be used to model various real-world systems, such as social networks, subway routes.
 - A tree is a special case of a graph, and tree traversal is also a special case of graph traversal.
-- Breadth-first traversal of a graph is a search method that expands layer by layer from near to far, usually implemented with a queue.
-- Depth-first traversal of a graph is a search method that prefers to go as deep as possible and backtracks when no further paths are available, often based on recursion.
+- Breadth-first traversal of a graph is a search method that expands layer by layer from near to far, typically using a queue.
+- Depth-first traversal of a graph is a search method that prioritizes reaching the end before backtracking when no further path is available. It is often implemented using recursion.
 
 ### Q & A
 
 **Q**: Is a path defined as a sequence of vertices or a sequence of edges?
 
-Definitions vary between different language versions on Wikipedia: the English version defines a path as "a sequence of edges," while the Chinese version defines it as "a sequence of vertices." Here is the original text from the English version: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.
+In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.
 
 In this document, a path is considered a sequence of edges, rather than a sequence of vertices. This is because there might be multiple edges connecting two vertices, in which case each edge corresponds to a path.
 
-**Q**: In a disconnected graph, are there points that cannot be traversed to?
+**Q**: In a disconnected graph, are there points that cannot be traversed?
 
-In a disconnected graph, starting from a certain vertex, there is at least one vertex that cannot be reached. Traversing a disconnected graph requires setting multiple starting points to traverse all connected components of the graph.
+In a disconnected graph, there is at least one vertex that cannot be reached from a specific point. To traverse a disconnected graph, you need to set multiple starting points to traverse all the connected components of the graph.
 
 **Q**: In an adjacency list, does the order of "all vertices connected to that vertex" matter?
 
-It can be in any order. However, in practical applications, it might be necessary to sort according to certain rules, such as the order in which vertices are added, or the order of vertex values, etc., to facilitate the quick search for vertices with certain extremal values.
+It can be in any order. However, in real-world applications, it might be necessary to sort them according to certain rules, such as the order in which vertices are added, or the order of vertex values. This can help find vertices quickly with certain extreme values.