From c74f8293b9249cf879a97b8e41b3914461cc5599 Mon Sep 17 00:00:00 2001 From: krahets Date: Mon, 13 Feb 2023 16:20:08 +0800 Subject: [PATCH 1/8] Graph dev --- codes/java/chapter_graph/graph_traversal.java | 70 +++++++++++++++++++ docs/chapter_graph/graph_traversal.md | 31 ++++++++ docs/chapter_tree/binary_tree_traversal.md | 44 ++++++------ 3 files changed, 124 insertions(+), 21 deletions(-) create mode 100644 codes/java/chapter_graph/graph_traversal.java create mode 100644 docs/chapter_graph/graph_traversal.md diff --git a/codes/java/chapter_graph/graph_traversal.java b/codes/java/chapter_graph/graph_traversal.java new file mode 100644 index 000000000..3b152198d --- /dev/null +++ b/codes/java/chapter_graph/graph_traversal.java @@ -0,0 +1,70 @@ +/** + * File: graph_adjacency_list.java + * Created Time: 2023-02-12 + * Author: Krahets (krahets@163.com) + */ + +package chapter_graph; + +import java.util.*; + +public class graph_traversal { + /* 以顶点 vet 为起点,对图 graph 执行广度优先遍历 */ + // 采用 GraphAdjList 表示图,以方便获取指定结点的所有邻接结点 + static List graphBFS(GraphAdjList graph, Vertex startVet) { + // 顶点遍历序列 + List res = new ArrayList<>(); + // 用于记录顶点是否已被访问 + Set visited = new HashSet<>() {{ add(startVet); }}; + // 队列用于实现 BFS + Queue que = new LinkedList<>() {{ offer(startVet); }}; + // 循环直至访问完所有顶点 + while (!que.isEmpty()) { + Vertex vet = que.poll(); // 队首顶点出队 + res.add(vet); // 访问该顶点 + // 遍历该顶点的所有邻接顶点 + for (Vertex adjVet : graph.adjList.get(vet)) { + if (!visited.contains(adjVet)) { + que.offer(adjVet); // 只入队未访问的顶点 + visited.add(adjVet); // 标记该顶点已被访问 + } + } + } + // 返回顶点遍历序列 + return res; + } + + /* 输入值列表 vals ,返回顶点列表 vets */ + static Vertex[] valsToVets(int[] vals) { + Vertex[] vets = new Vertex[vals.length]; + for (int i = 0; i < vals.length; i++) { + vets[i] = new Vertex(vals[i]); + } + return vets; + } + + /* 输入顶点列表 vets ,返回值列表 vals */ + static List vetsToVals(List vets) { + List vals = new ArrayList<>(); + for (Vertex vet : vets) { + vals.add(vet.val); + } + return vals; + } + + public static void main(String[] args) { + /* 初始化无向图 */ + Vertex[] v = valsToVets(new int[] { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }); + Vertex[][] edges = { { v[0], v[1] }, { v[0], v[3] }, { v[1], v[2] }, { v[1], v[4] }, + { v[2], v[5] }, { v[3], v[4] }, { v[3], v[6] }, { v[4], v[5] }, + { v[4], v[7] }, { v[5], v[8] }, { v[6], v[7] }, { v[7], v[8] }}; + GraphAdjList graph = new GraphAdjList(edges); + System.out.println("\n初始化后,图为"); + graph.print(); + + /* 广度优先遍历 BFS */ + List res = graphBFS(graph, v[0]); + System.out.println("\n广度优先遍历(BFS)顶点序列为"); + System.out.println(vetsToVals(res)); + } +} diff --git a/docs/chapter_graph/graph_traversal.md b/docs/chapter_graph/graph_traversal.md new file mode 100644 index 000000000..fccd51509 --- /dev/null +++ b/docs/chapter_graph/graph_traversal.md @@ -0,0 +1,31 @@ +--- +comments: true +--- + +# 图的遍历 + +与遍历树类似,遍历图也需要通过搜索算法来实现,并也可根据遍历顺序来分为「广度优先遍历 Breadth-First Traversal」和「深度优先遍历 Depth-First Travsersal」,简称分别为 BFS 和 DFS 。 + +!!! tip 「树」与「图」的关系 + + 本质上,可以把树看作是图的一种特例,即树是一种限制条件下的图。 + +## 广度优先遍历 + +广度优先遍历(BFS)代表一种优先遍历最近的顶点、一层层向外扩张的遍历方式。具体来看,从某个顶点出发,则优先遍历该顶点的所有邻接顶点,随后遍历下个顶点的所有邻接顶点,以此类推…… + +(图) + +BFS 常借助「队列」来实现,队列具有“先入先出”的性质,这与 BFS 的“由近及远”的遍历方式是异曲同工的。具体地,在每轮迭代中弹出队首顶点且访问之,并将该顶点的所有邻接顶点加入到队列尾部,直到所有顶点访问完成即可。 + +为了防止重复遍历顶点,我们需要借助一个 HashSet 来记录哪些结点已被访问,从而避免走“回头路”。 + +```java + +``` + + + +## 深度优先遍历 + +深度优先遍历(DFS)代表一种优先走到底,无路可走再回头的遍历方式。从某个顶点出发,首先不断地通过指针向下一个顶点遍历,直到走到头开始回溯,再继续走到底 + 回溯,以此类推…… \ No newline at end of file diff --git a/docs/chapter_tree/binary_tree_traversal.md b/docs/chapter_tree/binary_tree_traversal.md index 04b641d00..4267b71b4 100755 --- a/docs/chapter_tree/binary_tree_traversal.md +++ b/docs/chapter_tree/binary_tree_traversal.md @@ -4,7 +4,9 @@ comments: true # 7.2. 二叉树遍历 -非线性数据结构的遍历操作比线性数据结构更加复杂,往往需要使用搜索算法来实现。常见的二叉树遍历方式有层序遍历、前序遍历、中序遍历、后序遍历。 +从物理结构角度看,树是一种基于链表的数据结构,因此遍历方式也是通过指针(即引用)逐个遍历结点。同时,树还是一种非线性数据结构,这导致遍历树比遍历链表更加复杂,需要使用搜索算法来实现。 + +常见的二叉树遍历方式有层序遍历、前序遍历、中序遍历、后序遍历。 ## 7.2.1. 层序遍历 @@ -102,9 +104,9 @@ comments: true ```java title="binary_tree_dfs.java" [class]{binary_tree_dfs}-[func]{preOrder} - + [class]{binary_tree_dfs}-[func]{inOrder} - + [class]{binary_tree_dfs}-[func]{postOrder} ``` @@ -112,9 +114,9 @@ comments: true ```cpp title="binary_tree_dfs.cpp" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -122,9 +124,9 @@ comments: true ```python title="binary_tree_dfs.py" [class]{}-[func]{pre_order} - + [class]{}-[func]{in_order} - + [class]{}-[func]{post_order} ``` @@ -132,9 +134,9 @@ comments: true ```go title="binary_tree_dfs.go" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -142,9 +144,9 @@ comments: true ```javascript title="binary_tree_dfs.js" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -152,9 +154,9 @@ comments: true ```typescript title="binary_tree_dfs.ts" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -162,9 +164,9 @@ comments: true ```c title="binary_tree_dfs.c" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -172,9 +174,9 @@ comments: true ```csharp title="binary_tree_dfs.cs" [class]{binary_tree_dfs}-[func]{preOrder} - + [class]{binary_tree_dfs}-[func]{inOrder} - + [class]{binary_tree_dfs}-[func]{postOrder} ``` @@ -182,9 +184,9 @@ comments: true ```swift title="binary_tree_dfs.swift" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -192,9 +194,9 @@ comments: true ```zig title="binary_tree_dfs.zig" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` From dc441928d92fcd98889f6c65817d979ae9c06cb7 Mon Sep 17 00:00:00 2001 From: krahets Date: Wed, 15 Feb 2023 03:22:57 +0800 Subject: [PATCH 2/8] Add the section of Graph Traversal. --- .../chapter_graph/graph_adjacency_list.java | 48 ++-- .../chapter_graph/graph_adjacency_matrix.java | 2 +- codes/java/chapter_graph/graph_bfs.java | 53 ++++ codes/java/chapter_graph/graph_dfs.java | 51 ++++ docs/chapter_graph/graph.md | 8 +- docs/chapter_graph/graph_operations.md | 28 +- .../graph_traversal.assets/graph_bfs.png | Bin 0 -> 76683 bytes .../graph_bfs_step1.png | Bin 0 -> 65158 bytes .../graph_bfs_step10.png | Bin 0 -> 106787 bytes .../graph_bfs_step11.png | Bin 0 -> 85391 bytes .../graph_bfs_step2.png | Bin 0 -> 91860 bytes .../graph_bfs_step3.png | Bin 0 -> 96944 bytes .../graph_bfs_step4.png | Bin 0 -> 98870 bytes .../graph_bfs_step5.png | Bin 0 -> 104297 bytes .../graph_bfs_step6.png | Bin 0 -> 104648 bytes .../graph_bfs_step7.png | Bin 0 -> 104998 bytes .../graph_bfs_step8.png | Bin 0 -> 107712 bytes .../graph_bfs_step9.png | Bin 0 -> 107147 bytes .../graph_traversal.assets/graph_dfs.png | Bin 0 -> 82469 bytes .../graph_dfs_step1.png | Bin 0 -> 55580 bytes .../graph_dfs_step10.png | Bin 0 -> 92175 bytes .../graph_dfs_step11.png | Bin 0 -> 104232 bytes .../graph_dfs_step2.png | Bin 0 -> 66431 bytes .../graph_dfs_step3.png | Bin 0 -> 69281 bytes .../graph_dfs_step4.png | Bin 0 -> 72486 bytes .../graph_dfs_step5.png | Bin 0 -> 73883 bytes .../graph_dfs_step6.png | Bin 0 -> 80210 bytes .../graph_dfs_step7.png | Bin 0 -> 78906 bytes .../graph_dfs_step8.png | Bin 0 -> 85570 bytes .../graph_dfs_step9.png | Bin 0 -> 101093 bytes docs/chapter_graph/graph_traversal.md | 247 +++++++++++++++++- 31 files changed, 383 insertions(+), 54 deletions(-) create mode 100644 codes/java/chapter_graph/graph_bfs.java create mode 100644 codes/java/chapter_graph/graph_dfs.java create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step1.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step10.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step11.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step2.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step3.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step4.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step5.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step6.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step7.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step8.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_bfs_step9.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step1.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step10.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step11.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step2.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step3.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step4.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step5.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step6.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step7.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step8.png create mode 100644 docs/chapter_graph/graph_traversal.assets/graph_dfs_step9.png diff --git a/codes/java/chapter_graph/graph_adjacency_list.java b/codes/java/chapter_graph/graph_adjacency_list.java index a1ca10173..481fa7b98 100644 --- a/codes/java/chapter_graph/graph_adjacency_list.java +++ b/codes/java/chapter_graph/graph_adjacency_list.java @@ -7,19 +7,13 @@ package chapter_graph; import java.util.*; - -/* 顶点类 */ -class Vertex { - int val; - public Vertex(int val) { - this.val = val; - } -} +import include.*; /* 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) + // 邻接表,使用哈希表来代替链表,以提升删除边、删除顶点的效率 + // 请注意,adjList 中的元素是 Vertex 对象 + Map> adjList; /* 构造方法 */ public GraphAdjList(Vertex[][] edges) { @@ -59,26 +53,26 @@ class GraphAdjList { 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<>()); + // 在邻接表中添加一个新链表 + adjList.put(vet, new ArrayList<>()); } /* 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) + // 在邻接表中删除顶点 vet 对应的链表 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); + // 遍历其它顶点的链表,删除所有包含 vet 的边 + for (List list : adjList.values()) { + list.remove(vet); } } /* 打印邻接表 */ public void print() { System.out.println("邻接表 ="); - for (Map.Entry> entry : adjList.entrySet()) { + for (Map.Entry> entry : adjList.entrySet()) { List tmp = new ArrayList<>(); for (Vertex vertex : entry.getValue()) tmp.add(vertex.val); @@ -90,25 +84,21 @@ class GraphAdjList { 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 } }; + Vertex[] v = Vertex.valsToVets(new int[] { 1, 3, 2, 5, 4 }); + Vertex[][] edges = { { v[0], v[1] }, { v[0], v[3] }, { v[1], v[2] }, { v[2], v[3] }, { v[2], v[4] }, { v[3], v[4] } }; 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); + // 顶点 1, 2 即 v[0], v[2] + graph.addEdge(v[0], v[2]); 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); + // 顶点 1, 3 即 v[0], v[1] + graph.removeEdge(v[0], v[1]); System.out.println("\n删除边 1-3 后,图为"); graph.print(); @@ -119,8 +109,8 @@ public class graph_adjacency_list { graph.print(); /* 删除顶点 */ - // 顶点 3 即 v1 - graph.removeVertex(v1); + // 顶点 3 即 v[1] + graph.removeVertex(v[1]); 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 index 5c7edb15c..bf054ca97 100644 --- a/codes/java/chapter_graph/graph_adjacency_matrix.java +++ b/codes/java/chapter_graph/graph_adjacency_matrix.java @@ -100,7 +100,7 @@ public class graph_adjacency_matrix { /* 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 }, { 1, 2 }, { 2, 3 }, { 0, 3 }, { 2, 4 }, { 3, 4 } }; + int[][] edges = { { 0, 1 }, { 0, 3 }, { 1, 2 }, { 2, 3 }, { 2, 4 }, { 3, 4 } }; GraphAdjMat graph = new GraphAdjMat(vertices, edges); System.out.println("\n初始化后,图为"); graph.print(); diff --git a/codes/java/chapter_graph/graph_bfs.java b/codes/java/chapter_graph/graph_bfs.java new file mode 100644 index 000000000..45c481d46 --- /dev/null +++ b/codes/java/chapter_graph/graph_bfs.java @@ -0,0 +1,53 @@ +/** + * File: graph_bfs.java + * Created Time: 2023-02-12 + * Author: Krahets (krahets@163.com) + */ + +package chapter_graph; + +import java.util.*; +import include.*; + +public class graph_bfs { + /* 广度优先遍历 BFS */ + // 使用邻接表来表示图,以便获取指定顶点的所有邻接顶点 + static List graphBFS(GraphAdjList graph, Vertex startVet) { + // 顶点遍历序列 + List res = new ArrayList<>(); + // 哈希表,用于记录已被访问过的顶点 + Set visited = new HashSet<>() {{ add(startVet); }}; + // 队列用于实现 BFS + Queue que = new LinkedList<>() {{ offer(startVet); }}; + // 以顶点 vet 为起点,循环直至访问完所有顶点 + while (!que.isEmpty()) { + Vertex vet = que.poll(); // 队首顶点出队 + res.add(vet); // 记录访问顶点 + // 遍历该顶点的所有邻接顶点 + for (Vertex adjVet : graph.adjList.get(vet)) { + if (visited.contains(adjVet)) + continue; // 跳过已被访问过的顶点 + que.offer(adjVet); // 只入队未访问的顶点 + visited.add(adjVet); // 标记该顶点已被访问 + } + } + // 返回顶点遍历序列 + return res; + } + + public static void main(String[] args) { + /* 初始化无向图 */ + Vertex[] v = Vertex.valsToVets(new int[] { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }); + Vertex[][] edges = { { v[0], v[1] }, { v[0], v[3] }, { v[1], v[2] }, { v[1], v[4] }, + { v[2], v[5] }, { v[3], v[4] }, { v[3], v[6] }, { v[4], v[5] }, + { v[4], v[7] }, { v[5], v[8] }, { v[6], v[7] }, { v[7], v[8] } }; + GraphAdjList graph = new GraphAdjList(edges); + System.out.println("\n初始化后,图为"); + graph.print(); + + /* 广度优先遍历 BFS */ + List res = graphBFS(graph, v[0]); + System.out.println("\n广度优先遍历(BFS)顶点序列为"); + System.out.println(Vertex.vetsToVals(res)); + } +} diff --git a/codes/java/chapter_graph/graph_dfs.java b/codes/java/chapter_graph/graph_dfs.java new file mode 100644 index 000000000..1c542f423 --- /dev/null +++ b/codes/java/chapter_graph/graph_dfs.java @@ -0,0 +1,51 @@ +/** + * File: graph_dfs.java + * Created Time: 2023-02-12 + * Author: Krahets (krahets@163.com) + */ + +package chapter_graph; + +import java.util.*; +import include.*; + +public class graph_dfs { + /* 深度优先遍历 DFS 辅助函数 */ + static void dfs(GraphAdjList graph, Set visited, List res, Vertex vet) { + res.add(vet); // 记录访问顶点 + visited.add(vet); // 标记该顶点已被访问 + // 遍历该顶点的所有邻接顶点 + for (Vertex adjVet : graph.adjList.get(vet)) { + if (visited.contains(adjVet)) + continue; // 跳过已被访问过的顶点 + // 递归访问邻接顶点 + dfs(graph, visited, res, adjVet); + } + } + + /* 深度优先遍历 DFS */ + // 使用邻接表来表示图,以便获取指定顶点的所有邻接顶点 + static List graphDFS(GraphAdjList graph, Vertex startVet) { + // 顶点遍历序列 + List res = new ArrayList<>(); + // 哈希表,用于记录已被访问过的顶点 + Set visited = new HashSet<>(); + dfs(graph, visited, res, startVet); + return res; + } + + public static void main(String[] args) { + /* 初始化无向图 */ + Vertex[] v = Vertex.valsToVets(new int[] { 0, 1, 2, 3, 4, 5, 6 }); + Vertex[][] edges = { { v[0], v[1] }, { v[0], v[3] }, { v[1], v[2] }, + { v[2], v[5] }, { v[4], v[5] }, { v[5], v[6] } }; + GraphAdjList graph = new GraphAdjList(edges); + System.out.println("\n初始化后,图为"); + graph.print(); + + /* 深度优先遍历 BFS */ + List res = graphDFS(graph, v[0]); + System.out.println("\n深度优先遍历(DFS)顶点序列为"); + System.out.println(Vertex.vetsToVals(res)); + } +} diff --git a/docs/chapter_graph/graph.md b/docs/chapter_graph/graph.md index 5f8d861e5..1de9f230d 100644 --- a/docs/chapter_graph/graph.md +++ b/docs/chapter_graph/graph.md @@ -22,15 +22,15 @@ $$ 根据边是否有方向,分为「无向图 Undirected Graph」和「有向图 Directed Graph」。 -- 在无向图中,边表示两结点之间“双向”的连接关系,例如微信或 QQ 中的“好友关系”; +- 在无向图中,边表示两顶点之间“双向”的连接关系,例如微信或 QQ 中的“好友关系”; - 在有向图中,边是有方向的,即 $A \rightarrow B$ 和 $A \leftarrow B$ 两个方向的边是相互独立的,例如微博或抖音上的“关注”与“被关注”关系; ![directed_graph](graph.assets/directed_graph.png) 根据所有顶点是否连通,分为「连通图 Connected Graph」和「非连通图 Disconnected Graph」。 -- 对于连通图,从某个结点出发,可以到达其余任意结点; -- 对于非连通图,从某个结点出发,至少有一个结点无法到达; +- 对于连通图,从某个顶点出发,可以到达其余任意顶点; +- 对于非连通图,从某个顶点出发,至少有一个顶点无法到达; ![connected_graph](graph.assets/connected_graph.png) @@ -52,6 +52,8 @@ $$ 设图的顶点数量为 $n$ ,「邻接矩阵 Adjacency Matrix」使用一个 $n \times n$ 大小的矩阵来表示图,每一行(列)代表一个顶点,矩阵元素代表边,使用 $1$ 或 $0$ 来表示两个顶点之间有边或无边。 +如下图所示,记邻接矩阵为 $M$ 、顶点列表为 $V$ ,则矩阵元素 $M[i][j] = 1$ 代表着顶点 $V[i]$ 到顶点 $V[j]$ 之间有边,相反地 $M[i][j] = 0$ 代表两顶点之间无边。 + ![adjacency_matrix](graph.assets/adjacency_matrix.png) 邻接矩阵具有以下性质: diff --git a/docs/chapter_graph/graph_operations.md b/docs/chapter_graph/graph_operations.md index a9f0c5c02..69e1df191 100644 --- a/docs/chapter_graph/graph_operations.md +++ b/docs/chapter_graph/graph_operations.md @@ -89,7 +89,7 @@ comments: true === "Zig" ```zig title="graph_adjacency_matrix.zig" - + ``` ## 9.2.2. 基于邻接表的实现 @@ -119,11 +119,17 @@ comments: true 基于邻接表实现图的代码如下所示。 +!!! question "为什么需要使用顶点类 `Vertex` ?" + + 如果我们直接通过顶点值来区分不同顶点,那么值重复的顶点将无法被区分。 + 如果建立一个顶点列表,用索引来区分不同顶点,那么假设我们想要删除索引为 `i` 的顶点,则需要遍历整个邻接表,将其中 $> i$ 的索引全部执行 $-1$ ,这样的操作是比较耗时的。 + 因此,通过引入顶点类 `Vertex` ,每个顶点都是唯一的对象,这样在删除操作时就无需改动其余顶点了。 + === "Java" ```java title="graph_adjacency_list.java" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -131,7 +137,7 @@ comments: true ```cpp title="graph_adjacency_list.cpp" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -139,7 +145,7 @@ comments: true ```python title="graph_adjacency_list.py" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -147,7 +153,7 @@ comments: true ```go title="graph_adjacency_list.go" [class]{vertex}-[func]{} - + [class]{graphAdjList}-[func]{} ``` @@ -155,7 +161,7 @@ comments: true ```javascript title="graph_adjacency_list.js" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -163,7 +169,7 @@ comments: true ```typescript title="graph_adjacency_list.ts" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -171,7 +177,7 @@ comments: true ```c title="graph_adjacency_list.c" [class]{vertex}-[func]{} - + [class]{graphAdjList}-[func]{} ``` @@ -179,7 +185,7 @@ comments: true ```csharp title="graph_adjacency_list.cs" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -187,7 +193,7 @@ comments: true ```swift title="graph_adjacency_list.swift" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -195,7 +201,7 @@ comments: true ```zig title="graph_adjacency_list.zig" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` diff --git a/docs/chapter_graph/graph_traversal.assets/graph_bfs.png b/docs/chapter_graph/graph_traversal.assets/graph_bfs.png new file mode 100644 index 0000000000000000000000000000000000000000..da53cbaad59f8fa41ac25e86d9ff13c20f807146 GIT binary patch literal 76683 zcmeFZWmFy67A*`12oi#m-~@MfCxPJZuEB%5O9%vqAO|OSfZ*;f;o$D>?k=zTcBk)q z-QS<@|AoOg#W+>FR_(p^TyxE}cZDd(iN8X^LxO^WdL=0#sssfE2ONS%5J13(FtHRg z@B!_pB>o<%WQbrF_$SarL()`M7K#=)Mu38Y#(;wP;}qZ*3L5V}7yogI6g2+d$4b!D zf1Lxg8EgRs`=9fnptOL`KYk2=zdx?|^AqMf^nY9Y4*GpI9MpH1-^ZX4;A|*g(RW_J 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      Fig. 二叉树的层序遍历

      +### 实现代码 + 广度优先遍历一般借助「队列」来实现。队列的规则是“先进先出”,广度优先遍历的规则是 ”一层层平推“ ,两者背后的思想是一致的。 === "Java" ```java title="binary_tree_bfs.java" - [class]{binary_tree_bfs}-[func]{hierOrder} + [class]{binary_tree_bfs}-[func]{levelOrder} ``` === "C++" ```cpp title="binary_tree_bfs.cpp" - [class]{}-[func]{hierOrder} + [class]{}-[func]{levelOrder} ``` === "Python" ```python title="binary_tree_bfs.py" - [class]{}-[func]{hier_order} + [class]{}-[func]{level_order} ``` === "Go" ```go title="binary_tree_bfs.go" - [class]{}-[func]{hierOrder} + [class]{}-[func]{levelOrder} ``` === "JavaScript" ```javascript title="binary_tree_bfs.js" - [class]{}-[func]{hierOrder} + [class]{}-[func]{levelOrder} ``` === "TypeScript" ```typescript title="binary_tree_bfs.ts" - [class]{}-[func]{hierOrder} + [class]{}-[func]{levelOrder} ``` === "C" ```c title="binary_tree_bfs.c" - [class]{}-[func]{hierOrder} + [class]{}-[func]{levelOrder} ``` === "C#" ```csharp title="binary_tree_bfs.cs" - [class]{binary_tree_bfs}-[func]{hierOrder} + [class]{binary_tree_bfs}-[func]{levelOrder} ``` === "Swift" ```swift title="binary_tree_bfs.swift" - [class]{}-[func]{hierOrder} + [class]{}-[func]{levelOrder} ``` === "Zig" ```zig title="binary_tree_bfs.zig" - [class]{}-[func]{hierOrder} + [class]{}-[func]{levelOrder} ``` +### 复杂度分析 + +**时间复杂度**:所有结点被访问一次,使用 $O(n)$ 时间,其中 $n$ 为结点数量。 + +**空间复杂度**:当为满二叉树时达到最差情况,遍历到最底层前,队列中最多同时存在 $\frac{n + 1}{2}$ 个结点,使用 $O(n)$ 空间。 + ## 7.2.2. 前序、中序、后序遍历 相对地,前、中、后序遍历皆属于「深度优先遍历 Depth-First Traversal」,其体现着一种“先走到尽头,再回头继续”的回溯遍历方式。 @@ -100,13 +108,15 @@ comments: true +### 实现代码 + === "Java" ```java title="binary_tree_dfs.java" [class]{binary_tree_dfs}-[func]{preOrder} - + [class]{binary_tree_dfs}-[func]{inOrder} - + [class]{binary_tree_dfs}-[func]{postOrder} ``` @@ -114,9 +124,9 @@ comments: true ```cpp title="binary_tree_dfs.cpp" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -124,9 +134,9 @@ comments: true ```python title="binary_tree_dfs.py" [class]{}-[func]{pre_order} - + [class]{}-[func]{in_order} - + [class]{}-[func]{post_order} ``` @@ -134,9 +144,9 @@ comments: true ```go title="binary_tree_dfs.go" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -144,9 +154,9 @@ comments: true ```javascript title="binary_tree_dfs.js" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -154,9 +164,9 @@ comments: true ```typescript title="binary_tree_dfs.ts" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -164,9 +174,9 @@ comments: true ```c title="binary_tree_dfs.c" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -174,9 +184,9 @@ comments: true ```csharp title="binary_tree_dfs.cs" [class]{binary_tree_dfs}-[func]{preOrder} - + [class]{binary_tree_dfs}-[func]{inOrder} - + [class]{binary_tree_dfs}-[func]{postOrder} ``` @@ -184,9 +194,9 @@ comments: true ```swift title="binary_tree_dfs.swift" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` @@ -194,12 +204,18 @@ comments: true ```zig title="binary_tree_dfs.zig" [class]{}-[func]{preOrder} - + [class]{}-[func]{inOrder} - + [class]{}-[func]{postOrder} ``` !!! note 使用循环一样可以实现前、中、后序遍历,但代码相对繁琐,有兴趣的同学可以自行实现。 + +### 复杂度分析 + +**时间复杂度**:所有结点被访问一次,使用 $O(n)$ 时间,其中 $n$ 为结点数量。 + +**空间复杂度**:当树退化为链表时达到最差情况,递归深度达到 $n$ ,系统使用 $O(n)$ 栈帧空间。 From 8e0080f003195d196ce1e0c2098a21dd746e9a66 Mon Sep 17 00:00:00 2001 From: krahets Date: Wed, 15 Feb 2023 03:35:54 +0800 Subject: [PATCH 7/8] Update graph codes --- .../chapter_graph/graph_adjacency_list.cpp | 11 +-- .../go/chapter_graph/graph_adjacency_list.go | 8 +-- codes/java/chapter_graph/graph_traversal.java | 70 ------------------- .../chapter_graph/graph_adjacency_list.js | 9 ++- .../chapter_graph/graph_adjacency_list.swift | 12 ++-- .../chapter_graph/graph_adjacency_list.ts | 9 ++- 6 files changed, 29 insertions(+), 90 deletions(-) delete mode 100644 codes/java/chapter_graph/graph_traversal.java diff --git a/codes/cpp/chapter_graph/graph_adjacency_list.cpp b/codes/cpp/chapter_graph/graph_adjacency_list.cpp index a7f249d70..5a38b3925 100644 --- a/codes/cpp/chapter_graph/graph_adjacency_list.cpp +++ b/codes/cpp/chapter_graph/graph_adjacency_list.cpp @@ -14,8 +14,9 @@ struct Vertex { /* 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 - unordered_map> adjList; // 邻接表(使用哈希表实现) + // 邻接表,使用哈希表来代替链表,以提升删除边、删除顶点的效率 + // 请注意,adjList 中的元素是 Vertex 对象 + unordered_map> adjList; public: /* 构造方法 */ @@ -52,7 +53,7 @@ public: /* 添加顶点 */ void addVertex(Vertex* vet) { if (adjList.count(vet)) return; - // 在邻接表中添加一个新链表(即 HashSet) + // 在邻接表中添加一个新链表 adjList[vet] = unordered_set(); } @@ -60,9 +61,9 @@ public: void removeVertex(Vertex* vet) { if (!adjList.count(vet)) throw invalid_argument("不存在顶点"); - // 在邻接表中删除顶点 vet 对应的链表(即 HashSet) + // 在邻接表中删除顶点 vet 对应的链表 adjList.erase(vet); - // 遍历其它顶点的链表(即 HashSet),删除所有包含 vet 的边 + // 遍历其它顶点的链表,删除所有包含 vet 的边 for (auto& [key, set_] : adjList) { set_.erase(vet); } diff --git a/codes/go/chapter_graph/graph_adjacency_list.go b/codes/go/chapter_graph/graph_adjacency_list.go index 887384dd8..0d5916e37 100644 --- a/codes/go/chapter_graph/graph_adjacency_list.go +++ b/codes/go/chapter_graph/graph_adjacency_list.go @@ -24,8 +24,8 @@ func newVertex(val int) vertex { /* 基于邻接表实现的无向图类 */ type graphAdjList struct { - // 请注意,vertices 和 adjList 中存储的都是 Vertex 对象 - // 邻接表(使用哈希表实现), 使用哈希表模拟集合 + // 邻接表,使用哈希表来代替链表,以提升删除边、删除顶点的效率 + // 请注意,adjList 中的元素是 Vertex 对象 adjList map[vertex]map[vertex]struct{} } @@ -78,7 +78,7 @@ func (g *graphAdjList) addVertex(vet vertex) { if ok { return } - // 在邻接表中添加一个新链表(即 set) + // 在邻接表中添加一个新链表 g.adjList[vet] = make(map[vertex]struct{}) } @@ -90,7 +90,7 @@ func (g *graphAdjList) removeVertex(vet vertex) { } // 在邻接表中删除顶点 vet 对应的链表 delete(g.adjList, vet) - // 遍历其它顶点的链表(即 Set),删除所有包含 vet 的边 + // 遍历其它顶点的链表,删除所有包含 vet 的边 for _, set := range g.adjList { // 操作 delete(set, vet) diff --git a/codes/java/chapter_graph/graph_traversal.java b/codes/java/chapter_graph/graph_traversal.java deleted file mode 100644 index 3b152198d..000000000 --- a/codes/java/chapter_graph/graph_traversal.java +++ /dev/null @@ -1,70 +0,0 @@ -/** - * File: graph_adjacency_list.java - * Created Time: 2023-02-12 - * Author: Krahets (krahets@163.com) - */ - -package chapter_graph; - -import java.util.*; - -public class graph_traversal { - /* 以顶点 vet 为起点,对图 graph 执行广度优先遍历 */ - // 采用 GraphAdjList 表示图,以方便获取指定结点的所有邻接结点 - static List graphBFS(GraphAdjList graph, Vertex startVet) { - // 顶点遍历序列 - List res = new ArrayList<>(); - // 用于记录顶点是否已被访问 - Set visited = new HashSet<>() {{ add(startVet); }}; - // 队列用于实现 BFS - Queue que = new LinkedList<>() {{ offer(startVet); }}; - // 循环直至访问完所有顶点 - while (!que.isEmpty()) { - Vertex vet = que.poll(); // 队首顶点出队 - res.add(vet); // 访问该顶点 - // 遍历该顶点的所有邻接顶点 - for (Vertex adjVet : graph.adjList.get(vet)) { - if (!visited.contains(adjVet)) { - que.offer(adjVet); // 只入队未访问的顶点 - visited.add(adjVet); // 标记该顶点已被访问 - } - } - } - // 返回顶点遍历序列 - return res; - } - - /* 输入值列表 vals ,返回顶点列表 vets */ - static Vertex[] valsToVets(int[] vals) { - Vertex[] vets = new Vertex[vals.length]; - for (int i = 0; i < vals.length; i++) { - vets[i] = new Vertex(vals[i]); - } - return vets; - } - - /* 输入顶点列表 vets ,返回值列表 vals */ - static List vetsToVals(List vets) { - List vals = new ArrayList<>(); - for (Vertex vet : vets) { - vals.add(vet.val); - } - return vals; - } - - public static void main(String[] args) { - /* 初始化无向图 */ - Vertex[] v = valsToVets(new int[] { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }); - Vertex[][] edges = { { v[0], v[1] }, { v[0], v[3] }, { v[1], v[2] }, { v[1], v[4] }, - { v[2], v[5] }, { v[3], v[4] }, { v[3], v[6] }, { v[4], v[5] }, - { v[4], v[7] }, { v[5], v[8] }, { v[6], v[7] }, { v[7], v[8] }}; - GraphAdjList graph = new GraphAdjList(edges); - System.out.println("\n初始化后,图为"); - graph.print(); - - /* 广度优先遍历 BFS */ - List res = graphBFS(graph, v[0]); - System.out.println("\n广度优先遍历(BFS)顶点序列为"); - System.out.println(vetsToVals(res)); - } -} diff --git a/codes/javascript/chapter_graph/graph_adjacency_list.js b/codes/javascript/chapter_graph/graph_adjacency_list.js index 586e4a06b..86b934ca1 100644 --- a/codes/javascript/chapter_graph/graph_adjacency_list.js +++ b/codes/javascript/chapter_graph/graph_adjacency_list.js @@ -14,7 +14,10 @@ class Vertex { /* 基于邻接表实现的无向图类 */ class GraphAdjList { + // 邻接表,使用哈希表来代替链表,以提升删除边、删除顶点的效率 + // 请注意,adjList 中的元素是 Vertex 对象 adjList; + /* 构造方法 */ constructor(edges) { this.adjList = new Map(); @@ -54,7 +57,7 @@ class GraphAdjList { /* 添加顶点 */ addVertex(vet) { if (this.adjList.has(vet)) return; - // 在邻接表中添加一个新链表(即 HashSet) + // 在邻接表中添加一个新链表 this.adjList.set(vet, new Set()); } @@ -63,9 +66,9 @@ class GraphAdjList { if (!this.adjList.has(vet)) { throw new Error("Illegal Argument Exception"); } - // 在邻接表中删除顶点 vet 对应的链表(即 HashSet) + // 在邻接表中删除顶点 vet 对应的链表 this.adjList.delete(vet); - // 遍历其它顶点的链表(即 HashSet),删除所有包含 vet 的边 + // 遍历其它顶点的链表,删除所有包含 vet 的边 for (let set of this.adjList.values()) { set.delete(vet); } diff --git a/codes/swift/chapter_graph/graph_adjacency_list.swift b/codes/swift/chapter_graph/graph_adjacency_list.swift index 7185c542d..c5cafeb7e 100644 --- a/codes/swift/chapter_graph/graph_adjacency_list.swift +++ b/codes/swift/chapter_graph/graph_adjacency_list.swift @@ -23,9 +23,11 @@ class Vertex: Hashable { /* 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 - private var adjList: [Vertex: Set] // 邻接表(使用哈希表实现) + // 邻接表,使用哈希表来代替链表,以提升删除边、删除顶点的效率 + // 请注意,adjList 中的元素是 Vertex 对象 + private var adjList: [Vertex: Set] + /* 构造方法 */ init(edges: [[Vertex]]) { adjList = [:] // 添加所有顶点和边 @@ -66,7 +68,7 @@ class GraphAdjList { if adjList[vet] != nil { return } - // 在邻接表中添加一个新链表(即 HashSet) + // 在邻接表中添加一个新链表 adjList[vet] = [] } @@ -75,9 +77,9 @@ class GraphAdjList { if adjList[vet] == nil { fatalError("参数错误") } - // 在邻接表中删除顶点 vet 对应的链表(即 HashSet) + // 在邻接表中删除顶点 vet 对应的链表 adjList.removeValue(forKey: vet) - // 遍历其它顶点的链表(即 HashSet),删除所有包含 vet 的边 + // 遍历其它顶点的链表,删除所有包含 vet 的边 for key in adjList.keys { adjList[key]?.remove(vet) } diff --git a/codes/typescript/chapter_graph/graph_adjacency_list.ts b/codes/typescript/chapter_graph/graph_adjacency_list.ts index 66fc8de9c..fd1e6fecc 100644 --- a/codes/typescript/chapter_graph/graph_adjacency_list.ts +++ b/codes/typescript/chapter_graph/graph_adjacency_list.ts @@ -14,7 +14,10 @@ class Vertex { /* 基于邻接表实现的无向图类 */ class GraphAdjList { + // 邻接表,使用哈希表来代替链表,以提升删除边、删除顶点的效率 + // 请注意,adjList 中的元素是 Vertex 对象 adjList: Map>; + /* 构造方法 */ constructor(edges: Vertex[][]) { this.adjList = new Map(); @@ -54,7 +57,7 @@ class GraphAdjList { /* 添加顶点 */ addVertex(vet: Vertex): void { if (this.adjList.has(vet)) return; - // 在邻接表中添加一个新链表(即 HashSet) + // 在邻接表中添加一个新链表 this.adjList.set(vet, new Set()); } @@ -63,9 +66,9 @@ class GraphAdjList { if (!this.adjList.has(vet)) { throw new Error("Illegal Argument Exception"); } - // 在邻接表中删除顶点 vet 对应的链表(即 HashSet) + // 在邻接表中删除顶点 vet 对应的链表 this.adjList.delete(vet); - // 遍历其它顶点的链表(即 HashSet),删除所有包含 vet 的边 + // 遍历其它顶点的链表,删除所有包含 vet 的边 for (let set of this.adjList.values()) { set.delete(vet); } From 7238c560d1b43bb1767ecefc45bf2d4bd05e931b Mon Sep 17 00:00:00 2001 From: krahets Date: Wed, 15 Feb 2023 03:36:22 +0800 Subject: [PATCH 8/8] Update binary_tree_bfs codes --- codes/cpp/chapter_tree/binary_tree_bfs.cpp | 4 ++-- codes/csharp/chapter_tree/binary_tree_bfs.cs | 4 ++-- codes/go/chapter_tree/binary_tree_bfs.go | 2 +- codes/go/chapter_tree/binary_tree_bfs_test.go | 4 ++-- codes/java/chapter_tree/binary_tree_bfs.java | 4 ++-- codes/javascript/chapter_tree/binary_tree_bfs.js | 4 ++-- codes/python/chapter_tree/binary_tree_bfs.py | 4 ++-- codes/swift/chapter_tree/binary_tree_bfs.swift | 4 ++-- codes/typescript/chapter_tree/binary_tree_bfs.ts | 4 ++-- codes/zig/chapter_tree/binary_tree_bfs.zig | 4 ++-- 10 files changed, 19 insertions(+), 19 deletions(-) diff --git a/codes/cpp/chapter_tree/binary_tree_bfs.cpp b/codes/cpp/chapter_tree/binary_tree_bfs.cpp index c3694eb9c..8b2ba2d2e 100644 --- a/codes/cpp/chapter_tree/binary_tree_bfs.cpp +++ b/codes/cpp/chapter_tree/binary_tree_bfs.cpp @@ -7,7 +7,7 @@ #include "../include/include.hpp" /* 层序遍历 */ -vector hierOrder(TreeNode* root) { +vector levelOrder(TreeNode* root) { // 初始化队列,加入根结点 queue queue; queue.push(root); @@ -35,7 +35,7 @@ int main() { PrintUtil::printTree(root); /* 层序遍历 */ - vector vec = hierOrder(root); + vector vec = levelOrder(root); cout << endl << "层序遍历的结点打印序列 = "; PrintUtil::printVector(vec); diff --git a/codes/csharp/chapter_tree/binary_tree_bfs.cs b/codes/csharp/chapter_tree/binary_tree_bfs.cs index 0cbbf62f5..2edd07eef 100644 --- a/codes/csharp/chapter_tree/binary_tree_bfs.cs +++ b/codes/csharp/chapter_tree/binary_tree_bfs.cs @@ -13,7 +13,7 @@ public class binary_tree_bfs { /* 层序遍历 */ - public List hierOrder(TreeNode root) + public List levelOrder(TreeNode root) { // 初始化队列,加入根结点 Queue queue = new(); @@ -41,7 +41,7 @@ public class binary_tree_bfs Console.WriteLine("\n初始化二叉树\n"); PrintUtil.PrintTree(root); - List list = hierOrder(root); + List list = levelOrder(root); Console.WriteLine("\n层序遍历的结点打印序列 = " + string.Join(",", list.ToArray())); } } diff --git a/codes/go/chapter_tree/binary_tree_bfs.go b/codes/go/chapter_tree/binary_tree_bfs.go index c10f618e0..8ec46fbe7 100644 --- a/codes/go/chapter_tree/binary_tree_bfs.go +++ b/codes/go/chapter_tree/binary_tree_bfs.go @@ -11,7 +11,7 @@ import ( ) /* 层序遍历 */ -func hierOrder(root *TreeNode) []int { +func levelOrder(root *TreeNode) []int { // 初始化队列,加入根结点 queue := list.New() queue.PushBack(root) diff --git a/codes/go/chapter_tree/binary_tree_bfs_test.go b/codes/go/chapter_tree/binary_tree_bfs_test.go index eafd51bcc..7b1bbf1f5 100644 --- a/codes/go/chapter_tree/binary_tree_bfs_test.go +++ b/codes/go/chapter_tree/binary_tree_bfs_test.go @@ -11,7 +11,7 @@ import ( . "github.com/krahets/hello-algo/pkg" ) -func TestHierOrder(t *testing.T) { +func TestLevelOrder(t *testing.T) { /* 初始化二叉树 */ // 这里借助了一个从数组直接生成二叉树的函数 root := ArrToTree([]any{1, 2, 3, 4, 5, 6, 7}) @@ -19,6 +19,6 @@ func TestHierOrder(t *testing.T) { PrintTree(root) // 层序遍历 - nums := hierOrder(root) + nums := levelOrder(root) fmt.Println("\n层序遍历的结点打印序列 =", nums) } diff --git a/codes/java/chapter_tree/binary_tree_bfs.java b/codes/java/chapter_tree/binary_tree_bfs.java index 578ed2b81..246921dc6 100644 --- a/codes/java/chapter_tree/binary_tree_bfs.java +++ b/codes/java/chapter_tree/binary_tree_bfs.java @@ -11,7 +11,7 @@ import java.util.*; public class binary_tree_bfs { /* 层序遍历 */ - static List hierOrder(TreeNode root) { + static List levelOrder(TreeNode root) { // 初始化队列,加入根结点 Queue queue = new LinkedList<>() {{ add(root); }}; // 初始化一个列表,用于保存遍历序列 @@ -35,7 +35,7 @@ public class binary_tree_bfs { PrintUtil.printTree(root); /* 层序遍历 */ - List list = hierOrder(root); + List list = levelOrder(root); System.out.println("\n层序遍历的结点打印序列 = " + list); } } diff --git a/codes/javascript/chapter_tree/binary_tree_bfs.js b/codes/javascript/chapter_tree/binary_tree_bfs.js index 008ace0ad..c400853ea 100644 --- a/codes/javascript/chapter_tree/binary_tree_bfs.js +++ b/codes/javascript/chapter_tree/binary_tree_bfs.js @@ -8,7 +8,7 @@ const { arrToTree } = require("../include/TreeNode"); const { printTree } = require("../include/PrintUtil"); /* 层序遍历 */ -function hierOrder(root) { +function levelOrder(root) { // 初始化队列,加入根结点 let queue = [root]; // 初始化一个列表,用于保存遍历序列 @@ -33,5 +33,5 @@ console.log("\n初始化二叉树\n"); printTree(root); /* 层序遍历 */ -let list = hierOrder(root); +let list = levelOrder(root); console.log("\n层序遍历的结点打印序列 = " + list); diff --git a/codes/python/chapter_tree/binary_tree_bfs.py b/codes/python/chapter_tree/binary_tree_bfs.py index 657fd51da..dff32bd82 100644 --- a/codes/python/chapter_tree/binary_tree_bfs.py +++ b/codes/python/chapter_tree/binary_tree_bfs.py @@ -10,7 +10,7 @@ from include import * """ 层序遍历 """ -def hier_order(root: Optional[TreeNode]): +def level_order(root: Optional[TreeNode]): # 初始化队列,加入根结点 queue = collections.deque() queue.append(root) @@ -35,6 +35,6 @@ if __name__ == "__main__": print_tree(root) # 层序遍历 - res = hier_order(root) + res = level_order(root) print("\n层序遍历的结点打印序列 = ", res) assert res == [1, 2, 3, 4, 5, 6, 7] diff --git a/codes/swift/chapter_tree/binary_tree_bfs.swift b/codes/swift/chapter_tree/binary_tree_bfs.swift index a0aa3cd00..326214a92 100644 --- a/codes/swift/chapter_tree/binary_tree_bfs.swift +++ b/codes/swift/chapter_tree/binary_tree_bfs.swift @@ -7,7 +7,7 @@ import utils /* 层序遍历 */ -func hierOrder(root: TreeNode) -> [Int] { +func levelOrder(root: TreeNode) -> [Int] { // 初始化队列,加入根结点 var queue: [TreeNode] = [root] // 初始化一个列表,用于保存遍历序列 @@ -36,7 +36,7 @@ enum BinaryTreeBFS { PrintUtil.printTree(root: node) /* 层序遍历 */ - let list = hierOrder(root: node) + let list = levelOrder(root: node) print("\n层序遍历的结点打印序列 = \(list)") } } diff --git a/codes/typescript/chapter_tree/binary_tree_bfs.ts b/codes/typescript/chapter_tree/binary_tree_bfs.ts index f723e0486..83fba5c8b 100644 --- a/codes/typescript/chapter_tree/binary_tree_bfs.ts +++ b/codes/typescript/chapter_tree/binary_tree_bfs.ts @@ -9,7 +9,7 @@ import { arrToTree } from '../module/TreeNode'; import { printTree } from '../module/PrintUtil'; /* 层序遍历 */ -function hierOrder(root: TreeNode | null): number[] { +function levelOrder(root: TreeNode | null): number[] { // 初始化队列,加入根结点 const queue = [root]; // 初始化一个列表,用于保存遍历序列 @@ -35,7 +35,7 @@ console.log('\n初始化二叉树\n'); printTree(root); /* 层序遍历 */ -const list = hierOrder(root); +const list = levelOrder(root); console.log('\n层序遍历的结点打印序列 = ' + list); export {}; diff --git a/codes/zig/chapter_tree/binary_tree_bfs.zig b/codes/zig/chapter_tree/binary_tree_bfs.zig index eaa0d0d5a..dc0cb6720 100644 --- a/codes/zig/chapter_tree/binary_tree_bfs.zig +++ b/codes/zig/chapter_tree/binary_tree_bfs.zig @@ -6,7 +6,7 @@ const std = @import("std"); const inc = @import("include"); // 层序遍历 -fn hierOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) { +fn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) { // 初始化队列,加入根结点 const L = std.TailQueue(*inc.TreeNode(T)); var queue = L{}; @@ -48,7 +48,7 @@ pub fn main() !void { try inc.PrintUtil.printTree(root, null, false); // 层序遍历 - var list = try hierOrder(i32, mem_allocator, root.?); + var list = try levelOrder(i32, mem_allocator, root.?); defer list.deinit(); std.debug.print("\n层序遍历的结点打印序列 = ", .{}); inc.PrintUtil.printList(i32, list);

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树代表的是“一对多”的关系,而图则自由度更高,可以代表任意“多对多”关系。本质上,**可以把树看作是图的一类特例**。那么显然,树遍历操作也是图遍历操作的一个特例,两者的方法是非常类似的,建议你在学习本章节的过程中将两者融会贯通。 - 本质上,可以把树看作是图的一种特例,即树是一种限制条件下的图。 +「图」与「树」都是非线性数据结构,都需要使用「搜索算法」来实现遍历操作。 + +类似地,图的遍历方式也分为两种,即「广度优先遍历 Breadth-First Traversal」和「深度优先遍历 Depth-First Travsersal」,也称「广度优先搜索 Breadth-First Search」和「深度优先搜索 Depth-First Search」,简称为 BFS 和 DFS 。 ## 广度优先遍历 -广度优先遍历(BFS)代表一种优先遍历最近的顶点、一层层向外扩张的遍历方式。具体来看,从某个顶点出发,则优先遍历该顶点的所有邻接顶点,随后遍历下个顶点的所有邻接顶点,以此类推…… +**广度优先遍历优是一种由近及远的遍历方式,从距离最近的顶点开始访问,并一层层向外扩张**。具体地,从某个顶点出发,先遍历该顶点的所有邻接顶点,随后遍历下个顶点的所有邻接顶点,以此类推…… + +![graph_bfs](graph_traversal.assets/graph_bfs.png) + +### 算法实现 + +BFS 常借助「队列」来实现。队列具有“先入先出”的性质,这与 BFS “由近及远”的思想是异曲同工的。 + +1. 将遍历起始顶点 `startVet` 加入队列,并开启循环; +2. 在循环的每轮迭代中,弹出队首顶点弹出并记录访问,并将该顶点的所有邻接顶点加入到队列尾部; +3. 循环 `2.` ,直到所有顶点访问完成后结束。 + +为了防止重复遍历顶点,我们需要借助一个哈希表 `visited` 来记录哪些结点已被访问。 + +=== "Java" + + ```java title="graph_bfs.java" + [class]{graph_bfs}-[func]{graphBFS} + ``` + +=== "C++" + + ```cpp title="graph_bfs.cpp" + + ``` + +=== "Python" + + ```python title="graph_bfs.py" + + ``` + +=== "Go" + + ```go title="graph_bfs.go" + + ``` + +=== "JavaScript" + + ```javascript title="graph_bfs.js" + + ``` + +=== "TypeScript" + + ```typescript title="graph_bfs.ts" + + ``` + +=== "C" + + ```c title="graph_bfs.c" -(图) + ``` -BFS 常借助「队列」来实现,队列具有“先入先出”的性质,这与 BFS 的“由近及远”的遍历方式是异曲同工的。具体地,在每轮迭代中弹出队首顶点且访问之,并将该顶点的所有邻接顶点加入到队列尾部,直到所有顶点访问完成即可。 +=== "C#" -为了防止重复遍历顶点,我们需要借助一个 HashSet 来记录哪些结点已被访问,从而避免走“回头路”。 + ```csharp title="graph_bfs.cs" -```java + ``` -``` +=== "Swift" + ```swift title="graph_bfs.swift" + ``` + +=== "Zig" + + ```zig title="graph_bfs.zig" + + ``` + +代码相对抽象,建议对照以下动画图示来加深理解。 + +=== "Step 1" + ![graph_bfs_step1](graph_traversal.assets/graph_bfs_step1.png) + +=== "Step 2" + ![graph_bfs_step2](graph_traversal.assets/graph_bfs_step2.png) + +=== "Step 3" + ![graph_bfs_step3](graph_traversal.assets/graph_bfs_step3.png) + +=== "Step 4" + ![graph_bfs_step4](graph_traversal.assets/graph_bfs_step4.png) + +=== "Step 5" + ![graph_bfs_step5](graph_traversal.assets/graph_bfs_step5.png) + +=== "Step 6" + ![graph_bfs_step6](graph_traversal.assets/graph_bfs_step6.png) + +=== "Step 7" + ![graph_bfs_step7](graph_traversal.assets/graph_bfs_step7.png) + +=== "Step 8" + ![graph_bfs_step8](graph_traversal.assets/graph_bfs_step8.png) + +=== "Step 9" + ![graph_bfs_step9](graph_traversal.assets/graph_bfs_step9.png) + +=== "Step 10" + ![graph_bfs_step10](graph_traversal.assets/graph_bfs_step10.png) + +=== "Step 11" + ![graph_bfs_step11](graph_traversal.assets/graph_bfs_step11.png) + +!!! question "广度优先遍历的序列是否唯一?" + + 不唯一。广度优先遍历只要求“由近及远”,而相同距离的多个顶点的遍历顺序允许任意被打乱。以上图为例,顶点 $1$ , $3$ 的访问顺序可以交换、顶点 $2$ , $4$ , $6$ 的访问顺序也可以任意交换、以此类推…… + +### 复杂度分析 + +**时间复杂度:** 所有顶点都会入队、出队一次,使用 $O(|V|)$ 时间;在遍历邻接顶点的过程中,由于是无向图,因此所有边都会被访问 $2$ 次,使用 $O(2|E|)$ 时间;总体使用 $O(|V| + |E|)$ 时间。 + +**空间复杂度:** 列表 `res` ,哈希表 `visited` ,队列 `que` 中的顶点数量最多为 $|V|$ ,使用 $O(|V|)$ 空间。 ## 深度优先遍历 -深度优先遍历(DFS)代表一种优先走到底,无路可走再回头的遍历方式。从某个顶点出发,首先不断地通过指针向下一个顶点遍历,直到走到头开始回溯,再继续走到底 + 回溯,以此类推…… \ No newline at end of file +**深度优先遍历是一种优先走到底、无路可走再回头的遍历方式**。具体地,从某个顶点出发,不断地访问当前结点的某个邻接顶点,直到走到尽头时回溯,再继续走到底 + 回溯,以此类推……直至所有顶点遍历完成时结束。 + +![graph_dfs](graph_traversal.assets/graph_dfs.png) + +### 算法实现 + +这种“走到头 + 回溯”的算法形式一般基于递归来实现。与 BFS 类似,在 DFS 中我们也需要借助一个哈希表 `visited` 来记录已被访问的顶点,以避免重复访问顶点。 + +=== "Java" + + ```java title="graph_dfs.java" + [class]{graph_dfs}-[func]{dfs} + + [class]{graph_dfs}-[func]{graphDFS} + ``` + +=== "C++" + + ```cpp title="graph_dfs.cpp" + + ``` + +=== "Python" + + ```python title="graph_dfs.py" + + ``` + +=== "Go" + + ```go title="graph_dfs.go" + + ``` + +=== "JavaScript" + + ```javascript title="graph_dfs.js" + + ``` + +=== "TypeScript" + + ```typescript title="graph_dfs.ts" + + ``` + +=== "C" + + ```c title="graph_dfs.c" + + ``` + +=== "C#" + + ```csharp title="graph_dfs.cs" + + ``` + +=== "Swift" + + ```swift title="graph_dfs.swift" + + ``` + +=== "Zig" + + ```zig title="graph_dfs.zig" + + ``` + +深度优先遍历的算法流程如下图所示,其中 + +- **直虚线代表向下递推**,代表开启了一个新的递归方法来访问新顶点; +- **曲虚线代表向上回溯**,代表此递归方法已经返回,回溯到了开启此递归方法的位置; + +为了加深理解,请你将图示与代码结合起来,在脑中(或者用笔画下来)模拟整个 DFS 过程,包括每个递归方法何时开启、何时返回。 + +=== "Step 1" + ![graph_dfs_step1](graph_traversal.assets/graph_dfs_step1.png) + +=== "Step 2" + ![graph_dfs_step2](graph_traversal.assets/graph_dfs_step2.png) + +=== "Step 3" + ![graph_dfs_step3](graph_traversal.assets/graph_dfs_step3.png) + +=== "Step 4" + ![graph_dfs_step4](graph_traversal.assets/graph_dfs_step4.png) + +=== "Step 5" + ![graph_dfs_step5](graph_traversal.assets/graph_dfs_step5.png) + +=== "Step 6" + ![graph_dfs_step6](graph_traversal.assets/graph_dfs_step6.png) + +=== "Step 7" + ![graph_dfs_step7](graph_traversal.assets/graph_dfs_step7.png) + +=== "Step 8" + ![graph_dfs_step8](graph_traversal.assets/graph_dfs_step8.png) + +=== "Step 9" + ![graph_dfs_step9](graph_traversal.assets/graph_dfs_step9.png) + +=== "Step 10" + ![graph_dfs_step10](graph_traversal.assets/graph_dfs_step10.png) + +=== "Step 11" + ![graph_dfs_step11](graph_traversal.assets/graph_dfs_step11.png) + +!!! question "深度优先遍历的序列是否唯一?" + + 与广度优先遍历类似,深度优先遍历序列的顺序也不是唯一的。给定某顶点,先往哪个方向探索都行,都是深度优先遍历。 + + 以树的遍历为例,“根 $\rightarrow$ 左 $\rightarrow$ 右”、“左 $\rightarrow$ 根 $\rightarrow$ 右”、“左 $\rightarrow$ 右 $\rightarrow$ 根”分别对应前序、中序、后序遍历,体现三种不同的遍历优先级,而三者都属于深度优先遍历。 + +### 复杂度分析 + +**时间复杂度:** 所有顶点都被访问一次;所有边都被访问了 $2$ 次,使用 $O(2|E|)$ 时间;总体使用 $O(|V| + |E|)$ 时间。 + +**空间复杂度:** 列表 `res` ,哈希表 `visited` 顶点数量最多为 $|V|$ ,递归深度最大为 $|V|$ ,因此使用 $O(|V|)$ 空间。 From 9ad87617a05ab622b665401969f560da37db5fad Mon Sep 17 00:00:00 2001 From: krahets Date: Wed, 15 Feb 2023 03:24:26 +0800 Subject: [PATCH 3/8] Add missing Vertex.java --- codes/java/include/Vertex.java | 35 ++++++++++++++++++++++++++++++++++ 1 file changed, 35 insertions(+) create mode 100644 codes/java/include/Vertex.java diff --git a/codes/java/include/Vertex.java b/codes/java/include/Vertex.java new file mode 100644 index 000000000..80c59cfd8 --- /dev/null +++ b/codes/java/include/Vertex.java @@ -0,0 +1,35 @@ +/** + * File: TreeNode.java + * Created Time: 2022-11-25 + * Author: Krahets (krahets@163.com) + */ + +package include; + +import java.util.*; + +/* 顶点类 */ +public class Vertex { + public int val; + public Vertex(int val) { + this.val = val; + } + + /* 输入值列表 vals ,返回顶点列表 vets */ + public static Vertex[] valsToVets(int[] vals) { + Vertex[] vets = new Vertex[vals.length]; + for (int i = 0; i < vals.length; i++) { + vets[i] = new Vertex(vals[i]); + } + return vets; + } + + /* 输入顶点列表 vets ,返回值列表 vals */ + public static List vetsToVals(List vets) { + List vals = new ArrayList<>(); + for (Vertex vet : vets) { + vals.add(vet.val); + } + return vals; + } +} \ No newline at end of file From 416a72259439057247c8622de2ee84415002b59f Mon Sep 17 00:00:00 2001 From: krahets Date: Wed, 15 Feb 2023 03:25:08 +0800 Subject: [PATCH 4/8] Add mkdocs.yml --- mkdocs.yml | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/mkdocs.yml b/mkdocs.yml index d52f01d37..aa57e0942 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -166,7 +166,8 @@ nav: - 8.1. 堆(Heap): chapter_heap/heap.md - 9. 图: - 9.1. 图(Graph): chapter_graph/graph.md - - 9.2. 图基础操作: chapter_graph/graph_operations.md + - 9.2. 图基础操作: chapter_graph/graph_operations.md + - 9.3. 图的遍历: chapter_graph/graph_traversal.md - 10. 查找算法: - 10.1. 线性查找: chapter_searching/linear_search.md - 10.2. 二分查找: chapter_searching/binary_search.md From ae53c59183e1093d44905a84120ee96cee0208bf Mon Sep 17 00:00:00 2001 From: krahets Date: Wed, 15 Feb 2023 03:26:09 +0800 Subject: [PATCH 5/8] Update numbering --- docs/chapter_graph/graph_traversal.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs/chapter_graph/graph_traversal.md b/docs/chapter_graph/graph_traversal.md index a9c953672..fba7bb3e9 100644 --- a/docs/chapter_graph/graph_traversal.md +++ b/docs/chapter_graph/graph_traversal.md @@ -2,7 +2,7 @@ comments: true --- -# 图的遍历 +# 9.3. 图的遍历 !!! note "图与树的关系" @@ -12,7 +12,7 @@ comments: true 类似地,图的遍历方式也分为两种,即「广度优先遍历 Breadth-First Traversal」和「深度优先遍历 Depth-First Travsersal」,也称「广度优先搜索 Breadth-First Search」和「深度优先搜索 Depth-First Search」,简称为 BFS 和 DFS 。 -## 广度优先遍历 +## 9.3.1. 广度优先遍历 **广度优先遍历优是一种由近及远的遍历方式,从距离最近的顶点开始访问,并一层层向外扩张**。具体地,从某个顶点出发,先遍历该顶点的所有邻接顶点,随后遍历下个顶点的所有邻接顶点,以此类推…… @@ -133,7 +133,7 @@ BFS 常借助「队列」来实现。队列具有“先入先出”的性质, **空间复杂度:** 列表 `res` ,哈希表 `visited` ,队列 `que` 中的顶点数量最多为 $|V|$ ,使用 $O(|V|)$ 空间。 -## 深度优先遍历 +## 9.3.2. 深度优先遍历 **深度优先遍历是一种优先走到底、无路可走再回头的遍历方式**。具体地,从某个顶点出发,不断地访问当前结点的某个邻接顶点,直到走到尽头时回溯,再继续走到底 + 回溯,以此类推……直至所有顶点遍历完成时结束。 From 5541fe59649d84ab448a4a4b5ea7a9da7912b8d1 Mon Sep 17 00:00:00 2001 From: krahets Date: Wed, 15 Feb 2023 03:32:59 +0800 Subject: [PATCH 6/8] Fix indentation and update array.md --- docs/chapter_array_and_linkedlist/array.md | 10 ++- docs/chapter_graph/graph_operations.md | 20 +++--- docs/chapter_tree/binary_tree_traversal.md | 78 +++++++++++++--------- 3 files changed, 64 insertions(+), 44 deletions(-) diff --git a/docs/chapter_array_and_linkedlist/array.md b/docs/chapter_array_and_linkedlist/array.md index e0abe19be..a19369d84 100755 --- a/docs/chapter_array_and_linkedlist/array.md +++ b/docs/chapter_array_and_linkedlist/array.md @@ -14,7 +14,7 @@ comments: true 观察上图,我们发现 **数组首元素的索引为 $0$** 。你可能会想,这并不符合日常习惯,首个元素的索引为什么不是 $1$ 呢,这不是更加自然吗?我认同你的想法,但请先记住这个设定,后面讲内存地址计算时,我会尝试解答这个问题。 -**数组有多种初始化写法**。根据实际需要,选代码最短的那一种就好。 +**数组初始化**。一般会用到无初始值、给定初始值两种写法,可根据需求选取。在不给定初始值的情况下,一般所有元素会被初始化为默认值 $0$ 。 === "Java" @@ -28,8 +28,12 @@ comments: true ```cpp title="array.cpp" /* 初始化数组 */ - int* arr = new int[5]; - int* nums = new int[5] { 1, 3, 2, 5, 4 }; + // 存储在栈上 + int arr[5]; + int nums[5] { 1, 3, 2, 5, 4 }; + // 存储在堆上 + int* arr1 = new int[5]; + int* nums1 = new int[5] { 1, 3, 2, 5, 4 }; ``` === "Python" diff --git a/docs/chapter_graph/graph_operations.md b/docs/chapter_graph/graph_operations.md index 69e1df191..0042a32d9 100644 --- a/docs/chapter_graph/graph_operations.md +++ b/docs/chapter_graph/graph_operations.md @@ -129,7 +129,7 @@ comments: true ```java title="graph_adjacency_list.java" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -137,7 +137,7 @@ comments: true ```cpp title="graph_adjacency_list.cpp" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -145,7 +145,7 @@ comments: true ```python title="graph_adjacency_list.py" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -153,7 +153,7 @@ comments: true ```go title="graph_adjacency_list.go" [class]{vertex}-[func]{} - + [class]{graphAdjList}-[func]{} ``` @@ -161,7 +161,7 @@ comments: true ```javascript title="graph_adjacency_list.js" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -169,7 +169,7 @@ comments: true ```typescript title="graph_adjacency_list.ts" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -177,7 +177,7 @@ comments: true ```c title="graph_adjacency_list.c" [class]{vertex}-[func]{} - + [class]{graphAdjList}-[func]{} ``` @@ -185,7 +185,7 @@ comments: true ```csharp title="graph_adjacency_list.cs" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -193,7 +193,7 @@ comments: true ```swift title="graph_adjacency_list.swift" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` @@ -201,7 +201,7 @@ comments: true ```zig title="graph_adjacency_list.zig" [class]{Vertex}-[func]{} - + [class]{GraphAdjList}-[func]{} ``` diff --git a/docs/chapter_tree/binary_tree_traversal.md b/docs/chapter_tree/binary_tree_traversal.md index 4267b71b4..f770d70f0 100755 --- a/docs/chapter_tree/binary_tree_traversal.md +++ b/docs/chapter_tree/binary_tree_traversal.md @@ -10,7 +10,7 @@ comments: true ## 7.2.1. 层序遍历 -「层序遍历 Hierarchical-Order Traversal」从顶至底、一层一层地遍历二叉树,并在每层中按照从左到右的顺序访问结点。 +「层序遍历 Level-Order Traversal」从顶至底、一层一层地遍历二叉树,并在每层中按照从左到右的顺序访问结点。 层序遍历本质上是「广度优先搜索 Breadth-First Traversal」,其体现着一种“一圈一圈向外”的层进遍历方式。 @@ -18,68 +18,76 @@ comments: true