算法：用Java實現圖論算法中的拓撲排序，並應用拓撲排序安排課程表

本文我打算用Java實現圖論算法中的拓撲排序，並應用拓撲排序安排課程表。課程表就是很常規的那種Java學習的簡單路線圖，首先下圖就是Java課程學習的一個拓撲順序圖：

上面這張圖是什麼意思呢？黑色的字就代表要學習的課程，而紅色的箭頭就代表課程學習的拓撲順序，課程之間是有依賴的，箭頭代表課程學習順序：

1. 最左邊的Java、SQL、HTML、CSS這類課程則不需要提前學習任何課程，即可開始學習
2. 想學習Servlet課程，你就需要先學習Java課程，然後才能學習Servlet課程
3. 想學習JSP課程，你就得先學習Java課程和JavaScript課程，而學習JavaScript課程，就得先學習HTML和CSS課程

從上面的解讀例子中，你應該明白了，課程學習是有依賴關係的，不是你想按照什麼順序來學都可以的。你只學了Servlet和JSP，沒學過SQL，是無法學習SSM的（大家都懂的）。從上面的課程拓撲順序可以看出，部分課程的學習都是在其他的課程學習完畢後，才能開始學習的，其依賴關係如下：

• Java：先決條件：無
• HTML：先決條件：無
• CSS：先決條件：無
• SQL：先決條件：無
• JavaScript：先決條件：HTML、CSS
• Servlet：先決條件：Java
• JSP：先決條件：Java、JavaScript
• SSM：先決條件：JSP、Servlet、SQL
• SpringBoot：先決條件：SSM

其實上面的課程拓撲順序，也可以看做是一個有向無環圖，也就是傳說中的DAG圖。如何按照上面的學習依賴順序，安排所有課程的學習順序呢？這裏就用圖論算法中的拓撲排序算法。

首先，我們構建上面的課程拓撲順序圖的鄰接矩陣：

從上面的鄰接矩陣圖中可以看出，豎着的列，是1的就指着自己。比如SSM，豎着下來有3個1，橫過去看，就是SQL、Servlet和JSP指着它。所以說，我們可以通過豎着數1的個數，來判斷有多少課程指着自己，在圖論中，就叫一個頂點的入度。在有向圖中，如果有N條邊的箭頭指向某個頂點，就說這個頂點的入度是N。當然某一列的所有數字都是0，則代表這個頂點的入度是0，沒有任何其他頂點指着它。

拓撲排序的思路如下：

1. 先找到入度爲0的頂點（排序List中不能存在），就是頂點的那一列全是0，如果有多個，就取其一，放入排序List中
2. 放入後，在鄰接矩陣中刪除該頂點及其出去的邊，也就是將頂點的那一行全部弄成0
3. 再繼續找入度爲0的頂點，重複3，然後再接着重複4，直到排序List中的排序元素個數與頂點個數相等，或者說找不到入度爲0且排序List中不存在的頂點了，完成拓撲排序

接下來就是我用Java實現的拓撲排序，實現上面課程的順序安排方案（所有可行的方案），算法的代碼和精髓都在下面的代碼和其間詳盡的註釋中：

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

/**
 * @author LiYang
 * @ClassName TopologicalSort
 * @Description 圖論算法中的拓撲排序
 * @date 2019/11/18 18:33
 */
public class TopologicalSort {

    /**
     * 返回示例圖的鄰接矩陣
     * @return 示例圖鄰接矩陣
     */
    public static int[][] initCourseMatrix() {
        //創建示例圖鄰接矩陣
        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}
        };
    }

    /**
     * 拷貝matrix矩陣
     * @param matrix 源鄰接矩陣
     * @return 拷貝的鄰接矩陣
     */
    public static int[][] copyMatrix(int[][] matrix) {
        //新的鄰接矩陣，和原來的一樣大，正方形
        int[][] newMatrix = new int[matrix.length][matrix.length];
        
        //雙重遍歷拷貝
        for (int i = 0; i < matrix.length; i++) {
            for (int j = 0; j < matrix.length; j++) {
                newMatrix[i][j] = matrix[i][j];
            }
        }
        
        //返回拷貝的副本鄰接矩陣
        return newMatrix;
    }

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

    /**
     * 打印課程順序表
     * @param order 課程排序表
     * @param courseName 課程名字
     */
    public static void printCourseOrderTable(List<Integer> order, String[] courseName) {
        //課程表名字List
        List<String> courseTable = new ArrayList<>();
        
        //依據排好序的orderList，提取並加入課程名
        for (Integer item : order) {
            courseTable.add(courseName[item]);
        }
        
        //組裝課程順序表字符串
        String courseList = courseTable.toString().replaceAll("\\[", "")
                .replaceAll("]", "").replaceAll(", ", " -> ");
        
        //打印課程順序表
        System.out.println(courseList);
    }

    /**
     * 拓撲排序（安排示例課程圖的課程表）
     * @param matrix 課程拓撲順序的鄰接矩陣
     * @param order 課程順序List
     * @param courseName 課程名稱字符串數組
     */
    public static void topologicalSort(int[][] matrix, List<Integer> order, String[] courseName) {
        //如果所有課程已經排完序了
        if (order.size() == matrix.length) {
            
            //打印排好序的課程順序表
            printCourseOrderTable(order, courseName);
            
            //結束遞歸
            return;
        }

        //遍歷找入度爲0的課程
        for (int i = 0; i < matrix.length; i++) {
            //如果當前課程已經在拓撲排序列表裏面了 
            if (order.contains(i)) {
                //略過，不作處理
                continue;
            }
            
             //判斷課程是否全是0，也就是入度爲0
            boolean allZero = true;
            
            //遍歷查看是否是入度爲0
            for (int j = 0; j < matrix.length; j++) {
                if (matrix[j][i] > 0) {
                    allZero = false;
                    break;
                }
            }
            
            //如果課程入度爲0，則可以用於拓撲排序的下一門課程
            if (allZero) {
                
                //複製當前排序表
                List<Integer> newOrder = copyList(order);
                
                //加入當前課程
                newOrder.add(i);
                
                //複製當前的鄰接矩陣
                int[][] newMatrix = copyMatrix(matrix);

                //將當前已加入的課程刪除，同時刪除頂點和邊
                for (int k = 0; k < matrix.length; k++) {
                    newMatrix[i][k] = 0;
                }
                
                //繼續往下，遞歸進行拓撲排序
                topologicalSort(newMatrix, newOrder, courseName);
            }
        }
    }

    /**
     * 拓撲排序的驅動程序
     * @param matrix 課程拓撲順序的鄰接矩陣
     * @param courseName 課程名字
     */
    public static void topologicalSort(int[][] matrix, String[] courseName) {
        //新建一個課程排序表
        List<Integer> order = new ArrayList<>();
        
        //啓動課程表的拓撲排序
        topologicalSort(matrix, order, courseName);
    }

    /**
     * 運行拓撲排序算法，求出示例圖中的所有課程安排方案
     * @param args
     */
    public static void main(String[] args) {
        //獲得示例圖的鄰接矩陣
        int[][] matrix = initCourseMatrix();
        
        //課程名稱的字符串數組
        String[] courseName = new String[]{"Java", "HTML", "CSS", "SQL", "JavaScript", "Servlet", "JSP", "SSM", "SpringBoot"};
        
        //運行拓撲排序，求出示例圖中的所有課程安排方案
        topologicalSort(matrix, courseName);
    }
    
}

運行TopologicalSort類的main方法，執行課程的拓撲排序算法，控制檯打印了所有的可行的課程安排方案，測試通過：

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