Directed Graph Traversal - All paths

Question

Given a directed graph with

• A root node
• Some leaves nodes
• Multiple nodes can be connected to the same node
• Cycles can exist

We need to print all the paths from the root node to all the leaves nodes. This is the closest question I got to this problem Find all paths between two graph nodes

Answer 1

If you actually care about ordering your paths from shortest path to longest path then it would be far better to use a modified A* or Dijkstra Algorithm. With a slight modification the algorithm will return as many of the possible paths as you want in order of shortest path first. So if what you really want are all possible paths ordered from shortest to longest then this is the way to go. The code I suggested above would be much slower than it needs to be if you care about ordering from shortest to longest, not to mention would take up more space then you'd want in order to store every possible path at once.

If you want an A* based implementation capable of returning all paths ordered from the shortest to the longest, the following will accomplish that. It has several advantages. First off it is efficient at sorting from shortest to longest. Also it computes each additional path only when needed, so if you stop early because you dont need every single path you save some processing time. It also reuses data for subsequent paths each time it calculates the next path so it is more efficient. Finally if you find some desired path you can abort early saving some computation time. Overall this should be the most efficient algorithm if you care about sorting by path length.

import java.util.*;

public class AstarSearch {
    private final Map<Integer, Set<Neighbor>> adjacency;
    private final int destination;

    private final NavigableSet<Step> pending = new TreeSet<>();

    public AstarSearch(Map<Integer, Set<Neighbor>> adjacency, int source, int destination) {
        this.adjacency = adjacency;
        this.destination = destination;

        this.pending.add(new Step(source, null, 0));
    }

    public List<Integer> nextShortestPath() {
        Step current = this.pending.pollFirst();
        while( current != null) {
            if( current.getId() == this.destination )
                return current.generatePath();
            for (Neighbor neighbor : this.adjacency.get(current.id)) {
                if(!current.seen(neighbor.getId())) {
                    final Step nextStep = new Step(neighbor.getId(), current, current.cost + neighbor.cost + predictCost(neighbor.id, this.destination));
                    this.pending.add(nextStep);
                }
            }
            current = this.pending.pollFirst();
        }
        return null;
    }

    protected int predictCost(int source, int destination) {
        return 0; //Behaves identical to Dijkstra's algorithm, override to make it A*
    }

    private static class Step implements Comparable<Step> {
        final int id;
        final Step parent;
        final int cost;

        public Step(int id, Step parent, int cost) {
            this.id = id;
            this.parent = parent;
            this.cost = cost;
        }

        public int getId() {
            return id;
        }

        public Step getParent() {
            return parent;
        }

        public int getCost() {
            return cost;
        }

        public boolean seen(int node) {
            if(this.id == node)
                return true;
            else if(parent == null)
                return false;
            else
                return this.parent.seen(node);
        }

        public List<Integer> generatePath() {
            final List<Integer> path;
            if(this.parent != null)
                path = this.parent.generatePath();
            else
                path = new ArrayList<>();
            path.add(this.id);
            return path;
        }

        @Override
        public int compareTo(Step step) {
            if(step == null)
                return 1;
            if( this.cost != step.cost)
                return Integer.compare(this.cost, step.cost);
            if( this.id != step.id )
                return Integer.compare(this.id, step.id);
            if( this.parent != null )
                this.parent.compareTo(step.parent);
            if(step.parent == null)
                return 0;
            return -1;
        }

        @Override
        public boolean equals(Object o) {
            if (this == o) return true;
            if (o == null || getClass() != o.getClass()) return false;
            Step step = (Step) o;
            return id == step.id &&
                cost == step.cost &&
                Objects.equals(parent, step.parent);
        }

        @Override
        public int hashCode() {
            return Objects.hash(id, parent, cost);
        }
    }

   /*******************************************************
   *   Everything below here just sets up your adjacency  *
   *   It will just be helpful for you to be able to test *
   *   It isnt part of the actual A* search algorithm     *
   ********************************************************/

    private static class Neighbor {
        final int id;
        final int cost;

        public Neighbor(int id, int cost) {
            this.id = id;
            this.cost = cost;
        }

        public int getId() {
            return id;
        }

        public int getCost() {
            return cost;
        }
    }

    public static void main(String[] args) {
        final Map<Integer, Set<Neighbor>> adjacency = createAdjacency();
        final AstarSearch search = new AstarSearch(adjacency, 1, 4);
        System.out.println("printing all paths from shortest to longest...");
        List<Integer> path = search.nextShortestPath();
        while(path != null) {
            System.out.println(path);
            path = search.nextShortestPath();
        }
    }

    private static Map<Integer, Set<Neighbor>> createAdjacency() {
        final Map<Integer, Set<Neighbor>> adjacency = new HashMap<>();

        //This sets up the adjacencies. In this case all adjacencies have a cost of 1, but they dont need to.
        addAdjacency(adjacency, 1,2,1,5,1);         //{1 | 2,5}
        addAdjacency(adjacency, 2,1,1,3,1,4,1,5,1); //{2 | 1,3,4,5}
        addAdjacency(adjacency, 3,2,1,5,1);         //{3 | 2,5}
        addAdjacency(adjacency, 4,2,1);             //{4 | 2}
        addAdjacency(adjacency, 5,1,1,2,1,3,1);     //{5 | 1,2,3}

        return Collections.unmodifiableMap(adjacency);
    }

    private static void addAdjacency(Map<Integer, Set<Neighbor>> adjacency, int source, Integer... dests) {
        if( dests.length % 2 != 0)
            throw new IllegalArgumentException("dests must have an equal number of arguments, each pair is the id and cost for that traversal");

        final Set<Neighbor> destinations = new HashSet<>();
        for(int i = 0; i < dests.length; i+=2)
            destinations.add(new Neighbor(dests[i], dests[i+1]));
        adjacency.put(source, Collections.unmodifiableSet(destinations));
    }
}

The output from the above code is the following:

[1, 2, 4]
[1, 5, 2, 4]
[1, 5, 3, 2, 4]

Notice that each time you call nextShortestPath() it generates the next shortest path for you on demand. It only calculates the extra steps needed and doesnt traverse any old paths twice. Moreover if you decide you dont need all the paths and end execution early you've saved yourself considerable computation time. You only compute up to the number of paths you need and no more.

Finally it should be noted that the A* and Dijkstra algorithms do have some minor limitations, though I dont think it would effect you. Namely it will not work right on a graph that has negative weights.

Here is a link to JDoodle where you can run the code yourself in the browser and see it working. You can also change around the graph to show it works on other graphs as well: http://jdoodle.com/a/ukx