给定一个图和图中的一个源顶点，找到从源到给定图中所有顶点的最短路径。
Dijkstra的算法与最小生成树的Prim算法非常相似。像Prim的MST一样，我们以给定的源为根来生成SPT(最短路径树)。我们维护两组，一组包含最短路径树中包含的顶点，另一组包含尚未包含在最短路径树中的顶点。在算法的每个步骤中，我们都找到一个顶点，该顶点在另一个集合(尚未包括的集合)中，并且与源的距离最小。
以下是Dijkstra算法中用于查找从单个源顶点到给定图中所有其他顶点的最短路径的详细步骤。
算法
1)创建一个set sptSet (最短路径树集)，该集合跟踪最短路径树中包含的顶点，即，其与源的最小距离已被计算并确定。最初，此集合为空。
2)为输入图中的所有顶点分配一个距离值。将所有距离值初始化为INFINITE。将源顶点的距离值指定为0，以便首先选择它。
3)虽然sptSet不包括所有顶点
…。 a)选择一个顶点u，该顶点在sptSet中不存在，并且具有最小距离值。
…。 b)将u包含到sptSet中。
…。 c)更新u的所有相邻顶点的距离值。要更新距离值，请遍历所有相邻的顶点。对于每个相邻顶点v，如果u(距源)的距离值和边缘uv的权重之和小于v的距离值，则更新v的距离值。
C++
// A C++ program for Dijkstra's single source shortest path algorithm.
// The program is for adjacency matrix representation of the graph
#include
#include
// Number of vertices in the graph
#define V 9
// A utility function to find the vertex with minimum distance value, from
// the set of vertices not yet included in shortest path tree
int minDistance(int dist[], bool sptSet[])
{
// Initialize min value
int min = INT_MAX, min_index;
for (int v = 0; v < V; v++)
if (sptSet[v] == false && dist[v] <= min)
min = dist[v], min_index = v;
return min_index;
}
// A utility function to print the constructed distance array
int printSolution(int dist[], int n)
{
printf("Vertex Distance from Source\n");
for (int i = 0; i < V; i++)
printf("%d tt %d\n", i, dist[i]);
}
// Function that implements Dijkstra's single source shortest path algorithm
// for a graph represented using adjacency matrix representation
void dijkstra(int graph[V][V], int src)
{
int dist[V]; // The output array. dist[i] will hold the shortest
// distance from src to i
bool sptSet[V]; // sptSet[i] will be true if vertex i is included in shortest
// path tree or shortest distance from src to i is finalized
// Initialize all distances as INFINITE and stpSet[] as false
for (int i = 0; i < V; i++)
dist[i] = INT_MAX, sptSet[i] = false;
// Distance of source vertex from itself is always 0
dist[src] = 0;
// Find shortest path for all vertices
for (int count = 0; count < V - 1; count++) {
// Pick the minimum distance vertex from the set of vertices not
// yet processed. u is always equal to src in the first iteration.
int u = minDistance(dist, sptSet);
// Mark the picked vertex as processed
sptSet[u] = true;
// Update dist value of the adjacent vertices of the picked vertex.
for (int v = 0; v < V; v++)
// Update dist[v] only if is not in sptSet, there is an edge from
// u to v, and total weight of path from src to v through u is
// smaller than current value of dist[v]
if (!sptSet[v] && graph[u][v] && dist[u] != INT_MAX
&& dist[u] + graph[u][v] < dist[v])
dist[v] = dist[u] + graph[u][v];
}
// print the constructed distance array
printSolution(dist, V);
}
// driver program to test above function
int main()
{
/* Let us create the example graph discussed above */
int graph[V][V] = { { 0, 4, 0, 0, 0, 0, 0, 8, 0 },
{ 4, 0, 8, 0, 0, 0, 0, 11, 0 },
{ 0, 8, 0, 7, 0, 4, 0, 0, 2 },
{ 0, 0, 7, 0, 9, 14, 0, 0, 0 },
{ 0, 0, 0, 9, 0, 10, 0, 0, 0 },
{ 0, 0, 4, 14, 10, 0, 2, 0, 0 },
{ 0, 0, 0, 0, 0, 2, 0, 1, 6 },
{ 8, 11, 0, 0, 0, 0, 1, 0, 7 },
{ 0, 0, 2, 0, 0, 0, 6, 7, 0 } };
dijkstra(graph, 0);
return 0;
} 输出：
Vertex Distance from Source
0 tt 0
1 tt 4
2 tt 12
3 tt 19
4 tt 21
5 tt 11
6 tt 9
7 tt 8
8 tt 14
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