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< span > Welcome to contribute to Chinese-to-English translation! Please visit < a href = "https://github.com/krahets/hello-algo/issues/914" > #914< / a > for more details.< / span >
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0.1 About this book
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0.2 How to read
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Chapter 1. Encounter with algorithms
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1.1 Algorithms are everywhere
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1.2 What is an algorithm
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Chapter 2. Complexity analysis
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2.1 Algorithm efficiency assessment
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2.2 Iteration and recursion
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2.3 Time complexity
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2.4 Space complexity
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Chapter 3. Data structures
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Chapter 3. Data structures
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3.1 Classification of data structures
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3.2 Basic data types
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3.3 Number encoding *
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3.4 Character encoding *
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Chapter 4. Array and linked list
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Chapter 4. Array and linked list
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4.1 Array
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4.2 Linked list
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4.4 Memory and cache *
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Chapter 5. Stack and queue
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Chapter 5. Stack and queue
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5.3 Double-ended queue
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Chapter 6. Hash table
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Chapter 6. Hash table
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6.1 Hash table
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6.2 Hash collision
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6.3 Hash algorithm
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Chapter 7. Tree
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Chapter 7. Tree
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7.1 Binary tree
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7.2 Binary tree traversal
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7.3 Array Representation of tree
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7.4 Binary Search tree
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7.5 AVL tree *
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7.6 Summary
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Chapter 8. Heap
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Chapter 8. Heap
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8.1 Heap
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8.2 Building a heap
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8.3 Top-k problem
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8.4 Summary
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Chapter 9. Graph
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9.1 Graph
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9.1 Graph
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Table of contents
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< a href = "#911-common-types-and-terminologies-of-graphs" class = "md-nav__link" >
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9.1.1 Common types and terminologies of graphs
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9.1.2 Representation of graphs
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1. Adjacency matrix
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9.1.3 Common applications of graphs
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9.2 Basic graph operations
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9.3 Graph traversal
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Chapter 10. Searching
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10.1 Binary search
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10.2 Binary search insertion
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10.3 Binary search boundaries
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10.4 Hashing optimization strategies
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10.5 Search algorithms revisited
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10.6 Summary
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Chapter 11. Sorting
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Chapter 11. Sorting
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11.1 Sorting algorithms
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11.2 Selection sort
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11.3 Bubble sort
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11.4 Insertion sort
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11.5 Quick sort
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11.6 Merge sort
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11.7 Heap sort
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11.8 Bucket sort
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11.9 Counting sort
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11.10 Radix sort
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11.11 Summary
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Chapter 12. Divide and conquer
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Chapter 12. Divide and conquer
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12.1 Divide and conquer algorithms
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12.2 Divide and conquer search strategy
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12.3 Building binary tree problem
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12.4 Tower of Hanoi Problem
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12.5 Summary
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Chapter 13. Backtracking
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Chapter 13. Backtracking
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13.1 Backtracking algorithms
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13.2 Permutation problem
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13.3 Subset sum problem
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13.4 n queens problem
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13.5 Summary
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Chapter 14. Dynamic programming
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Chapter 14. Dynamic programming
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14.1 Introduction to dynamic programming
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14.2 Characteristics of DP problems
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14.3 DP problem-solving approach¶
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14.4 0-1 Knapsack problem
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14.5 Unbounded knapsack problem
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14.6 Edit distance problem
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14.7 Summary
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Chapter 15. Greedy
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Chapter 15. Greedy
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15.1 Greedy algorithms
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15.2 Fractional knapsack problem
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15.3 Maximum capacity problem
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15.4 Maximum product cutting problem
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15.5 Summary
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Chapter 16. Appendix
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Chapter 16. Appendix
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16.1 Installation
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16.2 Contributing
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16.3 Terminology
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References
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References
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Table of contents
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9.1.1 Common types and terminologies of graphs
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9.1.2 Representation of graphs
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1. Adjacency matrix
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2. Adjacency list
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9.1.3 Common applications of graphs
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< h1 id = "91-graph" > 9.1 Graph< a class = "headerlink" href = "#91-graph" title = "Permanent link" > ¶ < / a > < / h1 >
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< p > A < u > graph< / u > is a type of nonlinear data structure, consisting of < u > vertices< / u > and < u > edges< / u > . A graph < span class = "arithmatex" > \(G\)< / span > can be abstractly represented as a collection of a set of vertices < span class = "arithmatex" > \(V\)< / span > and a set of edges < span class = "arithmatex" > \(E\)< / span > . The following example shows a graph containing 5 vertices and 7 edges.< / p >
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< div class = "arithmatex" > \[
\begin{aligned}
V & = \{ 1, 2, 3, 4, 5 \} \newline
E & = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline
G & = \{ V, E \} \newline
\end{aligned}
\]< / div >
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< p > If vertices are viewed as nodes and edges as references (pointers) connecting the nodes, graphs can be seen as a data structure that extends from linked lists. As shown in Figure 9-1, < strong > compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) are more complex due to their higher degree of freedom< / strong > .< / p >
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< p > < a class = "glightbox" href = "../graph.assets/linkedlist_tree_graph.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Relationship between linked lists, trees, and graphs" class = "animation-figure" src = "../graph.assets/linkedlist_tree_graph.png" / > < / a > < / p >
< p align = "center" > Figure 9-1 Relationship between linked lists, trees, and graphs < / p >
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< h2 id = "911-common-types-and-terminologies-of-graphs" > 9.1.1 Common types and terminologies of graphs< a class = "headerlink" href = "#911-common-types-and-terminologies-of-graphs" title = "Permanent link" > ¶ < / a > < / h2 >
< p > Graphs can be divided into < u > undirected graphs< / u > and < u > directed graphs< / u > depending on whether edges have direction, as shown in Figure 9-2.< / p >
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< ul >
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< li > In undirected graphs, edges represent a "bidirectional" connection between two vertices, for example, the "friends" in Facebook.< / li >
< li > In directed graphs, edges have directionality, that is, the edges < span class = "arithmatex" > \(A \rightarrow B\)< / span > and < span class = "arithmatex" > \(A \leftarrow B\)< / span > are independent of each other. For example, the "follow" and "followed" relationship on Instagram or TikTok.< / li >
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< / ul >
< p > < a class = "glightbox" href = "../graph.assets/directed_graph.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Directed and undirected graphs" class = "animation-figure" src = "../graph.assets/directed_graph.png" / > < / a > < / p >
< p align = "center" > Figure 9-2 Directed and undirected graphs < / p >
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< p > Depending on whether all vertices are connected, graphs can be divided into < u > connected graphs< / u > and < u > disconnected graphs< / u > , as shown in Figure 9-3.< / p >
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< ul >
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< li > For connected graphs, it is possible to reach any other vertex starting from an arbitrary vertex.< / li >
< li > For disconnected graphs, there is at least one vertex that cannot be reached from an arbitrary starting vertex.< / li >
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< / ul >
< p > < a class = "glightbox" href = "../graph.assets/connected_graph.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Connected and disconnected graphs" class = "animation-figure" src = "../graph.assets/connected_graph.png" / > < / a > < / p >
< p align = "center" > Figure 9-3 Connected and disconnected graphs < / p >
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< 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 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.< / p >
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< p > < a class = "glightbox" href = "../graph.assets/weighted_graph.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Weighted and unweighted graphs" class = "animation-figure" src = "../graph.assets/weighted_graph.png" / > < / a > < / p >
< p align = "center" > Figure 9-4 Weighted and unweighted graphs < / p >
< p > Graph data structures include the following commonly used terms.< / p >
< ul >
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< li > < u > Adjacency< / u > : When there is an edge connecting two vertices, these two vertices are said to be "adjacent". In Figure 9-4, the adjacent vertices of vertex 1 are vertices 2, 3, and 5.< / li >
< li > < u > Path< / u > : The sequence of edges passed 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.< / li >
< li > < u > Degree< / u > : The number of edges a vertex has. For directed graphs, < u > in-degree< / u > refers to how many edges point to the vertex, and < u > out-degree< / u > refers to how many edges point out from the vertex.< / li >
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< / ul >
< h2 id = "912-representation-of-graphs" > 9.1.2 Representation of graphs< a class = "headerlink" href = "#912-representation-of-graphs" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > Common representations of graphs include "adjacency matrix" and "adjacency list". The following examples use undirected graphs.< / p >
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< h3 id = "1-adjacency-matrix" > 1. Adjacency matrix< a class = "headerlink" href = "#1-adjacency-matrix" title = "Permanent link" > ¶ < / a > < / h3 >
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< p > Let the number of vertices in the graph be < span class = "arithmatex" > \(n\)< / span > , the < u > adjacency matrix< / u > uses an < span class = "arithmatex" > \(n \times n\)< / span > matrix to represent the graph, where each row (column) represents a vertex, and the matrix elements represent edges, with < span class = "arithmatex" > \(1\)< / span > or < span class = "arithmatex" > \(0\)< / span > indicating whether there is an edge between two vertices.< / p >
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< p > As shown in Figure 9-5, let the adjacency matrix be < span class = "arithmatex" > \(M\)< / span > , and the list of vertices be < span class = "arithmatex" > \(V\)< / span > , then the matrix element < span class = "arithmatex" > \(M[i, j] = 1\)< / span > indicates there is an edge between vertex < span class = "arithmatex" > \(V[i]\)< / span > and vertex < span class = "arithmatex" > \(V[j]\)< / span > , conversely < span class = "arithmatex" > \(M[i, j] = 0\)< / span > indicates there is no edge between the two vertices.< / p >
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< p > < a class = "glightbox" href = "../graph.assets/adjacency_matrix.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Representation of a graph with an adjacency matrix" class = "animation-figure" src = "../graph.assets/adjacency_matrix.png" / > < / a > < / p >
< p align = "center" > Figure 9-5 Representation of a graph with an adjacency matrix < / p >
< p > Adjacency matrices have the following characteristics.< / p >
< ul >
< li > A vertex cannot be connected to itself, so the elements on the main diagonal of the adjacency matrix are meaningless.< / li >
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< li > For undirected graphs, edges in both directions are equivalent, thus the adjacency matrix is symmetric with regard to the main diagonal.< / li >
< li > By replacing the elements of the adjacency matrix from < span class = "arithmatex" > \(1\)< / span > and < span class = "arithmatex" > \(0\)< / span > to weights, we can represent weighted graphs.< / li >
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< / ul >
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< p > 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 < span class = "arithmatex" > \(O(1)\)< / span > . However, the space complexity of the matrix is < span class = "arithmatex" > \(O(n^2)\)< / span > , which consumes more memory.< / p >
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< h3 id = "2-adjacency-list" > 2. Adjacency list< a class = "headerlink" href = "#2-adjacency-list" title = "Permanent link" > ¶ < / a > < / h3 >
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< p > The < u > adjacency list< / u > uses < span class = "arithmatex" > \(n\)< / span > linked lists to represent the graph, with each linked list node representing a vertex. The < span class = "arithmatex" > \(i\)< / span > -th linked list corresponds to vertex < span class = "arithmatex" > \(i\)< / span > and contains all adjacent vertices (vertices connected to that vertex). Figure 9-6 shows an example of a graph stored using an adjacency list.< / p >
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< p > < a class = "glightbox" href = "../graph.assets/adjacency_list.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Representation of a graph with an adjacency list" class = "animation-figure" src = "../graph.assets/adjacency_list.png" / > < / a > < / p >
< p align = "center" > Figure 9-6 Representation of a graph with an adjacency list < / p >
< p > The adjacency list only stores actual edges, and the total number of edges is often much less than < span class = "arithmatex" > \(n^2\)< / span > , making it more space-efficient. However, finding edges in the adjacency list requires traversing the linked list, so its time efficiency is not as good as that of the adjacency matrix.< / p >
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< p > Observing Figure 9-6, < strong > the structure of the adjacency list is very similar to the "chaining" in hash tables, hence we can use similar methods to optimize efficiency< / strong > . For example, when the linked list is long, it can be transformed into an AVL tree or red-black tree, thus optimizing the time efficiency from < span class = "arithmatex" > \(O(n)\)< / span > to < span class = "arithmatex" > \(O(\log n)\)< / span > ; the linked list can also be transformed into a hash table, thus reducing the time complexity to < span class = "arithmatex" > \(O(1)\)< / span > .< / p >
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< h2 id = "913-common-applications-of-graphs" > 9.1.3 Common applications of graphs< a class = "headerlink" href = "#913-common-applications-of-graphs" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > As shown in Table 9-1, many real-world systems can be modeled with graphs, and corresponding problems can be reduced to graph computing problems.< / p >
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< p align = "center" > Table 9-1 Common graphs in real life < / p >
< div class = "center-table" >
< table >
< thead >
< tr >
< th > < / th >
< th > Vertices< / th >
< th > Edges< / th >
< th > Graph Computing Problem< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td > Social Networks< / td >
< td > Users< / td >
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< td > Follow / Followed< / td >
< td > Potential Following Recommendations< / td >
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< / tr >
< tr >
< td > Subway Lines< / td >
< td > Stations< / td >
< td > Connectivity Between Stations< / td >
< td > Shortest Route Recommendations< / td >
< / tr >
< tr >
< td > Solar System< / td >
< td > Celestial Bodies< / td >
< td > Gravitational Forces Between Celestial Bodies< / td >
< td > Planetary Orbit Calculations< / td >
< / tr >
< / tbody >
< / table >
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