Find all cycles in undirected graph

Question

I am using Algorithms 4th edition to polish up my graph theory a bit. The books comes with a lot of code for graph processing.
Currently, I am stuck with the following problems: How to find all cycles in an undirected graph? I was looking to modify the existing code for cycle detection to do that.

Here is the important part:

private void dfs(Graph G, int u, int v) {
        marked[v] = true;
        for (int w : G.adj(v)) {

            // short circuit if cycle already found
            if (cycle != null) return;

            if (!marked[w]) {
                edgeTo[w] = v;
                dfs(G, v, w);
            }

            // check for cycle (but disregard reverse of edge leading to v)
            else if (w != u) {
                cycle = new Stack<Integer>();
                for (int x = v; x != w; x = edgeTo[x]) {
                    cycle.push(x);
                }
                cycle.push(w);
                cycle.push(v);
            }
        }
    }

Now, if I were to find ALL cycles, I should remove the line that returns when a cycle is found and each time a cycle is created I would store it. The part I cannot figure out is: when does the algorithm stop? How can I be sure I have found all cycles?

Can the above code even be modified in a way to allow me to find all cycles?

Answer 1

Cycle detection is much easier than finding all cycles. Cycle detection can be done in linear time using a DFS like you've linked, but the number of cycles in a graph can be exponential, ruling out an polytime algorithm altogether. If you don't see how this could be possible, consider this graph:

1 -- 2
|  / |
| /  |
3 -- 4

There are three distinct cycles, but a DFS would find only two back-edges.

As such, modifying your algorithm to find all cycles will take a fair bit more work than simply changing a line or two. Instead, you have to find a set of base cycles, then combine them to form the set of all cycles. You can find an implementation of an algorithm that'll does this in this question .

Answer 2

/**
 * In this program we create a list of edges which is an ordered pair of two       
 * integers representing two vertices.
 *
 * We iterate through each edge and apply Union Find algorithm to detect 
 * cycle.
 *
 * This is a tested code and gives correct result for all inputs.
 */
package com.divyanshu.ds.disjointSet;

import java.util.HashMap;

/**
 * @author Divyanshu
 * DisjointSet is a data structure with three operations :
 * makeSet, union and findSet
 * 
 * Algorithms Used : Union by rank and path compression for detecting cycles    
 * in an undirected graph.
 */
public class DisjontSet {
    HashMap<Long, Node> map = new HashMap<>();

    class Node {
        long data;
        Node parent;
        int  rank;
    }

    public void makeSet(long data) {
        Node node = new Node();
        node.data = data;
        node.parent = node;
        node.rank = 0;
        map.put(data, node);
    }

    public void union(long firstSet,
                      long secondSet) {
        Node firstNode = map.get(firstSet);
        Node secondNode = map.get(secondSet);

        Node firstParent = findSet(firstNode);
        Node secondParent = findSet(secondNode);
        if (firstParent.data == secondParent.data) {
            return;
        }
        if (firstParent.rank >= secondParent.rank) {
            firstParent.rank = (firstParent.rank == secondParent.rank) ? firstParent.rank + 1 : firstParent.rank;
            secondParent.parent = firstParent;
        } else {
            firstParent.parent = secondParent;
        }
    }

    public long findSet(long data) {
        return findSet(map.get(data)).data;
    }

    private Node findSet(Node node) {
        if (node.parent == node) {
            return node;
        }
        node.parent = findSet(node.parent);
        return node.parent;
    }
}


=============================================================================

package com.divyanshu.ds.client;

import java.util.ArrayList;
import java.util.HashSet;
import java.util.List;
import java.util.Random;

import com.divyanshu.ds.disjointSet.DisjontSet;
import com.divyanshu.ds.disjointSet.Edge;

public class DisjointSetClient {

    public static void main(String[] args) {
        int edgeCount = 4;
        int vertexCount = 12;
        List<Edge> graph = generateGraph(edgeCount, vertexCount);
        System.out.println("Generated Graph : ");
        System.out.println(graph);
        DisjontSet disjontSet = getDisjointSet(graph);
        Boolean isGraphCyclic = isGraphCyclic(graph, disjontSet);
        System.out.println("Graph contains cycle : " + isGraphCyclic);
    }

    private static Boolean isGraphCyclic(List<Edge> graph,
                                         DisjontSet disjontSet) {
        Boolean isGraphCyclic = false;
        for (Edge edge : graph) {
            if (edge.getFirstVertex() != edge.getSecondVertex()) {
                Long first = disjontSet.findSet(edge.getFirstVertex());
                Long second = disjontSet.findSet(edge.getSecondVertex());
                if (first.equals(second)) {
                    isGraphCyclic = true;
                    break;
                } else {
                    disjontSet.union(first, second);
                }
            }
        }
        return isGraphCyclic;
    }

    private static DisjontSet getDisjointSet(List<Edge> graph) {
        DisjontSet disjontSet = new DisjontSet();
        for (Edge edge : graph) {
            disjontSet.makeSet(edge.getFirstVertex());
            disjontSet.makeSet(edge.getSecondVertex());
        }
        return disjontSet;
    }

    private static List<Edge> generateGraph(int edgeCount,
                                            int vertexCount) {
        List<Edge> graph = new ArrayList<>();
        HashSet<Edge> edgeSet = new HashSet<>();
        Random random = new Random();
        for (int j = 0; j < vertexCount; j++) {
            int first = random.nextInt(edgeCount);
            int second = random.nextInt(edgeCount);
            if (first != second) {
                edgeSet.add(new Edge(first, second));
            } else {
                j--;
            }
        }
        for (Edge edge : edgeSet) {
            graph.add(edge);
        }
        return graph;
    }

}

===================================================================

/**
 * 
 */
package com.divyanshu.ds.disjointSet;

/**
 * @author Divyanshu
 *
 */
public class Edge {
    private long firstVertex;
    private long secondVertex;

    public Edge(long firstVertex,
            long secondVertex) {
        this.firstVertex = firstVertex;
        this.secondVertex = secondVertex;
    }
    public long getFirstVertex() {
        return firstVertex;
    }

    public void setFirstVertex(long firstVertex) {
        this.firstVertex = firstVertex;
    }

    public long getSecondVertex() {
        return secondVertex;
    }

    public void setSecondVertex(long secondVertex) {
        this.secondVertex = secondVertex;
    }

    @Override
    public String toString() {
        return "(" + firstVertex + "," + secondVertex + ")";
    }

    @Override
    public int hashCode() {
        final int prime = 31;
        int result = 1;
        result = prime * result + (int) (firstVertex ^ (firstVertex >>> 32));
        result = prime * result + (int) (secondVertex ^ (secondVertex >>> 32));
        return result;
    }

    @Override
    public boolean equals(Object obj) {
        if (this == obj)
            return true;
        if (obj == null)
            return false;
        if (getClass() != obj.getClass())
            return false;
        Edge other = (Edge) obj;
        if (firstVertex != other.firstVertex)
            return false;
        if (secondVertex != other.secondVertex)
            return false;
        return true;
    }

}