Finding the shortest path in a network is a commonly encountered problem. For example you want to reach a target in the real world via the shortest path or in a computer network a network package should be efficiently routed through the network.

This tutorial describes the problem modeled as graph and the Dijkstra algorithm to solve the problem.

A graph is made of out nodes and directed edges which defines a connection from one node to another node.

A node (or vertex) is a discrete position in a graph. Edges can be directed an undirected. Edges have an associated distance (also called costs or weight). The distance between two nodes a and b is labeled as [a,b].

The mathematical description for graphs is G= {V,E}, meaning that a graph is defined by a set of vertexes (V) and a collection of edges.

The order of a graph is the number of nodes. The size of a graph is the number of edges.

Typical graph problems are described in the following list.

The following will focus on finding the shortest path from one node to another node in a graph.

The Dijkstra Algorithm finds the shortest path from a source to all destination in a directed graph (single source shortest path problem). During this process it will also determine a spanning tree for the graph.

The idea of Dijkstra is simple.

Dijkstra partitions all nodes into two distinct sets. Unsettled and settled. Initially all nodes are in the unsettled sets, e.g. they must be still evaluated. A node is moved to the settled set if a shortest path from the source to this node has been found.

Initially the distance of each node to the source is set to a very high value.

First only the source is in the set of unsettledNodes.

The algorithms runs until the unsettledNodes are empty. In earch iteration it selects the node with the lowest distance from the source out the unsettled nodes. If reads all edges which are outgoing from the source and evaluates for each destination node in these edges which is not yet settled if the known distance from the source to this node can be reduced if the selected edge is used. If this can be done then the distance is updated and the node is added to the nodes which need evaluation.

In pseudocode the algorithm can be described as follows. Please note that Dijkstra also determines the pre-successor of each node on its way to the source. I leave that out of the pseudo code to simplify it.

Foreach node set distance[node] = HIGH
SettledNodes = empty
UnSettledNodes = empty

Add sourceNode to UnSettledNodes
distance[sourceNode]= 0

while (UnSettledNodes is not empty) {
  evaluationNode = getNodeWithLowestDistance(UnSettledNodes)
  remove evaluationNode from UnSettledNodes 
    add evaluationNode to SettledNodes
    evaluatedNeighbors(evaluationNode)
}

getNodeWithLowestDistance(UnSettledNodes){
  find the node with the lowest distance in UnSettledNodes and return it 
}

evaluatedNeighbors(evaluationNode){
  Foreach destinationNode which can be reached via an edge from evaluationNode AND which is not in SettledNodes {
    edgeDistance = getDistance(edge(evaluationNode, destinationNode))
    newDistance = distance[evaluationNode] + edgeDistance
    if (distance[destinationNode]  > newDistance) {
      distance[destinationNode]  = newDistance 
      add destinationNode to UnSettledNodes
    }
  }
} 

The following is a simple implementation of Dijkstra's algorithm. It does not use any performance optimization (e.g. by using a PriorityQueue for the UnSettledNodes of does not cache the result of the target evaluation of the edges) to make the algorihms as simple as possible.

package de.vogella.algorithms.dijkstra.engine;

import java.util.ArrayList;
import java.util.Collections;
import java.util.HashMap;
import java.util.HashSet;
import java.util.LinkedList;
import java.util.List;
import java.util.Map;
import java.util.Set;

import de.vogella.algorithms.dijkstra.model.Edge;
import de.vogella.algorithms.dijkstra.model.Graph;
import de.vogella.algorithms.dijkstra.model.Vertex;

public class DijkstraAlgorithm {

  private final List<Vertex> nodes;
  private final List<Edge> edges;
  private Set<Vertex> settledNodes;
  private Set<Vertex> unSettledNodes;
  private Map<Vertex, Vertex> predecessors;
  private Map<Vertex, Integer> distance;

  public DijkstraAlgorithm(Graph graph) {
    // Create a copy of the array so that we can operate on this array
    this.nodes = new ArrayList<Vertex>(graph.getVertexes());
    this.edges = new ArrayList<Edge>(graph.getEdges());
  }

  public void execute(Vertex source) {
    settledNodes = new HashSet<Vertex>();
    unSettledNodes = new HashSet<Vertex>();
    distance = new HashMap<Vertex, Integer>();
    predecessors = new HashMap<Vertex, Vertex>();
    distance.put(source, 0);
    unSettledNodes.add(source);
    while (unSettledNodes.size() > 0) {
      Vertex node = getMinimum(unSettledNodes);
      settledNodes.add(node);
      unSettledNodes.remove(node);
      findMinimalDistances(node);
    }
  }

  private void findMinimalDistances(Vertex node) {
    List<Vertex> adjacentNodes = getNeighbors(node);
    for (Vertex target : adjacentNodes) {
      if (getShortestDistance(target) > getShortestDistance(node)
          + getDistance(node, target)) {
        distance.put(target, getShortestDistance(node)
            + getDistance(node, target));
        predecessors.put(target, node);
        unSettledNodes.add(target);
      }
    }

  }

  private int getDistance(Vertex node, Vertex target) {
    for (Edge edge : edges) {
      if (edge.getSource().equals(node)
          && edge.getDestination().equals(target)) {
        return edge.getWeight();
      }
    }
    throw new RuntimeException("Should not happen");
  }

  private List<Vertex> getNeighbors(Vertex node) {
    List<Vertex> neighbors = new ArrayList<Vertex>();
    for (Edge edge : edges) {
      if (edge.getSource().equals(node)
          && !isSettled(edge.getDestination())) {
        neighbors.add(edge.getDestination());
      }
    }
    return neighbors;
  }

  private Vertex getMinimum(Set<Vertex> vertexes) {
    Vertex minimum = null;
    for (Vertex vertex : vertexes) {
      if (minimum == null) {
        minimum = vertex;
      } else {
        if (getShortestDistance(vertex) < getShortestDistance(minimum)) {
          minimum = vertex;
        }
      }
    }
    return minimum;
  }

  private boolean isSettled(Vertex vertex) {
    return settledNodes.contains(vertex);
  }

  private int getShortestDistance(Vertex destination) {
    Integer d = distance.get(destination);
    if (d == null) {
      return Integer.MAX_VALUE;
    } else {
      return d;
    }
  }

  /*
   * This method returns the path from the source to the selected target and
   * NULL if no path exists
   */
  public LinkedList<Vertex> getPath(Vertex target) {
    LinkedList<Vertex> path = new LinkedList<Vertex>();
    Vertex step = target;
    // Check if a path exists
    if (predecessors.get(step) == null) {
      return null;
    }
    path.add(step);
    while (predecessors.get(step) != null) {
      step = predecessors.get(step);
      path.add(step);
    }
    // Put it into the correct order
    Collections.reverse(path);
    return path;
  }

} 

The following is a small JUnit Test to validate the correctness of the algorithm.

package de.vogella.algorithms.dijkstra.test;

import java.util.ArrayList;
import java.util.LinkedList;
import java.util.List;

import org.junit.Test;

import de.vogella.algorithms.dijkstra.engine.DijkstraAlgorithm;
import de.vogella.algorithms.dijkstra.model.Edge;
import de.vogella.algorithms.dijkstra.model.Graph;
import de.vogella.algorithms.dijkstra.model.Vertex;

import static org.junit.Assert.assertNotNull;
import static org.junit.Assert.assertTrue;

public class TestDijkstraAlgorithm {

  private List<Vertex> nodes;
  private List<Edge> edges;

  @Test
  public void testExcute() {
    nodes = new ArrayList<Vertex>();
    edges = new ArrayList<Edge>();
    for (int i = 0; i < 11; i++) {
      Vertex location = new Vertex("Node_" + i, "Node_" + i);
      nodes.add(location);
    }

    addLane("Edge_0", 0, 1, 85);
    addLane("Edge_1", 0, 2, 217);
    addLane("Edge_2", 0, 4, 173);
    addLane("Edge_3", 2, 6, 186);
    addLane("Edge_4", 2, 7, 103);
    addLane("Edge_5", 3, 7, 183);
    addLane("Edge_6", 5, 8, 250);
    addLane("Edge_7", 8, 9, 84);
    addLane("Edge_8", 7, 9, 167);
    addLane("Edge_9", 4, 9, 502);
    addLane("Edge_10", 9, 10, 40);
    addLane("Edge_11", 1, 10, 600);

    // Lets check from location Loc_1 to Loc_10
    Graph graph = new Graph(nodes, edges);
    DijkstraAlgorithm dijkstra = new DijkstraAlgorithm(graph);
    dijkstra.execute(nodes.get(0));
    LinkedList<Vertex> path = dijkstra.getPath(nodes.get(10));
    
    assertNotNull(path);
    assertTrue(path.size() > 0);
    
    for (Vertex vertex : path) {
      System.out.println(vertex);
    }
    
  }

  private void addLane(String laneId, int sourceLocNo, int destLocNo,
      int duration) {
    Edge lane = new Edge(laneId,nodes.get(sourceLocNo), nodes.get(destLocNo), duration);
    edges.add(lane);
  }
} 

http://renaud.waldura.com/doc/java/dijkstra/ Dijstras algorithm in Java