djikstra算法

### Dijkstra算法概述 Dijkstra算法是一种著名的图论算法，专门用于解决单源最短路径问题。该算法由荷兰计算机科学家Edsger W. Dijkstra于1956年提出[^1]。 #### 算法适用范围 此算法适用于带权重的有向图或无向图中的边权值非负的情况，在网络路由、地图导航等领域有着广泛的应用[^2]。 ### 工作原理 核心思想是从起始节点出发逐步探索最近邻接点并更新距离直到访问到目标顶点为止；每次从未处理过的结点集合里挑选当前已知离起点最近的一个作为新的考察对象，并据此调整其他未定节点的距离估计值直至遍历完成整个图结构。 具体来说： - 初始化阶段设定初始节点S至各目的节点V_i(i=0,1,...n)之间的临时最短路径长度d[V_i]=∞(除了自身外)，并将所有顶点标记为未确认状态； - 将起点加入优先队列Q中准备参与后续迭代运算过程； - 当前轮次内选取具有最小代价评估函数f(n)=g(n)+h(n)(这里特指仅考虑实际开销即令启发式项h≡0情形下简化版A*搜索策略也就是纯粹基于累积耗费度量标准来决定先后顺序) 的候选项u∈Q执行扩展操作——对于每一个与之相连且尚未被最终确定下来的邻居v而言： - 如果通过u到达v的新测得总成本lower than 原先记录下的预估值，则采用这条更优路线替换旧方案并相应修改关联属性字段parent[v]:=u以及dist[v]:=alt; - 循环上述流程不断重复直至找到终点t或者待检视列表为空表示不存在可行解结束程序运行。 ### C语言实现示例 下面给出一段简单的C语言代码实现了基本功能框架： ```c #include <stdio.h> #include <limits.h> #define V 9 // Number of vertices in the graph // 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[], int sptSet[]) { // Initialize min value int min = INT_MAX, min_index; for (int v = 0; v < V; v++) if (sptSet[v] == 0 && dist[v] <= min) min = dist[v], min_index = v; return min_index; } void printSolution(int dist[]) { printf("Vertex \t\t Distance from Source\n"); for (int i = 0; i < V; i++) printf("%d \t\t %d\n", i, dist[i]); } 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 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 it's 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); } ```