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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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5.1 Stack
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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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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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9.1.1 Common types 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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9.2 Basic graph operations
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9.3 Graph traversal
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Chapter 10. Searching
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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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2024-04-02 19:00:08 +08:00
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Table of contents
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9.1.1 Common types 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 >
< p > A "graph" is a type of nonlinear data structure, consisting of "vertices" and "edges". 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 >
< 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 >
< h2 id = "911-common-types-of-graphs" > 9.1.1 Common types of graphs< a class = "headerlink" href = "#911-common-types-of-graphs" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > Based on whether edges have direction, graphs can be divided into "undirected graphs" and "directed graphs", as shown in Figure 9-2.< / p >
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< ul >
< li > In undirected graphs, edges represent a "bidirectional" connection between two vertices, for example, the "friendship" in WeChat or QQ.< / 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 "be followed" relationship on Weibo or TikTok.< / li >
< / 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 > Based on whether all vertices are connected, graphs can be divided into "connected graphs" and "disconnected graphs", as shown in Figure 9-3.< / p >
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< ul >
< li > For connected graphs, it is possible to reach any other vertex starting from a certain vertex.< / li >
< li > For disconnected graphs, there is at least one vertex that cannot be reached from a certain starting vertex.< / li >
< / 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 "weighted graphs" as shown in Figure 9-4. 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.< / 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 > "Adjacency": 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 > "Path": 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 >
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< li > "Degree": The number of edges a vertex has. For directed graphs, "in-degree" refers to how many edges point to the vertex, and "out-degree" refers to how many edges point out from the vertex.< / li >
< / 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 >
< p > Common representations of graphs include "adjacency matrices" and "adjacency lists". The following examples use undirected graphs.< / p >
< h3 id = "1-adjacency-matrix" > 1. Adjacency matrix< a class = "headerlink" href = "#1-adjacency-matrix" title = "Permanent link" > ¶ < / a > < / h3 >
< p > Let the number of vertices in the graph be < span class = "arithmatex" > \(n\)< / span > , the "adjacency matrix" 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 >
< li > For undirected graphs, edges in both directions are equivalent, thus the adjacency matrix is symmetric about 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, it can represent weighted graphs.< / li >
< / ul >
< p > 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 < 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 >
< 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 "adjacency list" 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 >
< td > Friendships< / td >
< td > Potential Friend Recommendations< / td >
< / 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 >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 640 512" > <!-- ! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc. --> < path d = "M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z" / > < / svg >
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< script id = "__config" type = "application/json" > { "base" : "../.." , "features" : [ "announce.dismiss" , "content.action.edit" , "content.code.annotate" , "content.code.copy" , "content.tabs.link" , "content.tooltips" , "navigation.indexes" , "navigation.top" , "navigation.footer" , "navigation.tracking" , "search.highlight" , "search.share" , "search.suggest" , "toc.follow" ] , "search" : "../../assets/javascripts/workers/search.b8dbb3d2.min.js" , "translations" : { "clipboard.copied" : "Copied to clipboard" , "clipboard.copy" : "Copy to clipboard" , "search.result.more.one" : "1 more on this page" , "search.result.more.other" : "# more on this page" , "search.result.none" : "No matching documents" , "search.result.one" : "1 matching document" , "search.result.other" : "# matching documents" , "search.result.placeholder" : "Type to start searching" , "search.result.term.missing" : "Missing" , "select.version" : "Select version" } } < / script >
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< script > document $ . subscribe ( ( ) => { const lightbox = GLightbox ( { "touchNavigation" : true , "loop" : false , "zoomable" : true , "draggable" : false , "openEffect" : "zoom" , "closeEffect" : "zoom" , "slideEffect" : "none" } ) ; } ) < / script > < / body >
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