Documentation / Algorithms / Shortest path

Dijkstra's Shortest Path Algorithm

Dijkstra’s algorithm computes the shortest paths from a given node called source to all the other nodes in a graph. It produces a shortest path tree rooted in the source. This algorithm works only for nonnegative lengths. Check Dijkstra’s algorithm article on the Wikipedia for more details.

Complexity

O(n log(n) + m) with n the number of nodes and m the number of edges.

Reference

  • E W Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1:269–271, 1959.

Length of a path

Traditionally the length of a path is defined as the sum of the lengths of its edges. This implementation allows to take into account also the “lengths” of the nodes. This is done by a parameter of type Element passed in the constructors.

The lengths of individual elements (edges or/and nodes) are defined using another constructor parameter called lengthAttribute. If this parameter is null, the elements are considered to have unit lengths. In other words, the length of a path is the number of its edges or/and nodes. If the parameter is not null, the elements are supposed to have a numeric attribute named lengthAttribute used to store their lengths.

Solutions

Internal solution data is stored in attributes of the nodes of the underlying graph. The name of this attribute is another constructor parameter called resultAttribute. This name must be specified in order to avoid conflicts with existing attributes, but also to distinguish between solutions produced by different instances of this class working on the same graph (for example when computing shortest paths from two different sources. If not specified, a unique name is chosen automatically based on the hash code of the Dijkstra instance. The attributes store opaque internal objects and must not be accessed, modified or deleted. The only way to retrieve the solution is to use the following solution access methods:

  • getPathLength(Node) returns the length of the shortest path from the source to a given node
  • getParent(Node) and getEdgeFromParent(Node) are used to get the previous node or edge in the shortest path from the source to a given node
  • getPathEdges(Node), getPathEdgesIterator(Node), getPathNodes(Node) and getPathNodesIterator(Node) are used to iterate over the shortest paths computed by the algorithm
  • getTreeEdges(), getTreeEdgesIterator() and getTreeLength() are used to access the shortest path tree computed by the algorithm
  • getAllPathsIterator(Node) enumerates all the shortest paths from the source to a given node

Usage

A typical usage of this class involves the following steps:

  • Instantiation using one of the constructors with appropriate parameters
  • Initialization of the algorithm using init(Graph)
  • Setting the source node using setSource(Node)
  • Computation of the shortest paths using compute()
  • Retrieving the solution using different solution access methods
  • Cleaning up using clear()

Note that if the graph changes after the call of compute() the computed solution is no longer valid. In this case the behavior of the different solution access methods is undefined.

Complete Example

 import java.util.ArrayList;
 import java.util.Iterator;
 import java.util.List;
 
 import org.graphstream.algorithm.Dijkstra;
 import org.graphstream.graph.Edge;
 import org.graphstream.graph.Graph;
 import org.graphstream.graph.Node;
 import org.graphstream.graph.Path;
 import org.graphstream.graph.implementations.SingleGraph;
 
 public class DijkstraExample {
 //        B---9--E
 //       /|      |
 //      / |      |
 //     /  |      |
 //    14  2      6
 //   /    |      |
 //  /     |      |
 // A---9--C--11--F
 //  \     |     /
 //   \    |    /
 //    7  10   15
 //     \  |  /
 //      \ | /
 //       \|/
 //        D      G
  	public static Graph exampleGraph() {
		Graph g = new SingleGraph("example");
		g.addNode("A").addAttribute("xy", 0, 1);
		g.addNode("B").addAttribute("xy", 1, 2);
		g.addNode("C").addAttribute("xy", 1, 1);
		g.addNode("D").addAttribute("xy", 1, 0);
		g.addNode("E").addAttribute("xy", 2, 2);
		g.addNode("F").addAttribute("xy", 2, 1);
		g.addNode("G").addAttribute("xy", 2, 0);
		g.addEdge("AB", "A", "B").addAttribute("length", 14);
		g.addEdge("AC", "A", "C").addAttribute("length", 9);
		g.addEdge("AD", "A", "D").addAttribute("length", 7);
		g.addEdge("BC", "B", "C").addAttribute("length", 2);
		g.addEdge("CD", "C", "D").addAttribute("length", 10);
		g.addEdge("BE", "B", "E").addAttribute("length", 9);
		g.addEdge("CF", "C", "F").addAttribute("length", 11);
		g.addEdge("DF", "D", "F").addAttribute("length", 15);
		g.addEdge("EF", "E", "F").addAttribute("length", 6);
		for (Node n : g)
			n.addAttribute("label", n.getId());
		for (Edge e : g.getEachEdge())
			e.addAttribute("label", "" + (int) e.getNumber("length"));
		return g;
	}

	public static void main(String[] args) {
		Graph g = exampleGraph();
		g.display(false);

		// Edge lengths are stored in an attribute called "length"
		// The length of a path is the sum of the lengths of its edges
		Dijkstra dijkstra = new Dijkstra(Dijkstra.Element.EDGE, null, "length");

		// Compute the shortest paths in g from A to all nodes
		dijkstra.init(g);
		dijkstra.setSource(g.getNode("A"));
		dijkstra.compute();

		// Print the lengths of all the shortest paths
		for (Node node : g)
			System.out.printf("%s->%s:%10.2f%n", dijkstra.getSource(), node,
					dijkstra.getPathLength(node));

		// Color in blue all the nodes on the shortest path form A to B
		for (Node node : dijkstra.getPathNodes(g.getNode("B")))
			node.addAttribute("ui.style", "fill-color: blue;");

		// Color in red all the edges in the shortest path tree
		for (Edge edge : dijkstra.getTreeEdges())
			edge.addAttribute("ui.style", "fill-color: red;");

		// Print the shortest path from A to B
		System.out.println(dijkstra.getPath(g.getNode("B")));

		// Build a list containing the nodes in the shortest path from A to B
		// Note that nodes are added at the beginning of the list
		// because the iterator traverses them in reverse order, from B to A
		List<Node> list1 = new ArrayList<Node>();
		for (Node node : dijkstra.getPathNodes(g.getNode("B")))
			list1.add(0, node);

		// A shorter but less efficient way to do the same thing
		List<Node> list2 = dijkstra.getPath(g.getNode("B")).getNodePath();

		// cleanup to save memory if solutions are no longer needed
		dijkstra.clear();

		// Now compute the shortest path from A to all the other nodes
		// but taking the number of nodes in the path as its length
		dijkstra = new Dijkstra(Dijkstra.Element.NODE, null, null);
		dijkstra.init(g);
		dijkstra.setSource(g.getNode("A"));
		dijkstra.compute();

		// Print the lengths of the new shortest paths
		for (Node node : g)
			System.out.printf("%s->%s:%10.2f%n", dijkstra.getSource(), node,
					dijkstra.getPathLength(node));

		// Print all the shortest paths between A and F
		Iterator<Path> pathIterator = dijkstra.getAllPathsIterator(g
				.getNode("F"));
		while (pathIterator.hasNext())
			System.out.println(pathIterator.next());

	}
 }

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