Worst Case Running Time Time Complexity

Every time the main loop executes, one vertex is extracted from the queue. Assuming that there are V vertices in the graph, the queue may contain O(V) vertices. Each pop operation takes O(log V) time assuming the heap implementation of priority queues. So the total time required to execute the main loop itself is O(V log V). In addition, we must consider the time spent in the function expand, which applies the function handle_edge to each outgoing edge. Because expand is only called once per vertex, handle_edge is only called once per edge. It might call push(v'), but there can be at most V such calls during the entire execution, so the total cost of that case arm is at most O(V log V). The other case arm may be called O(E) times, however, and each call to increase_priority takes O(log V) time with the heap implementation. Therefore the total run time is O(V log V + E log V), which is O(E log V) because V is O(E) assuming a connected graph.

(There is another more complicated priority-queue implementation called a Fibonacci heap that implements increase_priority in O(1) time, so that the asymptotic complexity of Dijkstra's algorithm becomes O(V log V + E); however, large constant factors make Fibonacci heaps impractical for most uses.)

O(|E|+|V|log|V|)

Java Code: Dijkstras Shortest Path Algorithm

import java.util.PriorityQueue;
import java.util.List;
import java.util.ArrayList;
import java.util.Collections;

class Vertex implements Comparable
{
    public final String name;
    public Edge[] adjacencies;
    public double minDistance = Double.POSITIVE_INFINITY;
    public Vertex previous;
    public Vertex(String argName) { name = argName; }
    public String toString() { return name; }
    public int compareTo(Vertex other)
    {
        return Double.compare(minDistance, other.minDistance);
    }

}


class Edge
{
    public final Vertex target;
    public final double weight;
    public Edge(Vertex argTarget, double argWeight)
    { target = argTarget; weight = argWeight; }
}

public class Dijkstra
{
    public static void computePaths(Vertex source)
    {
        source.minDistance = 0.;
        PriorityQueue vertexQueue = new PriorityQueue();
    vertexQueue.add(source);

    while (!vertexQueue.isEmpty()) {
        Vertex u = vertexQueue.poll();

            // Visit each edge exiting u
            for (Edge e : u.adjacencies)
            {
                Vertex v = e.target;
                double weight = e.weight;
                double distanceThroughU = u.minDistance + weight;
        if (distanceThroughU < v.minDistance) {
            vertexQueue.remove(v);

            v.minDistance = distanceThroughU ;
            v.previous = u;
            vertexQueue.add(v);
        }
            }
        }
    }

    public static List getShortestPathTo(Vertex target)
    {
        List path = new ArrayList();
        for (Vertex vertex = target; vertex != null; vertex = vertex.previous)
            path.add(vertex);

        Collections.reverse(path);
        return path;
    }

    public static void main(String[] args)
    {
        // mark all the vertices 
        Vertex A = new Vertex("A");
        Vertex B = new Vertex("B");
        Vertex D = new Vertex("D");
        Vertex F = new Vertex("F");
        Vertex K = new Vertex("K");
        Vertex J = new Vertex("J");
        Vertex M = new Vertex("M");
        Vertex O = new Vertex("O");
        Vertex P = new Vertex("P");
        Vertex R = new Vertex("R");
        Vertex Z = new Vertex("Z");

        // set the edges and weight
        A.adjacencies = new Edge[]{ new Edge(M, 8) };
        B.adjacencies = new Edge[]{ new Edge(D, 11) };
        D.adjacencies = new Edge[]{ new Edge(B, 11) };
        F.adjacencies = new Edge[]{ new Edge(K, 23) };
        K.adjacencies = new Edge[]{ new Edge(O, 40) };
        J.adjacencies = new Edge[]{ new Edge(K, 25) };
        M.adjacencies = new Edge[]{ new Edge(R, 8) };
        O.adjacencies = new Edge[]{ new Edge(K, 40) };
        P.adjacencies = new Edge[]{ new Edge(Z, 18) };
        R.adjacencies = new Edge[]{ new Edge(P, 15) };
        Z.adjacencies = new Edge[]{ new Edge(P, 18) };


        computePaths(A); // run Dijkstra
        System.out.println("Distance to " + Z + ": " + Z.minDistance);
        List path = getShortestPathTo(Z);
        System.out.println("Path: " + path);
    }
}