Related
Imagine a directed acyclic graph as follows, where:
"A" is the root (there is always exactly one root)
each node knows its parent(s)
the node names are arbitrary - nothing can be inferred from them
we know from another source that the nodes were added to the tree in the order A to G (e.g. they are commits in a version control system)
What algorithm could I use to determine the lowest common ancestor (LCA) of two arbitrary nodes, for example, the common ancestor of:
B and E is B
D and F is B
Note:
There is not necessarily a single path to a given node from the root (e.g. "G" has two paths), so you can't simply traverse paths from root to the two nodes and look for the last equal element
I've found LCA algorithms for trees, especially binary trees, but they do not apply here because a node can have multiple parents (i.e. this is not a tree)
Den Roman's link (Archived version) seems promising, but it seemed a little bit complicated to me, so I tried another approach. Here is a simple algorithm I used:
Let say you want to compute LCA(x,y) with x and y two nodes.
Each node must have a value color and count, resp. initialized to white and 0.
Color all ancestors of x as blue (can be done using BFS)
Color all blue ancestors of y as red (BFS again)
For each red node in the graph, increment its parents' count by one
Each red node having a count value set to 0 is a solution.
There can be more than one solution, depending on your graph. For instance, consider this graph:
LCA(4,5) possible solutions are 1 and 2.
Note it still work if you want find the LCA of 3 nodes or more, you just need to add a different color for each of them.
I was looking for a solution to the same problem and I found a solution in the following paper:
http://dx.doi.org/10.1016/j.ipl.2010.02.014
In short, you are not looking for the lowest common ancestor, but for the lowest SINGLE common ancestor, which they define in this paper.
I know it's and old question and pretty good discussion, but since I had some similar problem to solve I came across JGraphT's Lowest Common Ancestor algorithms, thought this might be of help:
NativeLcaFinder
TarjanLowestCommonAncestor
Just some wild thinking. What about using both input nodes as roots, and doing two BFS simultaneously step by step. At a certain step, when there are overlapping in their BLACK sets (recording visited nodes), algorithm stops and the overlapped nodes are their LCA(s). In this way, any other common ancestors will have longer distances than what we have discovered.
Assume that you want to find the ancestors of x and y in a graph.
Maintain an array of vectors- parents (storing parents of each node).
Firstly do a bfs(keep storing parents of each vertex) and find all the ancestors of x (find parents of x and using parents, find all the ancestors of x) and store them in a vector. Also, store the depth of each parent in the vector.
Find the ancestors of y using same method and store them in another vector. Now, you have two vectors storing the ancestors of x and y respectively along with their depth.
LCA would be common ancestor with greatest depth. Depth is defined as longest distance from root(vertex with in_degree=0). Now, we can sort the vectors in decreasing order of their depths and find out the LCA. Using this method, we can even find multiple LCA's (if there).
This link (Archived version) describes how it is done in Mercurial - the basic idea is to find all parents for the specified nodes, group them per distance from the root, then do a search on those groups.
If the graph has cycles then 'ancestor' is loosely defined. Perhaps you mean the ancestor on the tree output of a DFS or BFS? Or perhaps by 'ancestor' you mean the node in the digraph that minimizes the number of hops from E and B?
If you're not worried about complexity, then you could compute an A* (or Dijkstra's shortest path) from every node to both E and B. For the nodes that can reach both E and B, you can find the node that minimizes PathLengthToE + PathLengthToB.
EDIT:
Now that you've clarified a few things, I think I understand what you're looking for.
If you can only go "up" the tree, then I suggest you perform a BFS from E and also a BFS from B. Every node in your graph will have two variables associated with it: hops from B and hops from E. Let both B and E have copies of the list of graph nodes. B's list is sorted by hops from B while E's list is sorted by hops from E.
For each element in B's list, attempt to find it in E's list. Place matches in a third list, sorted by hops from B + hops from E. After you've exhausted B's list, your third sorted list should contain the LCA at its head. This allows for one solution, multiple solutions(arbitrarily chosen among by their BFS ordering for B), or no solution.
I also need exactly same thing , to find LCA in a DAG (directed acyclic graph). LCA problem is related to RMQ (Range Minimum Query Problem).
It is possible to reduce LCA to RMQ and find desired LCA of two arbitrary node from a directed acyclic graph.
I found THIS TUTORIAL detail and good. I am also planing to implement this.
I am proposing O(|V| + |E|) time complexity solution, and i think this approach is correct otherwise please correct me.
Given directed acyclic graph, we need to find LCA of two vertices v and w.
Step1: Find shortest distance of all vertices from root vertex using bfs http://en.wikipedia.org/wiki/Breadth-first_search with time complexity O(|V| + |E|) and also find the parent of each vertices.
Step2: Find the common ancestors of both the vertices by using parent until we reach root vertex Time complexity- 2|v|
Step3: LCA will be that common ancestor which have maximum shortest distance.
So, this is O(|V| + |E|) time complexity algorithm.
Please, correct me if i am wrong or any other suggestions are welcome.
package FB;
import java.util.*;
public class commomAnsectorForGraph {
public static void main(String[] args){
commomAnsectorForGraph com = new commomAnsectorForGraph();
graphNode g = new graphNode('g');
graphNode d = new graphNode('d');
graphNode f = new graphNode('f');
graphNode c = new graphNode('c');
graphNode e = new graphNode('e');
graphNode a = new graphNode('a');
graphNode b = new graphNode('b');
List<graphNode> gc = new ArrayList<>();
gc.add(d);
gc.add(f);
g.children = gc;
List<graphNode> dc = new ArrayList<>();
dc.add(c);
d.children = dc;
List<graphNode> cc = new ArrayList<>();
cc.add(b);
c.children = cc;
List<graphNode> bc = new ArrayList<>();
bc.add(a);
b.children = bc;
List<graphNode> fc = new ArrayList<>();
fc.add(e);
f.children = fc;
List<graphNode> ec = new ArrayList<>();
ec.add(b);
e.children = ec;
List<graphNode> ac = new ArrayList<>();
a.children = ac;
graphNode gn = com.findAncestor(g, c, d);
System.out.println(gn.value);
}
public graphNode findAncestor(graphNode root, graphNode a, graphNode b){
if(root == null) return null;
if(root.value == a.value || root.value == b.value) return root;
List<graphNode> list = root.children;
int count = 0;
List<graphNode> temp = new ArrayList<>();
for(graphNode node : list){
graphNode res = findAncestor(node, a, b);
temp.add(res);
if(res != null) {
count++;
}
}
if(count == 2) return root;
for(graphNode t : temp){
if(t != null) return t;
}
return null;
}
}
class graphNode{
char value;
graphNode parent;
List<graphNode> children;
public graphNode(char value){
this.value = value;
}
}
Everyone.
Try please in Java.
static String recentCommonAncestor(String[] commitHashes, String[][] ancestors, String strID, String strID1)
{
HashSet<String> setOfAncestorsLower = new HashSet<String>();
HashSet<String> setOfAncestorsUpper = new HashSet<String>();
String[] arrPair= {strID, strID1};
Arrays.sort(arrPair);
Comparator<String> comp = new Comparator<String>(){
#Override
public int compare(String s1, String s2) {
return s2.compareTo(s1);
}};
int indexUpper = Arrays.binarySearch(commitHashes, arrPair[0], comp);
int indexLower = Arrays.binarySearch(commitHashes, arrPair[1], comp);
setOfAncestorsLower.addAll(Arrays.asList(ancestors[indexLower]));
setOfAncestorsUpper.addAll(Arrays.asList(ancestors[indexUpper]));
HashSet<String>[] sets = new HashSet[] {setOfAncestorsLower, setOfAncestorsUpper};
for (int i = indexLower + 1; i < commitHashes.length; i++)
{
for (int j = 0; j < 2; j++)
{
if (sets[j].contains(commitHashes[i]))
{
if (i > indexUpper)
if(sets[1 - j].contains(commitHashes[i]))
return commitHashes[i];
sets[j].addAll(Arrays.asList(ancestors[i]));
}
}
}
return null;
}
The idea is very simple. We suppose that commitHashes ordered in downgrade sequence.
We find lowest and upper indexes of strings(hashes-does not mean).
It is clearly that (considering descendant order) the common ancestor can be only after upper index (lower value among hashes).
Then we start enumerating the hashes of commit and build chain of descendent parent chains . For this purpose we have two hashsets are initialised by parents of lowest and upper hash of commit. setOfAncestorsLower, setOfAncestorsUpper. If next hash -commit belongs to any of chains(hashsets),
then if current index is upper than index of lowest hash, then if it is contained in another set (chain) we return the current hash as result. If not, we add its parents (ancestors[i]) to hashset, which traces set of ancestors of set,, where the current element contained. That is the all, basically
First of all, I got a N*N distance matrix, for each point, I calculated its nearest neighbor, so we had a N*2 matrix, It seems like this:
0 -> 1
1 -> 2
2 -> 3
3 -> 2
4 -> 2
5 -> 6
6 -> 7
7 -> 6
8 -> 6
9 -> 8
the second column was the nearest neighbor's index. So this was a special kind of directed
graph, with each vertex had and only had one out-degree.
Of course, we could first transform the N*2 matrix to a standard graph representation, and perform BFS/DFS to get the connected components.
But, given the characteristic of this special graph, is there any other fast way to do the job ?
I will be really appreciated.
Update:
I've implemented a simple algorithm for this case here.
Look, I did not use a union-find algorithm, because the data structure may make things not that easy, and I doubt whether It's the fastest way in my case(I meant practically).
You could argue that the _merge process could be time consuming, but if we swap the edges into the continuous place while assigning new label, the merging may cost little, but it need another N spaces to trace the original indices.
The fastest algorithm for finding connected components given an edge list is the union-find algorithm: for each node, hold the pointer to a node in the same set, with all edges converging to the same node, if you find a path of length at least 2, reconnect the bottom node upwards.
This will definitely run in linear time:
- push all edges into a union-find structure: O(n)
- store each node in its set (the union-find root)
and update the set of non-empty sets: O(n)
- return the set of non-empty sets (graph components).
Since the list of edges already almost forms a union-find tree, it is possible to skip the first step:
for each node
- if the node is not marked as collected
-- walk along the edges until you find an order-1 or order-2 loop,
collecting nodes en-route
-- reconnect all nodes to the end of the path and consider it a root for the set.
-- store all nodes in the set for the root.
-- update the set of non-empty sets.
-- mark all nodes as collected.
return the set of non-empty sets
The second algorithm is linear as well, but only a benchmark will tell if it's actually faster. The strength of the union-find algorithm is its optimization. This delays the optimization to the second step but removes the first step completely.
You can probably squeeze out a little more performance if you join the union step with the nearest neighbor calculation, then collect the sets in the second pass.
If you want to do it sequencially you can do it using weighted quick union and path compression .Complexity O(N+Mlog(log(N))).check this link .
Here is the pseudocode .honoring #pycho 's words
`
public class QuickUnion
{
private int[] id;
public QuickUnion(int N)
{
id = new int[N];
for (int i = 0; i < N; i++) id[i] = i;
}
public int root(int i)
{
while (i != id[i])
{
id[i] = id[id[i]];
i = id[i];
}
return i;
}
public boolean find(int p, int q)
{
return root(p) == root(q);
}
public void unite(int p, int q)
{
int i = root(p);
int j = root(q);
id[i] = j;
}
}
`
#reference https://www.cs.princeton.edu/~rs/AlgsDS07/01UnionFind.pdf
If you want to find connected components parallely, the asymptotic complexity can be reduced to O(log(log(N)) time using pointer jumping and weighted quick union with path compression. Check this link
https://vishwasshanbhog.wordpress.com/2016/05/04/efficient-parallel-algorithm-to-find-the-connected-components-of-the-graphs/
Since each node has only one outgoing edge, you can just traverse the graph one edge at a time until you get to a vertex you've already visited. An out-degree of 1 means any further traversal at this point will only take you where you've already been. The traversed vertices in that path are all in the same component.
In your example:
0->1->2->3->2, so [0,1,2,3] is a component
4->2, so update the component to [0,1,2,3,4]
5->6->7->6, so [5,6,7] is a component
8->6, so update the compoent to [5,6,7,8]
9->8, so update the compoent to [5,6,7,8,9]
You can visit each node exactly once, so time is O(n). Space is O(n) since all you need is a component id for each node, and a list of component ids.
Okay so here's my algorithm for finding a Cut in a graph (I'm not talking about a min cut here)
Say we're given an adjacency list of a non-directed graph.
Choose any vertice on the graph (let this be denoted by pivot)
Choose any other vertice on the graph (randomly). (denote this by x)
If the two vertices have an edge between them, then remove that edge from the graph. And dump all the vertices that x is connected to, onto pivot. (if not then go back to Step 2.
If any other vertices were connected to x, then change the adjacency list so that now x is replaced by pivot. Ie they're connected to Pivot.
If number of vertices is greater than 2 (go back to step 2)
If equal to 2. Just count number of vertices present in adjacency list of either of the 2 points. This will give the cut
My question is, is this algorithm correct?
That is a nice explanation of Krager's Min-Cut Algorithm for undirected graphs.
I think there might one detail you missed. Or perhaps I just mis-read your description.
You want to remove all self-loops.
For instance, after you remove a vertex and run through your algorithm, Vertex A may now have an edge that goes from Vertex A to Vertex A. This is called a self-loop. And they are generated frequently in process of contracting two vertices. As a first step, you can simply check the whole graph for self-loops, though there are some more sophisticated approaches.
Does that make sense?
I'll only change your randomization.
After choosing first vertex, choose another from his adjacency list. Now you are sure that two vertices have the edge between them. Next step is finding the vertex from adjancecy list.
Agree that you should definitely remove self-loop.
Also another point I want to add is after you randomly choose the first vertice, you don't have to randomly choose another node until you have one that is connected to the first node, you can simply choose from the ones that are connected to the first vertice because you know how many nodes are the first chosen one connects to. So a second random selection within a smaller range. This is just effectively randomly choosing an edge (determined by two nodes/vertices). I have some c# code implementing krager's algorithm you can play around. It's not the most efficient code (especially a more efficient data structure can be used) as I tested it on a 200 nodes graph, for 10000 iterations it takes about 30 seconds to run.
using System;
using System.Collections.Generic;
using System.Linq;
namespace MinCut
{
internal struct Graph
{
public int N { get; private set; }
public readonly List<int> Connections;
public Graph(int n) : this()
{
N = n;
Connections = new List<int>();
}
public override bool Equals(object obj)
{
return Equals((Graph)obj);
}
public override int GetHashCode()
{
return base.GetHashCode();
}
private bool Equals(Graph g)
{
return N == g.N;
}
}
internal sealed class GraphContraction
{
public static void Run(IList<Graph> graphs, int i)
{
var liveGraphs = graphs.Count;
if (i >= liveGraphs)
{
throw new Exception("Wrong random index generation; index cannot be larger than the number of nodes");
}
var leftV = graphs[i];
var r = new Random();
var index = r.Next(0, leftV.Connections.Count);
var rightV = graphs.Where(x=>x.N == leftV.Connections[index]).Single();
foreach (var v in graphs.Where(x => !x.Equals(leftV) && x.Connections.Contains(leftV.N)))
{
v.Connections.RemoveAll(x => x == leftV.N);
}
foreach (var c in leftV.Connections)
{
if (c != rightV.N)
{
rightV.Connections.Add(c);
int c1 = c;
graphs.Where(x=> x.N == c1).First().Connections.Add(rightV.N);
}
}
graphs.Remove(leftV);
}
}
}
Given two points A and B in a weighted graph, find all paths from A to B where the length of the path is between C1 and C2.
Ideally, each vertex should only be visited once, although this is not a hard requirement. I suppose I could use a heuristic to sort the results of the algorithm to weed out "silly" paths (e.g. a path that just visits the same two nodes over and over again)
I can think of simple brute force algorithms, but are there any more sophisticed algorithms that will make this more efficient? I can imagine as the graph grows this could become expensive.
In the application I am developing, A & B are actually the same point (i.e. the path must return to the start), if that makes any difference.
Note that this is an engineering problem, not a computer science problem, so I can use an algorithm that is fast but not necessarily 100% accurate. i.e. it is ok if it returns most of the possible paths, or if most of the paths returned are within the given length range.
[UPDATE]
This is what I have so far. I have this working on a small graph (30 nodes with around 100 edges). The time required is < 100ms
I am using a directed graph.
I do a depth first search of all possible paths.
At each new node
For each edge leaving the node
Reject the edge if the path we have already contains this edge (in other words, never go down the same edge in the same direction twice)
Reject the edge if it leads back to the node we just came from (in other words, never double back. This removes a lot of 'silly' paths)
Reject the edge if (minimum distance from the end node of the edge to the target node B + the distance travelled so far) > Maximum path length (C2)
If the end node of the edge is our target node B:
If the path fits within the length criteria, add it to the list of suitable paths.
Otherwise reject the edge (in other words, we only ever visit the target node B at the end of the path. It won't be an intermediate point on a path)
Otherwise, add the edge to our path and recurse into it's target node
I use Dijkstra to precompute the minimum distance of all nodes to the target node.
I wrote some java code to test the DFS approach I suggested: the code does not check for paths in range, but prints all paths. It should be simple to modify the code to only keep those in range. I also ran some simple tests. It seems to be giving correct results with 10 vertices and 50 edges or so, though I did not find time for any thorough testing. I also ran it for 100 vertices and 1000 edges. It doesn't run out of memory and keeps printing new paths till I kill it, of which there are a lot. This is not surprising for randomly generated dense graphs, but may not be the case for real world graphs, for example where vertex degrees follow a power law (specially with narrow weight ranges. Also, if you are just interested in how path lengths are distributed in a range, you can stop once you have generated a certain number.
The program outputs the following:
a) the adjacency list of a randomly generated graph.
b) Set of all paths it has found till now.
public class AllPaths {
int numOfVertices;
int[] status;
AllPaths(int numOfVertices){
this.numOfVertices = numOfVertices;
status = new int[numOfVertices+1];
}
HashMap<Integer,ArrayList<Integer>>adjList = new HashMap<Integer,ArrayList<Integer>>();
class FoundSubpath{
int pathWeight=0;
int[] vertices;
}
// For each vertex, a a list of all subpaths of length less than UB found.
HashMap<Integer,ArrayList<FoundSubpath>>allSubpathsFromGivenVertex = new HashMap<Integer,ArrayList<FoundSubpath>>();
public void printInputGraph(){
System.out.println("Random Graph Adjacency List:");
for(int i=1;i<=numOfVertices;i++){
ArrayList<Integer>toVtcs = adjList.get(new Integer(i));
System.out.print(i+ " ");
if(toVtcs==null){
continue;
}
for(int j=0;j<toVtcs.size();j++){
System.out.print(toVtcs.get(j)+ " ");
}
System.out.println(" ");
}
}
public void randomlyGenerateGraph(int numOfTrials){
Random rnd = new Random();
for(int i=1;i < numOfTrials;i++){
Integer fromVtx = new Integer(rnd.nextInt(numOfVertices)+1);
Integer toVtx = new Integer(rnd.nextInt(numOfVertices)+1);
if(fromVtx.equals(toVtx)){
continue;
}
ArrayList<Integer>toVtcs = adjList.get(fromVtx);
boolean alreadyAdded = false;
if(toVtcs==null){
toVtcs = new ArrayList<Integer>();
}else{
for(int j=0;j<toVtcs.size();j++){
if(toVtcs.get(j).equals(toVtx)){
alreadyAdded = true;
break;
}
}
}
if(!alreadyAdded){
toVtcs.add(toVtx);
adjList.put(fromVtx, toVtcs);
}
}
}
public void addAllViableSubpathsToMap(ArrayList<Integer>VerticesTillNowInPath){
FoundSubpath foundSpObj;
ArrayList<FoundSubpath>foundPathsList;
for(int i=0;i<VerticesTillNowInPath.size()-1;i++){
Integer startVtx = VerticesTillNowInPath.get(i);
if(allSubpathsFromGivenVertex.containsKey(startVtx)){
foundPathsList = allSubpathsFromGivenVertex.get(startVtx);
}else{
foundPathsList = new ArrayList<FoundSubpath>();
}
foundSpObj = new FoundSubpath();
foundSpObj.vertices = new int[VerticesTillNowInPath.size()-i-1];
int cntr = 0;
for(int j=i+1;j<VerticesTillNowInPath.size();j++){
foundSpObj.vertices[cntr++] = VerticesTillNowInPath.get(j);
}
foundPathsList.add(foundSpObj);
allSubpathsFromGivenVertex.put(startVtx,foundPathsList);
}
}
public void printViablePaths(Integer v,ArrayList<Integer>VerticesTillNowInPath){
ArrayList<FoundSubpath>foundPathsList;
foundPathsList = allSubpathsFromGivenVertex.get(v);
if(foundPathsList==null){
return;
}
for(int j=0;j<foundPathsList.size();j++){
for(int i=0;i<VerticesTillNowInPath.size();i++){
System.out.print(VerticesTillNowInPath.get(i)+ " ");
}
FoundSubpath fpObj = foundPathsList.get(j) ;
for(int k=0;k<fpObj.vertices.length;k++){
System.out.print(fpObj.vertices[k]+" ");
}
System.out.println("");
}
}
boolean DfsModified(Integer v,ArrayList<Integer>VerticesTillNowInPath,Integer source,Integer dest){
if(v.equals(dest)){
addAllViableSubpathsToMap(VerticesTillNowInPath);
status[v] = 2;
return true;
}
// If vertex v is already explored till destination, just print all subpaths that meet criteria, using hashmap.
if(status[v] == 1 || status[v] == 2){
printViablePaths(v,VerticesTillNowInPath);
}
// Vertex in current path. Return to avoid cycle.
if(status[v]==1){
return false;
}
if(status[v]==2){
return true;
}
status[v] = 1;
boolean completed = true;
ArrayList<Integer>toVtcs = adjList.get(v);
if(toVtcs==null){
status[v] = 2;
return true;
}
for(int i=0;i<toVtcs.size();i++){
Integer vDest = toVtcs.get(i);
VerticesTillNowInPath.add(vDest);
boolean explorationComplete = DfsModified(vDest,VerticesTillNowInPath,source,dest);
if(explorationComplete==false){
completed = false;
}
VerticesTillNowInPath.remove(VerticesTillNowInPath.size()-1);
}
if(completed){
status[v] = 2;
}else{
status[v] = 0;
}
return completed;
}
}
public class AllPathsCaller {
public static void main(String[] args){
int numOfVertices = 20;
/* This is the number of attempts made to create an edge. The edge is usually created but may not be ( eg, if an edge already exists between randomly attempted source and destination.*/
int numOfEdges = 200;
int src = 1;
int dest = 10;
AllPaths allPaths = new AllPaths(numOfVertices);
allPaths.randomlyGenerateGraph(numOfEdges);
allPaths.printInputGraph();
ArrayList<Integer>VerticesTillNowInPath = new ArrayList<Integer>();
VerticesTillNowInPath.add(new Integer(src));
System.out.println("List of Paths");
allPaths.DfsModified(new Integer(src),VerticesTillNowInPath,new Integer(src),new Integer(dest));
System.out.println("done");
}
}
I think you are on the right track with BFS. I came up with some rough vaguely java-like pseudo-code for a proposed solution using BFS. The idea is to store subpaths found during previous traversals, and their lengths, for reuse. I'll try to improve the code sometime today when I find the time, but hopefully it gives a clue as to where I am going with this. The complexity, I am guessing, should be order O(E).
,
Further comments:
This seems like a reasonable approach, though I am not sure I understand completely. I've constructed a simple example to make sure I do. Lets consider a simple graph with all edges weighted 1, and adjacency list representation as follows:
A->B,C
B->C
C->D,F
F->D
Say we wanted to find all paths from A to F, not just those in range, and destination vertices from a source vertex are explored in alphabetic order. Then the algorithm would work as follows:
First starting with B:
ABCDF
ABCF
Then starting with C:
ACDF
ACF
Is that correct?
A simple improvement in that case, would be to store for each vertex visited, the paths found after the first visit to that node. For example, in this example, once you visit C from B, you find that there are two paths to F from C: CF and CDF. You can save this information, and in the next iteration once you reach C, you can just append CF and CDF to the path you have found, and won't need to explore further.
To find edges in range, you can use the conditions you already described for paths generated as above.
A further thought: maybe you do not need to run Dijkstra's to find shortest paths at all. A subpath's length will be found the first time you traverse the subpath. So, in this example, the length of CDF and CF the first time you visit C via B. This information can be used for pruning the next time C is visited directly via A. This length will be more accurate than that found by Dijkstra's, as it would be the exact value, not the lower bound.
Further comments:
The algorithm can probably be improved with some thought. For example, each time the relaxation step is executed in Dijkstra's algorithm (steps 16-19 in the wikipedia description), the rejected older/newer subpath can be remembered using some data structure, if the older path is a plausible candidate (less than upper bound). In the end, it should be possible to reconstruct all the rejected paths, and keep the ones in range.
This algorithm should be O(V^2).
I think visiting each vertex only once may be too optimistic: algorithms such as Djikstra's shortest path have complexity v^2 for finding a single path, the shortest path. Finding all paths (including shortest path) is a harder problem, so should have complexity at least V^2.
My first thought on approaching the problem is a variation of Djikstra's shortest path algorithm. Applying this algorithm once would give you the length of the shortest path. This gives you a lower bound on the path length between the two vertices. Removing an edge at a time from this shortest path, and recalculating the shortest path should give you slightly longer paths.
In turn, edges can be removed from these slightly longer paths to generate more paths, and so on. You can stop once you have a sufficient number of paths, or if the paths you generate are over your upper bound.
This is my first guess. I am a newbie to stackoverflow: any feedback is welcome.
How can I find (iterate over) ALL the cycles in a directed graph from/to a given node?
For example, I want something like this:
A->B->A
A->B->C->A
but not:
B->C->B
I found this page in my search and since cycles are not same as strongly connected components, I kept on searching and finally, I found an efficient algorithm which lists all (elementary) cycles of a directed graph. It is from Donald B. Johnson and the paper can be found in the following link:
http://www.cs.tufts.edu/comp/150GA/homeworks/hw1/Johnson%2075.PDF
A java implementation can be found in:
http://normalisiert.de/code/java/elementaryCycles.zip
A Mathematica demonstration of Johnson's algorithm can be found here, implementation can be downloaded from the right ("Download author code").
Note: Actually, there are many algorithms for this problem. Some of them are listed in this article:
http://dx.doi.org/10.1137/0205007
According to the article, Johnson's algorithm is the fastest one.
Depth first search with backtracking should work here.
Keep an array of boolean values to keep track of whether you visited a node before. If you run out of new nodes to go to (without hitting a node you have already been), then just backtrack and try a different branch.
The DFS is easy to implement if you have an adjacency list to represent the graph. For example adj[A] = {B,C} indicates that B and C are the children of A.
For example, pseudo-code below. "start" is the node you start from.
dfs(adj,node,visited):
if (visited[node]):
if (node == start):
"found a path"
return;
visited[node]=YES;
for child in adj[node]:
dfs(adj,child,visited)
visited[node]=NO;
Call the above function with the start node:
visited = {}
dfs(adj,start,visited)
The simplest choice I found to solve this problem was using the python lib called networkx.
It implements the Johnson's algorithm mentioned in the best answer of this question but it makes quite simple to execute.
In short you need the following:
import networkx as nx
import matplotlib.pyplot as plt
# Create Directed Graph
G=nx.DiGraph()
# Add a list of nodes:
G.add_nodes_from(["a","b","c","d","e"])
# Add a list of edges:
G.add_edges_from([("a","b"),("b","c"), ("c","a"), ("b","d"), ("d","e"), ("e","a")])
#Return a list of cycles described as a list o nodes
list(nx.simple_cycles(G))
Answer: [['a', 'b', 'd', 'e'], ['a', 'b', 'c']]
First of all - you do not really want to try find literally all cycles because if there is 1 then there is an infinite number of those. For example A-B-A, A-B-A-B-A etc. Or it may be possible to join together 2 cycles into an 8-like cycle etc., etc... The meaningful approach is to look for all so called simple cycles - those that do not cross themselves except in the start/end point. Then if you wish you can generate combinations of simple cycles.
One of the baseline algorithms for finding all simple cycles in a directed graph is this: Do a depth-first traversal of all simple paths (those that do not cross themselves) in the graph. Every time when the current node has a successor on the stack a simple cycle is discovered. It consists of the elements on the stack starting with the identified successor and ending with the top of the stack. Depth first traversal of all simple paths is similar to depth first search but you do not mark/record visited nodes other than those currently on the stack as stop points.
The brute force algorithm above is terribly inefficient and in addition to that generates multiple copies of the cycles. It is however the starting point of multiple practical algorithms which apply various enhancements in order to improve performance and avoid cycle duplication. I was surprised to find out some time ago that these algorithms are not readily available in textbooks and on the web. So I did some research and implemented 4 such algorithms and 1 algorithm for cycles in undirected graphs in an open source Java library here : http://code.google.com/p/niographs/ .
BTW, since I mentioned undirected graphs : The algorithm for those is different. Build a spanning tree and then every edge which is not part of the tree forms a simple cycle together with some edges in the tree. The cycles found this way form a so called cycle base. All simple cycles can then be found by combining 2 or more distinct base cycles. For more details see e.g. this : http://dspace.mit.edu/bitstream/handle/1721.1/68106/FTL_R_1982_07.pdf .
The DFS-based variants with back edges will find cycles indeed, but in many cases it will NOT be minimal cycles. In general DFS gives you the flag that there is a cycle but it is not good enough to actually find cycles. For example, imagine 5 different cycles sharing two edges. There is no simple way to identify cycles using just DFS (including backtracking variants).
Johnson's algorithm is indeed gives all unique simple cycles and has good time and space complexity.
But if you want to just find MINIMAL cycles (meaning that there may be more then one cycle going through any vertex and we are interested in finding minimal ones) AND your graph is not very large, you can try to use the simple method below.
It is VERY simple but rather slow compared to Johnson's.
So, one of the absolutely easiest way to find MINIMAL cycles is to use Floyd's algorithm to find minimal paths between all the vertices using adjacency matrix.
This algorithm is nowhere near as optimal as Johnson's, but it is so simple and its inner loop is so tight that for smaller graphs (<=50-100 nodes) it absolutely makes sense to use it.
Time complexity is O(n^3), space complexity O(n^2) if you use parent tracking and O(1) if you don't.
First of all let's find the answer to the question if there is a cycle.
The algorithm is dead-simple. Below is snippet in Scala.
val NO_EDGE = Integer.MAX_VALUE / 2
def shortestPath(weights: Array[Array[Int]]) = {
for (k <- weights.indices;
i <- weights.indices;
j <- weights.indices) {
val throughK = weights(i)(k) + weights(k)(j)
if (throughK < weights(i)(j)) {
weights(i)(j) = throughK
}
}
}
Originally this algorithm operates on weighted-edge graph to find all shortest paths between all pairs of nodes (hence the weights argument). For it to work correctly you need to provide 1 if there is a directed edge between the nodes or NO_EDGE otherwise.
After algorithm executes, you can check the main diagonal, if there are values less then NO_EDGE than this node participates in a cycle of length equal to the value. Every other node of the same cycle will have the same value (on the main diagonal).
To reconstruct the cycle itself we need to use slightly modified version of algorithm with parent tracking.
def shortestPath(weights: Array[Array[Int]], parents: Array[Array[Int]]) = {
for (k <- weights.indices;
i <- weights.indices;
j <- weights.indices) {
val throughK = weights(i)(k) + weights(k)(j)
if (throughK < weights(i)(j)) {
parents(i)(j) = k
weights(i)(j) = throughK
}
}
}
Parents matrix initially should contain source vertex index in an edge cell if there is an edge between the vertices and -1 otherwise.
After function returns, for each edge you will have reference to the parent node in the shortest path tree.
And then it's easy to recover actual cycles.
All in all we have the following program to find all minimal cycles
val NO_EDGE = Integer.MAX_VALUE / 2;
def shortestPathWithParentTracking(
weights: Array[Array[Int]],
parents: Array[Array[Int]]) = {
for (k <- weights.indices;
i <- weights.indices;
j <- weights.indices) {
val throughK = weights(i)(k) + weights(k)(j)
if (throughK < weights(i)(j)) {
parents(i)(j) = parents(i)(k)
weights(i)(j) = throughK
}
}
}
def recoverCycles(
cycleNodes: Seq[Int],
parents: Array[Array[Int]]): Set[Seq[Int]] = {
val res = new mutable.HashSet[Seq[Int]]()
for (node <- cycleNodes) {
var cycle = new mutable.ArrayBuffer[Int]()
cycle += node
var other = parents(node)(node)
do {
cycle += other
other = parents(other)(node)
} while(other != node)
res += cycle.sorted
}
res.toSet
}
and a small main method just to test the result
def main(args: Array[String]): Unit = {
val n = 3
val weights = Array(Array(NO_EDGE, 1, NO_EDGE), Array(NO_EDGE, NO_EDGE, 1), Array(1, NO_EDGE, NO_EDGE))
val parents = Array(Array(-1, 1, -1), Array(-1, -1, 2), Array(0, -1, -1))
shortestPathWithParentTracking(weights, parents)
val cycleNodes = parents.indices.filter(i => parents(i)(i) < NO_EDGE)
val cycles: Set[Seq[Int]] = recoverCycles(cycleNodes, parents)
println("The following minimal cycle found:")
cycles.foreach(c => println(c.mkString))
println(s"Total: ${cycles.size} cycle found")
}
and the output is
The following minimal cycle found:
012
Total: 1 cycle found
To clarify:
Strongly Connected Components will find all subgraphs that have at least one cycle in them, not all possible cycles in the graph. e.g. if you take all strongly connected components and collapse/group/merge each one of them into one node (i.e. a node per component), you'll get a tree with no cycles (a DAG actually). Each component (which is basically a subgraph with at least one cycle in it) can contain many more possible cycles internally, so SCC will NOT find all possible cycles, it will find all possible groups that have at least one cycle, and if you group them, then the graph will not have cycles.
to find all simple cycles in a graph, as others mentioned, Johnson's algorithm is a candidate.
I was given this as an interview question once, I suspect this has happened to you and you are coming here for help. Break the problem into three questions and it becomes easier.
how do you determine the next valid
route
how do you determine if a point has
been used
how do you avoid crossing over the
same point again
Problem 1)
Use the iterator pattern to provide a way of iterating route results. A good place to put the logic to get the next route is probably the "moveNext" of your iterator. To find a valid route, it depends on your data structure. For me it was a sql table full of valid route possibilities so I had to build a query to get the valid destinations given a source.
Problem 2)
Push each node as you find them into a collection as you get them, this means that you can see if you are "doubling back" over a point very easily by interrogating the collection you are building on the fly.
Problem 3)
If at any point you see you are doubling back, you can pop things off the collection and "back up". Then from that point try to "move forward" again.
Hack: if you are using Sql Server 2008 there is are some new "hierarchy" things you can use to quickly solve this if you structure your data in a tree.
In the case of undirected graph, a paper recently published (Optimal listing of cycles and st-paths in undirected graphs) offers an asymptotically optimal solution. You can read it here http://arxiv.org/abs/1205.2766 or here http://dl.acm.org/citation.cfm?id=2627951
I know it doesn't answer your question, but since the title of your question doesn't mention direction, it might still be useful for Google search
Start at node X and check for all child nodes (parent and child nodes are equivalent if undirected). Mark those child nodes as being children of X. From any such child node A, mark it's children of being children of A, X', where X' is marked as being 2 steps away.). If you later hit X and mark it as being a child of X'', that means X is in a 3 node cycle. Backtracking to it's parent is easy (as-is, the algorithm has no support for this so you'd find whichever parent has X').
Note: If graph is undirected or has any bidirectional edges, this algorithm gets more complicated, assuming you don't want to traverse the same edge twice for a cycle.
If what you want is to find all elementary circuits in a graph you can use the EC algorithm, by JAMES C. TIERNAN, found on a paper since 1970.
The very original EC algorithm as I managed to implement it in php (hope there are no mistakes is shown below). It can find loops too if there are any. The circuits in this implementation (that tries to clone the original) are the non zero elements. Zero here stands for non-existence (null as we know it).
Apart from that below follows an other implementation that gives the algorithm more independece, this means the nodes can start from anywhere even from negative numbers, e.g -4,-3,-2,.. etc.
In both cases it is required that the nodes are sequential.
You might need to study the original paper, James C. Tiernan Elementary Circuit Algorithm
<?php
echo "<pre><br><br>";
$G = array(
1=>array(1,2,3),
2=>array(1,2,3),
3=>array(1,2,3)
);
define('N',key(array_slice($G, -1, 1, true)));
$P = array(1=>0,2=>0,3=>0,4=>0,5=>0);
$H = array(1=>$P, 2=>$P, 3=>$P, 4=>$P, 5=>$P );
$k = 1;
$P[$k] = key($G);
$Circ = array();
#[Path Extension]
EC2_Path_Extension:
foreach($G[$P[$k]] as $j => $child ){
if( $child>$P[1] and in_array($child, $P)===false and in_array($child, $H[$P[$k]])===false ){
$k++;
$P[$k] = $child;
goto EC2_Path_Extension;
} }
#[EC3 Circuit Confirmation]
if( in_array($P[1], $G[$P[$k]])===true ){//if PATH[1] is not child of PATH[current] then don't have a cycle
$Circ[] = $P;
}
#[EC4 Vertex Closure]
if($k===1){
goto EC5_Advance_Initial_Vertex;
}
//afou den ksana theoreitai einai asfales na svisoume
for( $m=1; $m<=N; $m++){//H[P[k], m] <- O, m = 1, 2, . . . , N
if( $H[$P[$k-1]][$m]===0 ){
$H[$P[$k-1]][$m]=$P[$k];
break(1);
}
}
for( $m=1; $m<=N; $m++ ){//H[P[k], m] <- O, m = 1, 2, . . . , N
$H[$P[$k]][$m]=0;
}
$P[$k]=0;
$k--;
goto EC2_Path_Extension;
#[EC5 Advance Initial Vertex]
EC5_Advance_Initial_Vertex:
if($P[1] === N){
goto EC6_Terminate;
}
$P[1]++;
$k=1;
$H=array(
1=>array(1=>0,2=>0,3=>0,4=>0,5=>0),
2=>array(1=>0,2=>0,3=>0,4=>0,5=>0),
3=>array(1=>0,2=>0,3=>0,4=>0,5=>0),
4=>array(1=>0,2=>0,3=>0,4=>0,5=>0),
5=>array(1=>0,2=>0,3=>0,4=>0,5=>0)
);
goto EC2_Path_Extension;
#[EC5 Advance Initial Vertex]
EC6_Terminate:
print_r($Circ);
?>
then this is the other implementation, more independent of the graph, without goto and without array values, instead it uses array keys, the path, the graph and circuits are stored as array keys (use array values if you like, just change the required lines). The example graph start from -4 to show its independence.
<?php
$G = array(
-4=>array(-4=>true,-3=>true,-2=>true),
-3=>array(-4=>true,-3=>true,-2=>true),
-2=>array(-4=>true,-3=>true,-2=>true)
);
$C = array();
EC($G,$C);
echo "<pre>";
print_r($C);
function EC($G, &$C){
$CNST_not_closed = false; // this flag indicates no closure
$CNST_closed = true; // this flag indicates closure
// define the state where there is no closures for some node
$tmp_first_node = key($G); // first node = first key
$tmp_last_node = $tmp_first_node-1+count($G); // last node = last key
$CNST_closure_reset = array();
for($k=$tmp_first_node; $k<=$tmp_last_node; $k++){
$CNST_closure_reset[$k] = $CNST_not_closed;
}
// define the state where there is no closure for all nodes
for($k=$tmp_first_node; $k<=$tmp_last_node; $k++){
$H[$k] = $CNST_closure_reset; // Key in the closure arrays represent nodes
}
unset($tmp_first_node);
unset($tmp_last_node);
# Start algorithm
foreach($G as $init_node => $children){#[Jump to initial node set]
#[Initial Node Set]
$P = array(); // declare at starup, remove the old $init_node from path on loop
$P[$init_node]=true; // the first key in P is always the new initial node
$k=$init_node; // update the current node
// On loop H[old_init_node] is not cleared cause is never checked again
do{#Path 1,3,7,4 jump here to extend father 7
do{#Path from 1,3,8,5 became 2,4,8,5,6 jump here to extend child 6
$new_expansion = false;
foreach( $G[$k] as $child => $foo ){#Consider each child of 7 or 6
if( $child>$init_node and isset($P[$child])===false and $H[$k][$child]===$CNST_not_closed ){
$P[$child]=true; // add this child to the path
$k = $child; // update the current node
$new_expansion=true;// set the flag for expanding the child of k
break(1); // we are done, one child at a time
} } }while(($new_expansion===true));// Do while a new child has been added to the path
# If the first node is child of the last we have a circuit
if( isset($G[$k][$init_node])===true ){
$C[] = $P; // Leaving this out of closure will catch loops to
}
# Closure
if($k>$init_node){ //if k>init_node then alwaya count(P)>1, so proceed to closure
$new_expansion=true; // $new_expansion is never true, set true to expand father of k
unset($P[$k]); // remove k from path
end($P); $k_father = key($P); // get father of k
$H[$k_father][$k]=$CNST_closed; // mark k as closed
$H[$k] = $CNST_closure_reset; // reset k closure
$k = $k_father; // update k
} } while($new_expansion===true);//if we don't wnter the if block m has the old k$k_father_old = $k;
// Advance Initial Vertex Context
}//foreach initial
}//function
?>
I have analized and documented the EC but unfortunately the documentation is in Greek.
There are two steps (algorithms) involved in finding all cycles in a DAG.
The first step is to use Tarjan's algorithm to find the set of strongly connected components.
Start from any arbitrary vertex.
DFS from that vertex. For each node x, keep two numbers, dfs_index[x] and dfs_lowval[x].
dfs_index[x] stores when that node is visited, while dfs_lowval[x] = min(dfs_low[k]) where
k is all the children of x that is not the directly parent of x in the dfs-spanning tree.
All nodes with the same dfs_lowval[x] are in the same strongly connected component.
The second step is to find cycles (paths) within the connected components. My suggestion is to use a modified version of Hierholzer's algorithm.
The idea is:
Choose any starting vertex v, and follow a trail of edges from that vertex until you return to v.
It is not possible to get stuck at any vertex other than v, because the even degree of all vertices ensures that, when the trail enters another vertex w there must be an unused edge leaving w. The tour formed in this way is a closed tour, but may not cover all the vertices and edges of the initial graph.
As long as there exists a vertex v that belongs to the current tour but that has adjacent edges not part of the tour, start another trail from v, following unused edges until you return to v, and join the tour formed in this way to the previous tour.
Here is the link to a Java implementation with a test case:
http://stones333.blogspot.com/2013/12/find-cycles-in-directed-graph-dag.html
I stumbled over the following algorithm which seems to be more efficient than Johnson's algorithm (at least for larger graphs). I am however not sure about its performance compared to Tarjan's algorithm.
Additionally, I only checked it out for triangles so far. If interested, please see "Arboricity and Subgraph Listing Algorithms" by Norishige Chiba and Takao Nishizeki (http://dx.doi.org/10.1137/0214017)
DFS from the start node s, keep track of the DFS path during traversal, and record the path if you find an edge from node v in the path to s. (v,s) is a back-edge in the DFS tree and thus indicates a cycle containing s.
Regarding your question about the Permutation Cycle, read more here:
https://www.codechef.com/problems/PCYCLE
You can try this code (enter the size and the digits number):
# include<cstdio>
using namespace std;
int main()
{
int n;
scanf("%d",&n);
int num[1000];
int visited[1000]={0};
int vindex[2000];
for(int i=1;i<=n;i++)
scanf("%d",&num[i]);
int t_visited=0;
int cycles=0;
int start=0, index;
while(t_visited < n)
{
for(int i=1;i<=n;i++)
{
if(visited[i]==0)
{
vindex[start]=i;
visited[i]=1;
t_visited++;
index=start;
break;
}
}
while(true)
{
index++;
vindex[index]=num[vindex[index-1]];
if(vindex[index]==vindex[start])
break;
visited[vindex[index]]=1;
t_visited++;
}
vindex[++index]=0;
start=index+1;
cycles++;
}
printf("%d\n",cycles,vindex[0]);
for(int i=0;i<(n+2*cycles);i++)
{
if(vindex[i]==0)
printf("\n");
else
printf("%d ",vindex[i]);
}
}
DFS c++ version for the pseudo-code in second floor's answer:
void findCircleUnit(int start, int v, bool* visited, vector<int>& path) {
if(visited[v]) {
if(v == start) {
for(auto c : path)
cout << c << " ";
cout << endl;
return;
}
else
return;
}
visited[v] = true;
path.push_back(v);
for(auto i : G[v])
findCircleUnit(start, i, visited, path);
visited[v] = false;
path.pop_back();
}
http://www.me.utexas.edu/~bard/IP/Handouts/cycles.pdf
The CXXGraph library give a set of algorithms and functions to detect cycles.
For a full algorithm explanation visit the wiki.