所有结点对的最短路径问题（Java语言描述）

所有结点对的最短路径问题

推荐阅读——《所有结点对的最短路径问题》

常量定义

private static final int NOT_A_VERTEX = -1;

算法实现

public static void allPairs(int[][] a, int[][] d, int[][] path) {
    int n = a.length;
    // Initialize d and path
    for(int i = 0; i < n; i++) {
        for(int j = 0; j < n; j++) {
            d[i][j] = a[i][j];
            path[i][j] = NOT_A_VERTEX;
        }
    }
    for(int k = 0; k < n; k++) {
        // Consider each vertex as an intermediate
        for(int i = 0; i < n; i++) {
            for(int j = 0; j < n; j++) {
                if(d[i][k] + d[k][j] < d[i][j]) {
                    // Update shortest path
                    d[i][j] = d[i][k] + d[k][j];
                    path[i][j] = k;
                }
            }
        }
    }
}

说明

上面是计算所有结点对的最短路径的算法。
a[][] 包含邻接矩阵，其中 a[i][i] 假定为零。
d[] 包含最短路径的值。
顶点从0开始编号； 所有数组的维数相等。
如果 d[i][i] 设置为负值，则存在负周期。
实际路径可以使用 path[][] 进行计算。

完整代码

public class Main {

    private static final int NOT_A_VERTEX = -1;

    /**
     * Compute all-shortest paths.
     * a[][] contains the adjacency matrix with a[i][i] presumed to be zero.
     * d[] contains the values of the shortest path.
     * Vertices are numbered starting at 0; all arrays have equal dimension.
     * A negative cycle exists if d[i][i] is set to a negative value.
     * Actual path can be computed using path[][].
     * NOT_A_VERTEX is -1
     */
    public static void allPairs(int[][] a, int[][] d, int[][] path) {
        int n = a.length;
        // Initialize d and path
        for(int i = 0; i < n; i++) {
            for(int j = 0; j < n; j++) {
                d[i][j] = a[i][j];
                path[i][j] = NOT_A_VERTEX;
            }
        }
        for(int k = 0; k < n; k++) {
            // Consider each vertex as an intermediate
            for(int i = 0; i < n; i++) {
                for(int j = 0; j < n; j++) {
                    if(d[i][k] + d[k][j] < d[i][j]) {
                        // Update shortest path
                        d[i][j] = d[i][k] + d[k][j];
                        path[i][j] = k;
                    }
                }
            }
        }
    }

    public static void main(String[] args) {
        int[][] a = { { 0, 2, -2, 2 }, { 1000, 0, -3, 1000 }, { 4, 1000, 0, 1000 }, { 1000, -2, 3, 0 } };
        int[][] d = new int[4][4];
        int[][] path = new int[4][4];
        allPairs(a, d, path);
        for(int i = 0; i < 4; i++) {
            for(int j = 0; j < 4; j++) {
                System.out.print(d[i][j] + "\t");
            }
            System.out.println();
        }
        for(int i = 0; i < 4; i++) {
            for(int j = 0; j < 4; j++) {
                System.out.print(path[i][j] + "\t");
            }
            System.out.println();
        }
    }

}

测试结果

0	0	-3	2	
1	0	-3	3	
4	4	0	6	
-1	-2	-5	0	
-1	3	3	-1	
2	-1	-1	2	
-1	3	-1	0	
2	-1	1	-1