算法：用Java实现图论算法中的拓扑排序，并应用拓扑排序安排课程表

本文我打算用Java实现图论算法中的拓扑排序，并应用拓扑排序安排课程表。课程表就是很常规的那种Java学习的简单路线图，首先下图就是Java课程学习的一个拓扑顺序图：

上面这张图是什么意思呢？黑色的字就代表要学习的课程，而红色的箭头就代表课程学习的拓扑顺序，课程之间是有依赖的，箭头代表课程学习顺序：

1. 最左边的Java、SQL、HTML、CSS这类课程则不需要提前学习任何课程，即可开始学习
2. 想学习Servlet课程，你就需要先学习Java课程，然后才能学习Servlet课程
3. 想学习JSP课程，你就得先学习Java课程和JavaScript课程，而学习JavaScript课程，就得先学习HTML和CSS课程

从上面的解读例子中，你应该明白了，课程学习是有依赖关系的，不是你想按照什么顺序来学都可以的。你只学了Servlet和JSP，没学过SQL，是无法学习SSM的（大家都懂的）。从上面的课程拓扑顺序可以看出，部分课程的学习都是在其他的课程学习完毕后，才能开始学习的，其依赖关系如下：

• Java：先决条件：无
• HTML：先决条件：无
• CSS：先决条件：无
• SQL：先决条件：无
• JavaScript：先决条件：HTML、CSS
• Servlet：先决条件：Java
• JSP：先决条件：Java、JavaScript
• SSM：先决条件：JSP、Servlet、SQL
• SpringBoot：先决条件：SSM

其实上面的课程拓扑顺序，也可以看做是一个有向无环图，也就是传说中的DAG图。如何按照上面的学习依赖顺序，安排所有课程的学习顺序呢？这里就用图论算法中的拓扑排序算法。

首先，我们构建上面的课程拓扑顺序图的邻接矩阵：

从上面的邻接矩阵图中可以看出，竖着的列，是1的就指着自己。比如SSM，竖着下来有3个1，横过去看，就是SQL、Servlet和JSP指着它。所以说，我们可以通过竖着数1的个数，来判断有多少课程指着自己，在图论中，就叫一个顶点的入度。在有向图中，如果有N条边的箭头指向某个顶点，就说这个顶点的入度是N。当然某一列的所有数字都是0，则代表这个顶点的入度是0，没有任何其他顶点指着它。

拓扑排序的思路如下：

1. 先找到入度为0的顶点（排序List中不能存在），就是顶点的那一列全是0，如果有多个，就取其一，放入排序List中
2. 放入后，在邻接矩阵中删除该顶点及其出去的边，也就是将顶点的那一行全部弄成0
3. 再继续找入度为0的顶点，重复3，然后再接着重复4，直到排序List中的排序元素个数与顶点个数相等，或者说找不到入度为0且排序List中不存在的顶点了，完成拓扑排序

接下来就是我用Java实现的拓扑排序，实现上面课程的顺序安排方案（所有可行的方案），算法的代码和精髓都在下面的代码和其间详尽的注释中：

import java.util.ArrayList;
import java.util.List;

/**
 * @author LiYang
 * @ClassName TopologicalSort
 * @Description 图论算法中的拓扑排序
 * @date 2019/11/18 18:33
 */
public class TopologicalSort {

    /**
     * 返回示例图的邻接矩阵
     * @return 示例图邻接矩阵
     */
    public static int[][] initCourseMatrix() {
        //创建示例图邻接矩阵
        return new int[][]{
                {0, 0, 0, 0, 0, 1, 1, 0, 0},
                {0, 0, 0, 0, 1, 0, 0, 0, 0},
                {0, 0, 0, 0, 1, 0, 0, 0, 0},
                {0, 0, 0, 0, 0, 0, 0, 1, 0},
                {0, 0, 0, 0, 0, 0, 1, 0, 0},
                {0, 0, 0, 0, 0, 0, 0, 1, 0},
                {0, 0, 0, 0, 0, 0, 0, 1, 0},
                {0, 0, 0, 0, 0, 0, 0, 0, 1},
                {0, 0, 0, 0, 0, 0, 0, 0, 0}
        };
    }

    /**
     * 拷贝matrix矩阵
     * @param matrix 源邻接矩阵
     * @return 拷贝的邻接矩阵
     */
    public static int[][] copyMatrix(int[][] matrix) {
        //新的邻接矩阵，和原来的一样大，正方形
        int[][] newMatrix = new int[matrix.length][matrix.length];
        
        //双重遍历拷贝
        for (int i = 0; i < matrix.length; i++) {
            for (int j = 0; j < matrix.length; j++) {
                newMatrix[i][j] = matrix[i][j];
            }
        }
        
        //返回拷贝的副本邻接矩阵
        return newMatrix;
    }

    /**
     * 拷贝List
     * @param order 源orderList
     * @return 拷贝的副本orderList
     */
    public static List<Integer> copyList(List<Integer> order) {
        //拷贝的List
        List<Integer> copyList = new ArrayList<>();
        
        //拷贝
        for (Integer item : order) {
            copyList.add(item);
        }
        
        //返回拷贝的副本
        return copyList;
    }

    /**
     * 打印课程顺序表
     * @param order 课程排序表
     * @param courseName 课程名字
     */
    public static void printCourseOrderTable(List<Integer> order, String[] courseName) {
        //课程表名字List
        List<String> courseTable = new ArrayList<>();
        
        //依据排好序的orderList，提取并加入课程名
        for (Integer item : order) {
            courseTable.add(courseName[item]);
        }
        
        //组装课程顺序表字符串
        String courseList = courseTable.toString().replaceAll("\\[", "")
                .replaceAll("]", "").replaceAll(", ", " -> ");
        
        //打印课程顺序表
        System.out.println(courseList);
    }

    /**
     * 拓扑排序（安排示例课程图的课程表）
     * @param matrix 课程拓扑顺序的邻接矩阵
     * @param order 课程顺序List
     * @param courseName 课程名称字符串数组
     */
    public static void topologicalSort(int[][] matrix, List<Integer> order, String[] courseName) {
        //如果所有课程已经排完序了
        if (order.size() == matrix.length) {
            
            //打印排好序的课程顺序表
            printCourseOrderTable(order, courseName);
            
            //结束递归
            return;
        }

        //遍历找入度为0的课程
        for (int i = 0; i < matrix.length; i++) {
            //如果当前课程已经在拓扑排序列表里面了 
            if (order.contains(i)) {
                //略过，不作处理
                continue;
            }
            
             //判断课程是否全是0，也就是入度为0
            boolean allZero = true;
            
            //遍历查看是否是入度为0
            for (int j = 0; j < matrix.length; j++) {
                if (matrix[j][i] > 0) {
                    allZero = false;
                    break;
                }
            }
            
            //如果课程入度为0，则可以用于拓扑排序的下一门课程
            if (allZero) {
                
                //复制当前排序表
                List<Integer> newOrder = copyList(order);
                
                //加入当前课程
                newOrder.add(i);
                
                //复制当前的邻接矩阵
                int[][] newMatrix = copyMatrix(matrix);

                //将当前已加入的课程删除，同时删除顶点和边
                for (int k = 0; k < matrix.length; k++) {
                    newMatrix[i][k] = 0;
                }
                
                //继续往下，递归进行拓扑排序
                topologicalSort(newMatrix, newOrder, courseName);
            }
        }
    }

    /**
     * 拓扑排序的驱动程序
     * @param matrix 课程拓扑顺序的邻接矩阵
     * @param courseName 课程名字
     */
    public static void topologicalSort(int[][] matrix, String[] courseName) {
        //新建一个课程排序表
        List<Integer> order = new ArrayList<>();
        
        //启动课程表的拓扑排序
        topologicalSort(matrix, order, courseName);
    }

    /**
     * 运行拓扑排序算法，求出示例图中的所有课程安排方案
     * @param args
     */
    public static void main(String[] args) {
        //获得示例图的邻接矩阵
        int[][] matrix = initCourseMatrix();
        
        //课程名称的字符串数组
        String[] courseName = new String[]{"Java", "HTML", "CSS", "SQL", "JavaScript", "Servlet", "JSP", "SSM", "SpringBoot"};
        
        //运行拓扑排序，求出示例图中的所有课程安排方案
        topologicalSort(matrix, courseName);
    }
    
}

运行TopologicalSort类的main方法，执行课程的拓扑排序算法，控制台打印了所有的可行的课程安排方案，测试通过：

Java -> HTML -> CSS -> SQL -> JavaScript -> Servlet -> JSP -> SSM -> SpringBoot
Java -> HTML -> CSS -> SQL -> JavaScript -> JSP -> Servlet -> SSM -> SpringBoot
Java -> HTML -> CSS -> SQL -> Servlet -> JavaScript -> JSP -> SSM -> SpringBoot
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