{"id":27655,"date":"2018-04-09T20:21:02","date_gmt":"2018-04-09T14:51:02","guid":{"rendered":"https:\/\/www.wikitechy.com\/technology\/?p=27655"},"modified":"2018-09-14T18:50:05","modified_gmt":"2018-09-14T13:20:05","slug":"detect-cycle-directed-graph-using-colors","status":"publish","type":"post","link":"https:\/\/www.wikitechy.com\/technology\/detect-cycle-directed-graph-using-colors\/","title":{"rendered":"Detect Cycle in a directed graph using colors"},"content":{"rendered":"

Given a directed graph, check whether the graph contains a cycle or not. Your function should return true if the given graph contains at least one cycle, else return false. For example, the following graph contains three cycles 0->2->0, 0->1->2->0 and 3->3, so your function must return true.<\/p>\n

Solution<\/strong>
\nDepth First Traversal can be used to detect cycle in a Graph. DFS for a connected graph produces a tree. There is a cycle in a graph only if there is a back edge<\/a> present in the graph. A back edge is an edge that is from a node to itself (selfloop) or one of its ancestor in the tree produced by DFS. In the following graph, there are 3 back edges, marked with cross sign. We can observe that these 3 back edges indicate 3 cycles present in the graph.<\/p>\n

\"Detect<\/p>\n

In the previous post<\/a>, we have discussed a solution that stores visited vertices in a separate array which stores vertices of current recursion call stack.<\/p>\n

In this post a different solution is discussed. The solution is from CLRS book<\/a>. The idea is to do DFS of given graph and while doing traversal, assign one of the below three colors to every vertex.<\/p>\n

WHITE<\/strong> : Vertex is not processed yet.  Initially\r\n        all vertices are WHITE.\r\n\r\nGRAY<\/strong> : Vertex is being processed (DFS for this \r\n       vertex has started, but not finished which means\r\n       that all descendants (ind DFS tree) of this vertex\r\n       are not processed yet (or this vertex is in function\r\n       call stack)\r\n\r\nBLACK<\/strong> : Vertex and all its descendants are \r\n        processed.\r\n\r\nWhile doing DFS, if we encounter an edge from current \r\nvertex to a GRAY vertex, then this edge is back edge \r\nand hence there is a cycle.\r\n<\/pre>\n
Below is C++ implementation based on above idea.<\/p>\n
 
 <\/div> 
\/\/ A DFS based approach to find if there is a cycle\/\/ in a directed graph.  This approach strictly follows\/\/ the algorithm given in CLRS book.#include <bits\/stdc++.h>using namespace std; enum Color {WHITE, GRAY, BLACK}; \/\/ Graph class represents a directed graph using\/\/ adjacency list representationclass Graph{    int V; \/\/ No. of vertices    list<int>* adj; \/\/ adjacency lists     \/\/ DFS traversal of the vertices reachable from v    bool DFSUtil(int v, int color[]);public:    Graph(int V);  \/\/ Constructor     \/\/ function to add an edge to graph    void addEdge(int v, int w);     bool isCyclic();}; \/\/ ConstructorGraph::Graph(int V){    this->V = V;    adj = new list<int>[V];} \/\/ Utility function to add an edgevoid Graph::addEdge(int v, int w){    adj[v].push_back(w); \/\/ Add w to v's list.} \/\/ Recursive function to find if there is back edge\/\/ in DFS subtree tree rooted with 'u'bool Graph::DFSUtil(int u, int color[]){    \/\/ GRAY :  This vertex is being processed (DFS    \/\/         for this vertex has started, but not    \/\/         ended (or this vertex is in function    \/\/         call stack)    color[u] = GRAY;     \/\/ Iterate through all adjacent vertices    list<int>::iterator i;    for (i = adj[u].begin(); i != adj[u].end(); ++i)    {        int v = *i;  \/\/ An adjacent of u         \/\/ If there is        if (color[v] == GRAY)          return true;         \/\/ If v is not processed and there is a back        \/\/ edge in subtree rooted with v        if (color[v] == WHITE && DFSUtil(v, color))          return true;    }     \/\/ Mark this vertex as processed    color[u] = BLACK;     return false;} \/\/ Returns true if there is a cycle in graphbool Graph::isCyclic(){    \/\/ Initialize color of all vertices as WHITE    int *color = new int[V];    for (int i = 0; i < V; i++)        color[i] = WHITE;     \/\/ Do a DFS traversal beginning with all    \/\/ vertices    for (int i = 0; i < V; i++)        if (color[i] == WHITE)           if (DFSUtil(i, color) == true)              return true;     return false;} \/\/ Driver code to test aboveint main(){    \/\/ Create a graph given in the above diagram    Graph g(4);    g.addEdge(0, 1);    g.addEdge(0, 2);    g.addEdge(1, 2);    g.addEdge(2, 0);    g.addEdge(2, 3);    g.addEdge(3, 3);     if (g.isCyclic())        cout << "Graph contains cycle";    else        cout << "Graph doesn't contain cycle";     return 0;}<\/code><\/pre> <\/div>\n
Output :<\/p>\n
Graph contains cycle<\/pre>\n
Time complexity of above solution is O(V + E) where V is number of vertices and E is number of edges in the graph.<\/p>\n","protected":false},"excerpt":{"rendered":"
Detect Cycle in a directed graph using colors-Graph cycle-Depth First Traversal can be used to detect cycle in a Graph. DFS for a connected graph.<\/p>\n","protected":false},"author":1,"featured_media":31270,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[81352],"tags":[76188,76186,76185,76184,81977,76183,76187,81978],"class_list":["post-27655","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-graph-cycle","tag-detect-cycle-in-a-graph-c-code","tag-detect-cycle-in-directed-graph-c","tag-detect-cycle-in-directed-graph-in-c","tag-detect-cycle-in-directed-graph-java","tag-detect-cycle-in-directed-graph-python","tag-detect-cycle-in-undirected-graph","tag-detect-cycle-in-undirected-graph-in-c","tag-find-cycle-in-directed-graph"],"_links":{"self":[{"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/posts\/27655","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/comments?post=27655"}],"version-history":[{"count":0,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/posts\/27655\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/media\/31270"}],"wp:attachment":[{"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/media?parent=27655"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/categories?post=27655"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/tags?post=27655"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}