Johnson算法是一种用于解决有向图最短路径问题的算法。它的基本思想是通过对图进行转换,将原图中的负权边转换为非负权边,然后利用Dijkstra算法或Bellman-Ford算法求解最短路径。
以下是使用C语言实现Johnson算法的基本步骤:
#define MAX_VERTEX 100
#define INF 9999
int graph[MAX_VERTEX][MAX_VERTEX];
void bellmanFord(int V, int start)
{
int dist[V];
for (int i = 0; i < V; i++)
dist[i] = INF;
dist[start] = 0;
for (int i = 0; i < V - 1; i++)
{
for (int u = 0; u < V; u++)
{
for (int v = 0; v < V; v++)
{
if (graph[u][v] != 0 && dist[u] + graph[u][v] < dist[v])
dist[v] = dist[u] + graph[u][v];
}
}
}
for (int u = 0; u < V; u++)
{
for (int v = 0; v < V; v++)
{
if (graph[u][v] != 0 && dist[u] + graph[u][v] < dist[v])
printf("图中存在负权环,无法计算最短路径");
}
}
// 将负权边转换为非负权边
for (int u = 0; u < V; u++)
{
for (int v = 0; v < V; v++)
{
if (graph[u][v] != 0)
graph[u][v] += dist[u] - dist[v];
}
}
}
void dijkstra(int V, int start)
{
int dist[V];
bool visited[V];
for (int i = 0; i < V; i++)
{
dist[i] = INF;
visited[i] = false;
}
dist[start] = 0;
for (int count = 0; count < V - 1; count++)
{
int u = -1;
for (int i = 0; i < V; i++)
{
if (!visited[i] && (u == -1 || dist[i] < dist[u]))
u = i;
}
visited[u] = true;
for (int v = 0; v < V; v++)
{
if (!visited[v] && graph[u][v] != 0 && dist[u] != INF && dist[u] + graph[u][v] < dist[v])
dist[v] = dist[u] + graph[u][v];
}
}
printf("顶点 最短路径\n");
for (int i = 0; i < V; i++)
{
if (dist[i] == INF)
printf("%d \t 无限远\n", i);
else
printf("%d \t %d\n", i, dist[i]);
}
}
int main()
{
int V;
int start;
printf("输入顶点数量:");
scanf("%d", &V);
printf("输入起始顶点:");
scanf("%d", &start);
printf("输入图的邻接矩阵:\n");
for (int i = 0; i < V; i++)
{
for (int j = 0; j < V; j++)
{
scanf("%d", &graph[i][j]);
}
}
bellmanFord(V, start);
dijkstra(V, start);
return 0;
}
上述代码实现了Johnson算法,在输入图的邻接矩阵后,根据起始顶点计算出图中各顶点的最短路径。