From f58068c838d4d746e360eee1461bab43321544e3 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Sat, 28 Jan 2023 23:56:41 +0800 Subject: [PATCH] Add the chapter of Graph (#303) * Update the chapter graph * Update the chapter graph * Update the chapter graph * Update the chapter graph --- .../chapter_graph/graph_adjacency_list.java | 127 ++++++ .../chapter_graph/graph_adjacency_matrix.java | 131 ++++++ codes/java/chapter_heap/my_heap.java | 1 + codes/java/include/PrintUtil.java | 27 ++ .../classification_of_data_structure.md | 4 +- .../graph.assets/adjacency_list.png | Bin 0 -> 67309 bytes .../graph.assets/adjacency_list_add_edge.png | Bin 0 -> 76356 bytes .../adjacency_list_add_vertex.png | Bin 0 -> 77309 bytes .../adjacency_list_initialization.png | Bin 0 -> 71151 bytes .../adjacency_list_remove_edge.png | Bin 0 -> 73547 bytes .../adjacency_list_remove_vertex.png | Bin 0 -> 68068 bytes .../graph.assets/adjacency_matrix.png | Bin 0 -> 66498 bytes .../adjacency_matrix_add_edge.png | Bin 0 -> 66523 bytes .../adjacency_matrix_add_vertex.png | Bin 0 -> 75529 bytes .../adjacency_matrix_initialization.png | Bin 0 -> 64058 bytes .../adjacency_matrix_remove_edge.png | Bin 0 -> 67272 bytes .../adjacency_matrix_remove_vertex.png | Bin 0 -> 64441 bytes .../graph.assets/connected_graph.png | Bin 0 -> 50690 bytes .../graph.assets/directed_graph.png | Bin 0 -> 56025 bytes .../graph.assets/linkedlist_tree_graph.png | Bin 0 -> 79351 bytes .../graph.assets/weighted_graph.png | Bin 0 -> 58609 bytes docs/chapter_graph/graph.md | 409 ++++++++++++++++++ mkdocs.yml | 2 + 23 files changed, 699 insertions(+), 2 deletions(-) create mode 100644 codes/java/chapter_graph/graph_adjacency_list.java create mode 100644 codes/java/chapter_graph/graph_adjacency_matrix.java create mode 100644 docs/chapter_graph/graph.assets/adjacency_list.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_list_add_edge.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_list_add_vertex.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_list_initialization.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_list_remove_edge.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_list_remove_vertex.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_matrix.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_matrix_add_edge.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_matrix_add_vertex.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_matrix_initialization.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_matrix_remove_edge.png create mode 100644 docs/chapter_graph/graph.assets/adjacency_matrix_remove_vertex.png create mode 100644 docs/chapter_graph/graph.assets/connected_graph.png create mode 100644 docs/chapter_graph/graph.assets/directed_graph.png create mode 100644 docs/chapter_graph/graph.assets/linkedlist_tree_graph.png create mode 100644 docs/chapter_graph/graph.assets/weighted_graph.png create mode 100644 docs/chapter_graph/graph.md diff --git a/codes/java/chapter_graph/graph_adjacency_list.java b/codes/java/chapter_graph/graph_adjacency_list.java new file mode 100644 index 000000000..417c84405 --- /dev/null +++ b/codes/java/chapter_graph/graph_adjacency_list.java @@ -0,0 +1,127 @@ +/** + * File: graph_adjacency_matrix.java + * Created Time: 2023-01-26 + * Author: Krahets (krahets@163.com) + */ + +package chapter_graph; + +import java.util.*; + +/* 顶点类 */ +class Vertex { + int val; + public Vertex(int val) { + this.val = val; + } +} + +/* 基于邻接表实现的无向图类 */ +class GraphAdjList { + // 请注意,vertices 和 adjList 中存储的都是 Vertex 对象 + Map> adjList; // 邻接表(使用哈希表实现) + + /* 构造函数 */ + public GraphAdjList(Vertex[][] edges) { + this.adjList = new HashMap<>(); + // 添加所有顶点和边 + for (Vertex[] edge : edges) { + addVertex(edge[0]); + addVertex(edge[1]); + addEdge(edge[0], edge[1]); + } + } + + /* 获取顶点数量 */ + public int size() { + return adjList.size(); + } + + /* 添加边 */ + public void addEdge(Vertex vet1, Vertex vet2) { + if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2) + throw new IllegalArgumentException(); + // 添加边 vet1 - vet2 + adjList.get(vet1).add(vet2); + adjList.get(vet2).add(vet1); + } + + /* 删除边 */ + public void removeEdge(Vertex vet1, Vertex vet2) { + if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2) + throw new IllegalArgumentException(); + // 删除边 vet1 - vet2 + adjList.get(vet1).remove(vet2); + adjList.get(vet2).remove(vet1); + } + + /* 添加顶点 */ + public void addVertex(Vertex vet) { + if (adjList.containsKey(vet)) + return; + // 在邻接表中添加一个新链表(即 HashSet) + adjList.put(vet, new HashSet<>()); + } + + /* 删除顶点 */ + public void removeVertex(Vertex vet) { + if (!adjList.containsKey(vet)) + throw new IllegalArgumentException(); + // 在邻接表中删除顶点 vet 对应的链表(即 HashSet) + adjList.remove(vet); + // 遍历其它顶点的链表(即 HashSet),删除所有包含 vet 的边 + for (Set set : adjList.values()) { + set.remove(vet); + } + } + + /* 打印邻接表 */ + public void print() { + System.out.println("邻接表 ="); + for (Map.Entry> entry : adjList.entrySet()) { + List tmp = new ArrayList<>(); + for (Vertex vertex : entry.getValue()) + tmp.add(vertex.val); + System.out.println(entry.getKey().val + ": " + tmp + ","); + } + } +} + +public class graph_adjacency_list { + public static void main(String[] args) { + /* 初始化无向图 */ + Vertex v0 = new Vertex(1), + v1 = new Vertex(3), + v2 = new Vertex(2), + v3 = new Vertex(5), + v4 = new Vertex(4); + Vertex[][] edges = { { v0, v1 }, { v1, v2 }, { v2, v3 }, { v0, v3 }, { v2, v4 }, { v3, v4 } }; + GraphAdjList graph = new GraphAdjList(edges); + System.out.println("\n初始化后,图为"); + graph.print(); + + /* 添加边 */ + // 顶点 1, 2 即 v0, v2 + graph.addEdge(v0, v2); + System.out.println("\n添加边 1-2 后,图为"); + graph.print(); + + /* 删除边 */ + // 顶点 1, 3 即 v0, v1 + graph.removeEdge(v0, v1); + System.out.println("\n删除边 1-3 后,图为"); + graph.print(); + + /* 添加顶点 */ + Vertex v5 = new Vertex(6); + graph.addVertex(v5); + System.out.println("\n添加顶点 6 后,图为"); + graph.print(); + + /* 删除顶点 */ + // 顶点 3 即 v1 + graph.removeVertex(v1); + System.out.println("\n删除顶点 3 后,图为"); + graph.print(); + } +} diff --git a/codes/java/chapter_graph/graph_adjacency_matrix.java b/codes/java/chapter_graph/graph_adjacency_matrix.java new file mode 100644 index 000000000..1fd83ff97 --- /dev/null +++ b/codes/java/chapter_graph/graph_adjacency_matrix.java @@ -0,0 +1,131 @@ +/** + * File: graph_adjacency_matrix.java + * Created Time: 2023-01-26 + * Author: Krahets (krahets@163.com) + */ + +package chapter_graph; + +import include.*; +import java.util.*; + +/* 基于邻接矩阵实现的无向图类 */ +class GraphAdjMat { + List vertices; // 顶点列表,元素代表“顶点值”,索引代表“顶点索引” + List> adjMat; // 邻接矩阵,行列索引对应“顶点索引” + + /* 构造函数 */ + public GraphAdjMat(int[] vertices, int[][] edges) { + this.vertices = new ArrayList<>(); + this.adjMat = new ArrayList<>(); + // 添加顶点 + for (int val : vertices) { + addVertex(val); + } + // 添加边 + // 请注意,edges 元素代表顶点索引,即对应 vertices 元素索引 + for (int[] e : edges) { + addEdge(e[0], e[1]); + } + } + + /* 获取顶点数量 */ + public int size() { + return vertices.size(); + } + + /* 添加顶点 */ + public void addVertex(int val) { + int n = size(); + // 向顶点列表中添加新顶点的值 + vertices.add(val); + // 在邻接矩阵中添加一行 + List newRow = new ArrayList<>(n); + for (int j = 0; j < n; j++) { + newRow.add(0); + } + adjMat.add(newRow); + // 在邻接矩阵中添加一列 + for (List row : adjMat) { + row.add(0); + } + } + + /* 删除顶点 */ + public void removeVertex(int index) { + if (index >= size()) + throw new IndexOutOfBoundsException(); + // 在顶点列表中移除索引 index 的顶点 + vertices.remove(index); + // 在邻接矩阵中删除索引 index 的行 + adjMat.remove(index); + // 在邻接矩阵中删除索引 index 的列 + for (List row : adjMat) { + row.remove(index); + } + } + + /* 添加边 */ + // 参数 i, j 对应 vertices 元素索引 + public void addEdge(int i, int j) { + // 索引越界与相等处理 + if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) + throw new IndexOutOfBoundsException(); + // 在无向图中,邻接矩阵沿主对角线对称,即满足 (i, j) == (j, i) + adjMat.get(i).set(j, 1); + adjMat.get(j).set(i, 1); + } + + /* 删除边 */ + // 参数 i, j 对应 vertices 元素索引 + public void removeEdge(int i, int j) { + // 索引越界与相等处理 + if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) + throw new IndexOutOfBoundsException(); + adjMat.get(i).set(j, 0); + adjMat.get(j).set(i, 0); + } + + /* 打印邻接矩阵 */ + public void print() { + System.out.print("顶点列表 = "); + System.out.println(vertices); + System.out.println("邻接矩阵 ="); + PrintUtil.printMatrix(adjMat); + } +} + +public class graph_adjacency_matrix { + public static void main(String[] args) { + /* 初始化无向图 */ + // 请注意,edges 元素代表顶点索引,即对应 vertices 元素索引 + int[] vertices = { 1, 3, 2, 5, 4 }; + int[][] edges = { { 0, 1 }, { 0, 2 }, { 1, 2 }, { 2, 3 }, { 0, 3 }, { 2, 4 }, { 3, 4 } }; + GraphAdjMat graph = new GraphAdjMat(vertices, edges); + System.out.println("\n初始化后,图为"); + graph.print(); + + /* 添加边 */ + // 顶点 1, 2 的索引分别为 0, 2 + graph.addEdge(0, 2); + System.out.println("\n添加边 1-2 后,图为"); + graph.print(); + + /* 删除边 */ + // 顶点 1, 3 的索引分别为 0, 1 + graph.removeEdge(0, 1); + System.out.println("\n删除边 1-3 后,图为"); + graph.print(); + + /* 添加顶点 */ + graph.addVertex(6); + System.out.println("\n添加顶点 6 后,图为"); + graph.print(); + + /* 删除顶点 */ + // 顶点 3 的索引为 1 + graph.removeVertex(1); + System.out.println("\n删除顶点 3 后,图为"); + graph.print(); + } +} diff --git a/codes/java/chapter_heap/my_heap.java b/codes/java/chapter_heap/my_heap.java index f823824e6..ba068c698 100644 --- a/codes/java/chapter_heap/my_heap.java +++ b/codes/java/chapter_heap/my_heap.java @@ -9,6 +9,7 @@ package chapter_heap; import include.*; import java.util.*; +/* 最大堆类 */ class MaxHeap { // 使用列表而非数组,这样无需考虑扩容问题 private List maxHeap; diff --git a/codes/java/include/PrintUtil.java b/codes/java/include/PrintUtil.java index 1ebc45ba0..c8bdf1f74 100755 --- a/codes/java/include/PrintUtil.java +++ b/codes/java/include/PrintUtil.java @@ -20,6 +20,33 @@ class Trunk { }; public class PrintUtil { + + /** + * Print a matrix (Array) + * @param + * @param matrix + */ + public static void printMatrix(T[][] matrix) { + System.out.println("["); + for (T[] row : matrix) { + System.out.println(" " + row + ","); + } + System.out.println("]"); + } + + /** + * Print a matrix (List) + * @param + * @param matrix + */ + public static void printMatrix(List> matrix) { + System.out.println("["); + for (List row : matrix) { + System.out.println(" " + row + ","); + } + System.out.println("]"); + } + /** * Print a linked list * @param head diff --git a/docs/chapter_data_structure/classification_of_data_structure.md b/docs/chapter_data_structure/classification_of_data_structure.md index b38ced9e9..a2f5221b8 100644 --- a/docs/chapter_data_structure/classification_of_data_structure.md +++ b/docs/chapter_data_structure/classification_of_data_structure.md @@ -33,8 +33,8 @@ comments: true **所有数据结构都是基于数组、或链表、或两者组合实现的**。例如栈和队列,既可以使用数组实现、也可以使用链表实现,而例如哈希表,其实现同时包含了数组和链表。 -- **基于数组可实现**:栈、队列、堆、哈希表、矩阵、张量(维度 $\geq 3$ 的数组)等; -- **基于链表可实现**:栈、队列、堆、哈希表、树、图等; +- **基于数组可实现**:栈、队列、哈希表、树、堆、图、矩阵、张量(维度 $\geq 3$ 的数组)等; +- **基于链表可实现**:栈、队列、哈希表、树、堆、图等; 基于数组实现的数据结构也被称为「静态数据结构」,这意味着该数据结构在在被初始化后,长度不可变。相反地,基于链表实现的数据结构被称为「动态数据结构」,该数据结构在被初始化后,我们也可以在程序运行中修改其长度。 diff --git a/docs/chapter_graph/graph.assets/adjacency_list.png b/docs/chapter_graph/graph.assets/adjacency_list.png new file mode 100644 index 0000000000000000000000000000000000000000..24bd4a4880d08565868459a8c60389de69d1da30 GIT binary patch literal 67309 zcmeFZWmr{P_dX1p25A+fyQRAYq(nkV>F!cGHlTEOmwQjNZP5q60o7Lcu{}LBTwn0{lQhWeE*Zk)QlMemQi0Pomv*Do9VIDt&27$AoJj7nP0S6=- zNwrTfG&ly8)R zib{{AaLE)$YYpuxHtP`mQxxICny`@_!LSqJWM`Z6;SIE<87hD=u84eo*rjrX$;9`KItg!B?+0vqNNtcfZq|Dd=-*+UGe&t^cu0z<6fZ!m~yw z2I%C3+gxW-@P6>?2Td^Ov&L!eWgOu}XDBPzuWvIx(ibfV)MC^xA+O`DTG7l`U6a0Q(@sJ2%oR7o^)=fb;pA1oF<; zLRce$*QJcHv*!N232(847nHZ&ew@tZ#40U5g6gbwrTw(|)z4@Hrm+Qv;24}gYRwI9 zMY<`w+l#0zrF=oa8Sx^!&TZa<@^2mdXG23Uy1IKMGr6J=`;c|a}`8!!og=>q-EYp7} 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zjc;p4jZf)luCs~6xcx110gHP3>+;AvJqTfk3c85cj*~J3LQ3lj)pR32V{ObgbcZQzwT^*d>I-Ebl^VbRe5x+ zyj)!K5{;hWreuQd=bWc*linX!|84^Mv6d-eGm6ieNbOs|@tgp`e8pSl=ED~;)0aAJ z-;vftrhbU7BHQ*$uO2k*r|lAtewR0SrO%ib0EhOck7vf_#B5p%@+O+tzV8^~PhD5H zg>#!ef1o+MG?mpq;%f4tVj@mAdWT_5^XgEr>aQ2)br}TGu8b;Xi|wbiI%NJ_PSotI z_$3UpU=4T7iLoi@CX_kA&&#j>t($I>@1x{@AN)U~PzU$f{*m3J9?Et+Pvw)(Rfi~D p$V37a vertices; // 顶点列表,元素代表“顶点值”,索引代表“顶点索引” + List> adjMat; // 邻接矩阵,行列索引对应“顶点索引” + + /* 构造函数 */ + public GraphAdjMat(int[] vertices, int[][] edges) { + this.vertices = new ArrayList<>(); + this.adjMat = new ArrayList<>(); + // 添加顶点 + for (int val : vertices) { + addVertex(val); + } + // 添加边 + // 请注意,edges 元素代表顶点索引,即对应 vertices 元素索引 + for (int[] e : edges) { + addEdge(e[0], e[1]); + } + } + + /* 获取顶点数量 */ + public int size() { + return vertices.size(); + } + + /* 添加顶点 */ + public void addVertex(int val) { + int n = size(); + // 向顶点列表中添加新顶点的值 + vertices.add(val); + // 在邻接矩阵中添加一行 + List newRow = new ArrayList<>(n); + for (int j = 0; j < n; j++) { + newRow.add(0); + } + adjMat.add(newRow); + // 在邻接矩阵中添加一列 + for (List row : adjMat) { + row.add(0); + } + } + + /* 删除顶点 */ + public void removeVertex(int index) { + if (index >= size()) + throw new IndexOutOfBoundsException(); + // 在顶点列表中移除索引 index 的顶点 + vertices.remove(index); + // 在邻接矩阵中删除索引 index 的行 + adjMat.remove(index); + // 在邻接矩阵中删除索引 index 的列 + for (List row : adjMat) { + row.remove(index); + } + } + + /* 添加边 */ + // 参数 i, j 对应 vertices 元素索引 + public void addEdge(int i, int j) { + // 索引越界与相等处理 + if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) + throw new IndexOutOfBoundsException(); + // 在无向图中,邻接矩阵沿主对角线对称,即满足 (i, j) == (j, i) + adjMat.get(i).set(j, 1); + adjMat.get(j).set(i, 1); + } + + /* 删除边 */ + // 参数 i, j 对应 vertices 元素索引 + public void removeEdge(int i, int j) { + // 索引越界与相等处理 + if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) + throw new IndexOutOfBoundsException(); + adjMat.get(i).set(j, 0); + adjMat.get(j).set(i, 0); + } + } + ``` + +=== "C++" + + ```cpp title="graph_adjacency_matrix.cpp" + + ``` + +=== "Python" + + ```python title="graph_adjacency_matrix.py" + + ``` + +=== "Go" + + ```go title="graph_adjacency_matrix.go" + + ``` + +=== "JavaScript" + + ```js title="graph_adjacency_matrix.js" + + ``` + +=== "TypeScript" + + ```typescript title="graph_adjacency_matrix.ts" + + ``` + +=== "C" + + ```c title="graph_adjacency_matrix.c" + + ``` + +=== "C#" + + ```csharp title="graph_adjacency_matrix.cs" + + ``` + +=== "Swift" + + ```swift title="graph_adjacency_matrix.swift" + + ``` + +### 基于邻接表的实现 + +设图的顶点总数为 $n$ 、边总数为 $m$ ,则有: + +- **添加边**:在顶点对应链表的尾部添加边即可,使用 $O(1)$ 时间。因为是无向图,所以需要同时添加两个方向的边。 +- **删除边**:在顶点对应链表中查询与删除指定边,使用 $O(m)$ 时间。与添加边一样,需要同时删除两个方向的边。 +- **添加顶点**:在邻接表中添加一个链表即可,并以新增顶点为链表头结点,使用 $O(1)$ 时间。 +- **删除顶点**:需要遍历整个邻接表,删除包含指定顶点的所有边,使用 $O(n + m)$ 时间。 +- **初始化**:需要在邻接表中建立 $n$ 个结点和 $2m$ 条边,使用 $O(n + m)$ 时间。 + +=== "初始化邻接表" + ![adjacency_list_initialization](graph.assets/adjacency_list_initialization.png) + +=== "添加边" + ![adjacency_list_add_edge](graph.assets/adjacency_list_add_edge.png) + +=== "删除边" + ![adjacency_list_remove_edge](graph.assets/adjacency_list_remove_edge.png) + +=== "添加顶点" + ![adjacency_list_add_vertex](graph.assets/adjacency_list_add_vertex.png) + +=== "删除顶点" + ![adjacency_list_remove_vertex](graph.assets/adjacency_list_remove_vertex.png) + +基于邻接表实现图的代码如下所示。 + +=== "Java" + + ```java title="graph_adjacency_list.java" + /* 顶点类 */ + class Vertex { + int val; + public Vertex(int val) { + this.val = val; + } + } + + /* 基于邻接表实现的无向图类 */ + class GraphAdjList { + // 请注意,vertices 和 adjList 中存储的都是 Vertex 对象 + Map> adjList; // 邻接表(使用哈希表实现) + + /* 构造函数 */ + public GraphAdjList(Vertex[][] edges) { + this.adjList = new HashMap<>(); + // 添加所有顶点和边 + for (Vertex[] edge : edges) { + addVertex(edge[0]); + addVertex(edge[1]); + addEdge(edge[0], edge[1]); + } + } + + /* 获取顶点数量 */ + public int size() { + return adjList.size(); + } + + /* 添加边 */ + public void addEdge(Vertex vet1, Vertex vet2) { + if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2) + throw new IllegalArgumentException(); + // 添加边 vet1 - vet2 + adjList.get(vet1).add(vet2); + adjList.get(vet2).add(vet1); + } + + /* 删除边 */ + public void removeEdge(Vertex vet1, Vertex vet2) { + if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2) + throw new IllegalArgumentException(); + // 删除边 vet1 - vet2 + adjList.get(vet1).remove(vet2); + adjList.get(vet2).remove(vet1); + } + + /* 添加顶点 */ + public void addVertex(Vertex vet) { + if (adjList.containsKey(vet)) + return; + // 在邻接表中添加一个新链表(即 HashSet) + adjList.put(vet, new HashSet<>()); + } + + /* 删除顶点 */ + public void removeVertex(Vertex vet) { + if (!adjList.containsKey(vet)) + throw new IllegalArgumentException(); + // 在邻接表中删除顶点 vet 对应的链表(即 HashSet) + adjList.remove(vet); + // 遍历其它顶点的链表(即 HashSet),删除所有包含 vet 的边 + for (Set set : adjList.values()) { + set.remove(vet); + } + } + } + ``` + +=== "C++" + + ```cpp title="graph_adjacency_list.cpp" + + ``` + +=== "Python" + + ```python title="graph_adjacency_list.py" + + ``` + +=== "Go" + + ```go title="graph_adjacency_list.go" + + ``` + +=== "JavaScript" + + ```js title="graph_adjacency_list.js" + + ``` + +=== "TypeScript" + + ```typescript title="graph_adjacency_list.ts" + + ``` + +=== "C" + + ```c title="graph_adjacency_list.c" + + ``` + +=== "C#" + + ```csharp title="graph_adjacency_list.cs" + + ``` + +=== "Swift" + + ```swift title="graph_adjacency_list.swift" + + ``` + +### 效率对比 + +设图中共有 $n$ 个顶点和 $m$ 条边,下表为邻接矩阵和邻接表的时间和空间效率对比。 + +

+ +观察上表,貌似邻接表(哈希表)的时间与空间效率最优。但实际上,在邻接矩阵中操作边的效率更高,只需要一次数组访问或赋值操作即可。总结以上,**邻接矩阵体现“以空间换时间”,邻接表体现“以时间换空间”**。 + +## 图常见应用 + +现实中的许多系统都可以使用图来建模,对应的待求解问题也可以被约化为图计算问题。 + +
+ +| | 顶点 | 边 | 图计算问题 | +| -------- | ---- | -------------------- | ------------ | +| 社交网络 | 用户 | 好友关系 | 潜在好友推荐 | +| 地铁线路 | 站点 | 站点间的连通性 | 最短路线推荐 | +| 太阳系 | 星体 | 星体间的万有引力作用 | 行星轨道计算 | + +
diff --git a/mkdocs.yml b/mkdocs.yml index ad836ae9f..6181cfa2c 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -163,6 +163,8 @@ nav: - 小结: chapter_tree/summary.md - 堆: - 堆(Heap): chapter_heap/heap.md + - 图: + - 图(Graph): chapter_graph/graph.md - 查找算法: - 线性查找: chapter_searching/linear_search.md - 二分查找: chapter_searching/binary_search.md

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