Dijkstra Algorithm
The Dijkstra Algorithm, named after its creator, Edsger W. Dijkstra, is a widely-used graph traversal algorithm that finds the shortest path between a starting node and all other nodes in a weighted graph. Often used in routing and navigation applications, Dijkstra's algorithm works by iteratively selecting the node with the smallest known distance from the starting node, and updating the distances of its neighboring nodes based on the edge weights connecting them. This process is repeated until all nodes have been visited or the shortest path to the destination node has been determined.
In order to efficiently keep track of distances and visited nodes, the algorithm typically uses a priority queue data structure. Initially, the starting node is assigned a distance of zero, while all other nodes are assigned an infinite distance. As the algorithm progresses, it extracts nodes with the smallest distance from the priority queue, marking them as visited and updating the distances of their neighbors. This ensures that nodes are visited in the order of their proximity to the starting node, guaranteeing the discovery of the shortest path. Once the destination node has been visited or all nodes have been processed, the path can be reconstructed by tracing back from the destination node to the starting node via the shortest path tree formed during the algorithm's execution.
package Others;
/**
* Dijkstra's algorithm,is a graph search algorithm that solves the single-source
* shortest path problem for a graph with nonnegative edge path costs, producing
* a shortest path tree.
* <p>
* NOTE: The inputs to Dijkstra's algorithm are a directed and weighted graph consisting
* of 2 or more nodes, generally represented by an adjacency matrix or list, and a start node.
* <p>
* Original source of code: https://rosettacode.org/wiki/Dijkstra%27s_algorithm#Java
* Also most of the comments are from RosettaCode.
*/
import java.util.*;
public class Dijkstra {
private static final Graph.Edge[] GRAPH = {
// Distance from node "a" to node "b" is 7.
// In the current Graph there is no way to move the other way (e,g, from "b" to "a"),
// a new edge would be needed for that
new Graph.Edge("a", "b", 7),
new Graph.Edge("a", "c", 9),
new Graph.Edge("a", "f", 14),
new Graph.Edge("b", "c", 10),
new Graph.Edge("b", "d", 15),
new Graph.Edge("c", "d", 11),
new Graph.Edge("c", "f", 2),
new Graph.Edge("d", "e", 6),
new Graph.Edge("e", "f", 9),
};
private static final String START = "a";
private static final String END = "e";
/**
* main function
* Will run the code with "GRAPH" that was defined above.
*/
public static void main(String[] args) {
Graph g = new Graph(GRAPH);
g.dijkstra(START);
g.printPath(END);
//g.printAllPaths();
}
}
class Graph {
// mapping of vertex names to Vertex objects, built from a set of Edges
private final Map<String, Vertex> graph;
/**
* One edge of the graph (only used by Graph constructor)
*/
public static class Edge {
public final String v1, v2;
public final int dist;
public Edge(String v1, String v2, int dist) {
this.v1 = v1;
this.v2 = v2;
this.dist = dist;
}
}
/**
* One vertex of the graph, complete with mappings to neighbouring vertices
*/
public static class Vertex implements Comparable<Vertex> {
public final String name;
// MAX_VALUE assumed to be infinity
public int dist = Integer.MAX_VALUE;
public Vertex previous = null;
public final Map<Vertex, Integer> neighbours = new HashMap<>();
public Vertex(String name) {
this.name = name;
}
private void printPath() {
if (this == this.previous) {
System.out.printf("%s", this.name);
} else if (this.previous == null) {
System.out.printf("%s(unreached)", this.name);
} else {
this.previous.printPath();
System.out.printf(" -> %s(%d)", this.name, this.dist);
}
}
public int compareTo(Vertex other) {
if (dist == other.dist)
return name.compareTo(other.name);
return Integer.compare(dist, other.dist);
}
@Override
public boolean equals(Object object) {
if (this == object) return true;
if (object == null || getClass() != object.getClass()) return false;
if (!super.equals(object)) return false;
Vertex vertex = (Vertex) object;
if (dist != vertex.dist) return false;
if (name != null ? !name.equals(vertex.name) : vertex.name != null) return false;
if (previous != null ? !previous.equals(vertex.previous) : vertex.previous != null) return false;
if (neighbours != null ? !neighbours.equals(vertex.neighbours) : vertex.neighbours != null) return false;
return true;
}
@Override
public int hashCode() {
int result = super.hashCode();
result = 31 * result + (name != null ? name.hashCode() : 0);
result = 31 * result + dist;
result = 31 * result + (previous != null ? previous.hashCode() : 0);
result = 31 * result + (neighbours != null ? neighbours.hashCode() : 0);
return result;
}
@Override
public String toString() {
return "(" + name + ", " + dist + ")";
}
}
/**
* Builds a graph from a set of edges
*/
public Graph(Edge[] edges) {
graph = new HashMap<>(edges.length);
// one pass to find all vertices
for (Edge e : edges) {
if (!graph.containsKey(e.v1)) graph.put(e.v1, new Vertex(e.v1));
if (!graph.containsKey(e.v2)) graph.put(e.v2, new Vertex(e.v2));
}
// another pass to set neighbouring vertices
for (Edge e : edges) {
graph.get(e.v1).neighbours.put(graph.get(e.v2), e.dist);
// graph.get(e.v2).neighbours.put(graph.get(e.v1), e.dist); // also do this for an undirected graph
}
}
/**
* Runs dijkstra using a specified source vertex
*/
public void dijkstra(String startName) {
if (!graph.containsKey(startName)) {
System.err.printf("Graph doesn't contain start vertex \"%s\"%n", startName);
return;
}
final Vertex source = graph.get(startName);
NavigableSet<Vertex> q = new TreeSet<>();
// set-up vertices
for (Vertex v : graph.values()) {
v.previous = v == source ? source : null;
v.dist = v == source ? 0 : Integer.MAX_VALUE;
q.add(v);
}
dijkstra(q);
}
/**
* Implementation of dijkstra's algorithm using a binary heap.
*/
private void dijkstra(final NavigableSet<Vertex> q) {
Vertex u, v;
while (!q.isEmpty()) {
// vertex with shortest distance (first iteration will return source)
u = q.pollFirst();
if (u.dist == Integer.MAX_VALUE)
break; // we can ignore u (and any other remaining vertices) since they are unreachable
// look at distances to each neighbour
for (Map.Entry<Vertex, Integer> a : u.neighbours.entrySet()) {
v = a.getKey(); // the neighbour in this iteration
final int alternateDist = u.dist + a.getValue();
if (alternateDist < v.dist) { // shorter path to neighbour found
q.remove(v);
v.dist = alternateDist;
v.previous = u;
q.add(v);
}
}
}
}
/**
* Prints a path from the source to the specified vertex
*/
public void printPath(String endName) {
if (!graph.containsKey(endName)) {
System.err.printf("Graph doesn't contain end vertex \"%s\"%n", endName);
return;
}
graph.get(endName).printPath();
System.out.println();
}
/**
* Prints the path from the source to every vertex (output order is not guaranteed)
*/
public void printAllPaths() {
for (Vertex v : graph.values()) {
v.printPath();
System.out.println();
}
}
}
