基于深度优先搜索的图遍历

基于深度优先搜索的无向图遍历

使用深度优先搜索遍历无向图，将无向图用邻接表存储：

算法流程图

1. 初始化起点 source，当前节点v为起点，终点 target，路径path为空，路径集合 paths 为空
2. 将当前节点v添加到 path 中
3. 判断当前节点v是否为终点，是转step4，否转step5
4. 保存 path 至 paths 中，转step7
5. 获取当前节点的所有邻接点，用集合N表示
6. 遍历N，若 N_i 不在 path 中，令v=N_i ，转step2；若N_i 在path 中，i +=1。
7. 删除 path 中最后一个节点,令v=path中最后一个节点，转step5
8. 以上步骤遍历了所有每一个点的邻接点，算法结束，输出起点到终点的所有路径paths

Python实现

from typing import List


def dfs(adjacent_list, source, target):
    """
    :param adjacent_list: 邻接表
    :param source: 起点
    :param target: 终点
    :return: 起点-终点的所有路径
    """

    def dfs_helper(adjacent_list, source, current_node, target):

        path.append(current_node)  # 压栈
        if current_node == target:
            paths.append(path.copy())
        else:
            neighbors = adjacent_list[current_node]
            for neighbor in neighbors:
                if neighbor not in path:
                    dfs_helper(adjacent_list, source, neighbor, target)
        path.pop()  # 弹栈

    paths = []
    path = []
    dfs_helper(adjacent_list, source, source, target)
    return paths


if __name__ == "__main__":
    # 邻接表
    adjacent_list = {
        1: [2, 3],
        2: [1, 4, 5],
        3: [1, 4, 7],
        4: [2, 3, 5, 6, 7],
        5: [2, 4, 6],
        6: [4, 5],
        7: [3, 4]
    }
    # 深搜
    paths: List[List] = dfs(adjacent_list, 1, 6)

    [print(path) for path in paths]

Java实现

package org.example;

import java.util.*;

public class DepthFirstSearch {
    //    List path = new ArrayList<>();
    Stack<Integer> path = new Stack<>();
    List<List<Integer>> paths = new ArrayList<>();

    void dfs(Map<Integer, List<Integer>> adjacent_list, int source, int current_node, int target) {
        path.push(current_node);
        if (current_node == target) {
            paths.add(new ArrayList<>(path));
            path.remove(path.size() - 1);
        } else {
            List<Integer> neighbors = adjacent_list.get(current_node);
            for (Integer neighbor : neighbors) {
                if (!path.contains(neighbor)) {
                    dfs(adjacent_list, source, neighbor, target);
                }
            }
            path.pop();
        }
    }

    public static void main(String[] args) {
        Map<Integer, List<Integer>> adjacent_list = new HashMap<>();
        adjacent_list.put(1, Arrays.asList(2, 3));
        adjacent_list.put(2, Arrays.asList(1, 4, 5));
        adjacent_list.put(3, Arrays.asList(1, 4, 7));
        adjacent_list.put(4, Arrays.asList(2, 3, 5, 6, 7));
        adjacent_list.put(5, Arrays.asList(2, 4, 6));
        adjacent_list.put(6, Arrays.asList(4, 5));
        adjacent_list.put(7, Arrays.asList(3, 4));
        System.out.println(adjacent_list);

        DepthFirstSearch dfs = new DepthFirstSearch();
        dfs.dfs(adjacent_list, 1, 1, 6);
        for (List<Integer> path : dfs.paths) {
            System.out.println(path);
        }

    }
}

基于深度优先搜索的有向图遍历

和无向图遍历一样，建立邻接矩阵即可。

Python实现

from typing import List, Tuple, Any, Dict
import networkx
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from typing import List


def paint_topological_graph(nodes,
                            edges: List[Tuple],
                            coordinates: Dict[Any, Tuple] = None,
                            directed=False
                            ):
    print(nodes)
    print(edges)
    print(coordinates)

    graph = networkx.DiGraph() if directed else networkx.Graph()  # 全连通 有向图
    graph.add_nodes_from(nodes)
    graph.add_edges_from(edges)
    networkx.draw(graph, pos=coordinates, with_labels=True, node_color='red', )

    plt.show()
    print(networkx.has_path(graph, 1, 12))
    return graph


def dfs(adjacent_list, source, target):
    """
    :param adjacent_list: 邻接表
    :param source: 起点
    :param target: 终点
    :return: 起点-终点的所有路径
    """

    def dfs_helper(adjacent_list, source, current_node, target):

        path.append(current_node)
        if current_node == target:
            paths.append(path.copy())
            path.pop()
        else:
            neighbors = adjacent_list[current_node]
            for neighbor in neighbors:
                if neighbor not in path:
                    dfs_helper(adjacent_list, source, neighbor, target)
            path.pop()

    paths = []
    path = []
    dfs_helper(adjacent_list, source, source, target)
    return paths


if __name__ == "__main__":
    # 点坐标
    node_coord = {
        1: (1, 0), 2: (1, 3), 3: (2.5, 3), 4: (2, 2.5), 5: (3, 2), 6: (2, 1.5), 7: (3, 0), 8: (6, 0), 9: (5.5, 2),
        10: (5.5, 3), 11: (6, 4), 12: (0, 0), 13: (0, 1), 14: (5.5, 0.5), 15: (4.5, 0.5), 16: (5, 5),
    }

    edges = [
        (13, 12), (1, 2), (2, 4), (2, 3), (4, 3), (4, 5), (1, 6), (1, 7), (6, 7), (6, 5), (7, 8), (5, 9), (5, 10),
        (3, 11), (11, 10), (9, 8), (10, 9), (8, 11), (14, 15), (8, 14), (12, 1), (11, 16),
    ]

    # 画图
    paint_topological_graph(nodes=np.arange(1, 17, 1),
                            edges=edges,
                            directed=True,
                            coordinates=node_coord
                            )
    # 邻接表
    adjacent_list = {
        1: [2, 6, 7],
        2: [3, 4],
        3: [11],
        4: [3, 5],
        5: [9, 10],
        6: [5, 7],
        7: [8],
        8: [11, 14],
        9: [8],
        10: [9],
        11: [10, 16],
        12: [1],
        13: [12],
        14: [15],
        15: [],
        16: [],
    }
    # 深搜
    paths: List[List] = dfs(adjacent_list, 1, 11)

    [print(path) for path in paths]