算法篇-Dijkstra算法

迪克斯特拉算法

迪杰斯特拉算法(Dijkstra)是由荷兰计算机科学家狄克斯特拉于1959 年提出的，因此又叫狄克斯特拉算法。是从一个顶点到其余各顶点的最短路径算法，解决的是有权图中最短路径问题。

迪杰斯特拉算法主要特点是

从起始点开始，采用贪心算法的策略，每次遍历到始点距离最近且未访问过的顶点的邻接节点，更新始点到其距离，直到扩展到终点为止。

算法思想

按路径长度递增次序产生算法：

把顶点集合V分成两组：

• S：已求出的顶点的集合（初始时只含有源点V0）
• V-S=T：尚未确定的顶点集合

将T中顶点按递增的次序加入到S中，保证：

• 从源点V0到S中其他各顶点的长度都不大于从V0到T中任何顶点的最短路径长度
• 每个顶点对应一个距离值

S中顶点：从V0到此顶点的长度
T中顶点：从V0到此顶点的只包括S中顶点作中间顶点的最短路径长度

依据：可以证明V0到T中顶点Vk的，或是从V0到Vk的直接路径的权值；或是从V0经S中顶点到Vk的路径权值之和。（反证法可证）

求最短路径步骤

G={V,E}

1. 初始时令 S={V0},T=V-S={其余顶点}，T中顶点对应的距离值,若存在<V0,Vi>，d(V0,Vi)为<V0,Vi>弧上的权值,若不存在<V0,Vi>，d(V0,Vi)为∞
2. 从T中选取一个与S中顶点有关联边且权值最小的顶点W，加入到S中
3. 对其余T中顶点的距离值进行修改：若加进W作中间顶点，从V0到Vi的距离值缩短，则修改此距离值,重复上述步骤2、3，直到S  中包含所有顶点，即W=Vi为止

代码实现

import java.io.*;
import java.lang.*;
import java.util.*;

class ShortestPath {
	// A utility function to find the vertex with minimum
	// distance value, from the set of vertices not yet
	// included in shortest path tree
	static final int V = 9;
	int minDistance(int dist[], Boolean sptSet[])
	{
		// Initialize min value
		int min = Integer.MAX_VALUE, min_index = -1;

		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
	void printSolution(int dist[])
	{
		System.out.println(
			"Vertex \t\t Distance from Source");
		for (int i = 0; i < V; i++)
			System.out.println(i + " \t\t " + dist[i]);
	}

	// Function that implements Dijkstra's single source
	// shortest path algorithm for a graph represented using
	// adjacency matrix representation
	void dijkstra(int graph[][], int src)
	{
		int dist[] = new int[V]; // The output array.
								// dist[i] will hold
		// the shortest distance from src to i

		// sptSet[i] will true if vertex i is included in
		// shortest path tree or shortest distance from src
		// to i is finalized
		Boolean sptSet[] = new Boolean[V];

		// Initialize all distances as INFINITE and stpSet[]
		// as false
		for (int i = 0; i < V; i++) {
			dist[i] = Integer.MAX_VALUE;
			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 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] != 0
					&& dist[u] != Integer.MAX_VALUE
					&& dist[u] + graph[u][v] < dist[v])
					dist[v] = dist[u] + graph[u][v];
		}

		// print the constructed distance array
		printSolution(dist);
	}

	// Driver's code
	public static void main(String[] args)
	{
		int graph[][]
			= new int[][] { { 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 } };
		ShortestPath t = new ShortestPath();
		t.dijkstra(graph, 0);
	}
}

参考资料