{"id":26017,"date":"2017-10-26T09:32:15","date_gmt":"2017-10-26T04:02:15","guid":{"rendered":"https:\/\/www.wikitechy.com\/technology\/?p=26017"},"modified":"2017-10-26T09:32:15","modified_gmt":"2017-10-26T04:02:15","slug":"longest-path-directed-acyclic-graph","status":"publish","type":"post","link":"https:\/\/www.wikitechy.com\/technology\/longest-path-directed-acyclic-graph\/","title":{"rendered":"Longest Path in a Directed Acyclic Graph"},"content":{"rendered":"

Given a Weighted D<\/strong>irected A<\/strong>cyclic G<\/strong>raph (DAG) and a source vertex s in it, find the longest distances from s to all other vertices in the given graph.<\/span><\/p>\n

The longest path problem for a general graph is not as easy as the shortest path problem because the longest path problem doesn\u2019t have optimal substructure property. In fact, the Longest Path problem is NP-Hard for a general graph. However, the longest path problem has a linear time solution for directed acyclic graphs. The idea is similar to linear time solution for shortest path in a directed acyclic graph., we use Tological Sorting.<\/p>\n

We initialize distances to all vertices as minus infinite and distance to source as 0, then we find a topological sorting of the graph. Topological Sorting of a graph represents a linear ordering of the graph (See below, figure (b) is a linear representation of figure (a) ). Once we have topological order (or linear representation), we one by one process all vertices in topological order. For every vertex being processed, we update distances of its adjacent using distance of current vertex.<\/p>\n[ad type=”banner”]\n

Following figure shows step by step process of finding longest paths.<\/p>\n

Following is complete algorithm for finding longest distances.
\n1)<\/strong> Initialize dist[] = {NINF, NINF, \u2026.} and dist[s] = 0 where s is the source vertex. Here NINF means negative infinite.
\n2)<\/strong> Create a toplogical order of all vertices.
\n3)<\/strong> Do following for every vertex u in topological order.
\n\u2026\u2026\u2026..Do following for every adjacent vertex v of u
\n\u2026\u2026\u2026\u2026\u2026\u2026if (dist[v] < dist[u] + weight(u, v))
\n\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026dist[v] = dist[u] + weight(u, v)<\/p>\n

Following is C++ implementation of the above algorithm.<\/p>\n

\n
<\/div> 
\/\/ A C++ program to find single source longest distances in a DAG#include <iostream>#include <list>#include <stack>#include <limits.h>#define NINF INT_MINusing namespace std;#inc\/\/ Graph is represented using adjacency list. Every node of adjacency list\/\/ contains vertex number of the vertex to which edge connects. It also\/\/ contains weight of the edgeclass AdjListNode{    int v;    int weight;public:    AdjListNode(int _v, int _w)  { v = _v;  weight = _w;}    int getV()       {  return v;  }    int getWeight()  {  return weight; }}; \/\/ Class to represent a graph using adjacency list representationclass Graph{    int V;    \/\/ No. of vertices'     \/\/ Pointer to an array containing adjacency lists    list<AdjListNode> *adj;     \/\/ A function used by longestPath    void topologicalSortUtil(int v, bool visited[], stack<int> &Stack);public:    Graph(int V);   \/\/ Constructor     \/\/ function to add an edge to graph    void addEdge(int u, int v, int weight);     \/\/ Finds longest distances from given source vertex    void longestPath(int s);}; Graph::Graph(int V) \/\/ Constructor{    this->V = V;    adj = new list<AdjListNode>[V];} void Graph::addEdge(int u, int v, int weight){    AdjListNode node(v, weight);    adj[u].push_back(node); \/\/ Add v to u's list} \/\/ A recursive function used by longestPath. See below link for details\/\/ http:\/\/www.geeksforgeeks.org\/topological-sorting\/void Graph::topologicalSortUtil(int v, bool visited[], stack<int> &Stack){    \/\/ Mark the current node as visited    visited[v] = true;     \/\/ Recur for all the vertices adjacent to this vertex    list<AdjListNode>::iterator i;    for (i = adj[v].begin(); i != adj[v].end(); ++i)    {        AdjListNode node = *i;        if (!visited[node.getV()])            topologicalSortUtil(node.getV(), visited, Stack);    }     \/\/ Push current vertex to stack which stores topological sort    Stack.push(v);} \/\/ The function to find longest distances from a given vertex. It uses\/\/ recursive topologicalSortUtil() to get topological sorting.void Graph::longestPath(int s){    stack<int> Stack;    int dist[V];     \/\/ Mark all the vertices as not visited    bool *visited = new bool[V];    for (int i = 0; i < V; i++)        visited[i] = false;     \/\/ Call the recursive helper function to store Topological Sort    \/\/ starting from all vertices one by one    for (int i = 0; i < V; i++)        if (visited[i] == false)            topologicalSortUtil(i, visited, Stack);     \/\/ Initialize distances to all vertices as infinite and distance    \/\/ to source as 0    for (int i = 0; i < V; i++)        dist[i] = NINF;    dist[s] = 0;     \/\/ Process vertices in topological order    while (Stack.empty() == false)    {        \/\/ Get the next vertex from topological order        int u = Stack.top();        Stack.pop();         \/\/ Update distances of all adjacent vertices        list<AdjListNode>::iterator i;        if (dist[u] != NINF)        {          for (i = adj[u].begin(); i != adj[u].end(); ++i)             if (dist[i->getV()] < dist[u] + i->getWeight())                dist[i->getV()] = dist[u] + i->getWeight();        }    }     \/\/ Print the calculated longest distances    for (int i = 0; i < V; i++)        (dist[i] == NINF)? cout << "INF ": cout << dist[i] << " ";} \/\/ Driver program to test above functionsint main(){    \/\/ Create a graph given in the above diagram.  Here vertex numbers are    \/\/ 0, 1, 2, 3, 4, 5 with following mappings:    \/\/ 0=r, 1=s, 2=t, 3=x, 4=y, 5=z    Graph g(6);    g.addEdge(0, 1, 5);    g.addEdge(0, 2, 3);    g.addEdge(1, 3, 6);    g.addEdge(1, 2, 2);    g.addEdge(2, 4, 4);    g.addEdge(2, 5, 2);    g.addEdge(2, 3, 7);    g.addEdge(3, 5, 1);    g.addEdge(3, 4, -1);    g.addEdge(4, 5, -2);     int s = 1;    cout << "Following are longest distances from source vertex " << s <<" \\n";    g.longestPath(s);     return 0;}<\/code><\/pre> <\/div>\n
Output:<\/strong><\/p>\n
Following are longest distances from source vertex 1\r\nINF 0 2 9 8 10<\/pre>\n
Time Complexity:<\/strong> Time complexity of topological sorting is O(V+E). After finding topological order, the algorithm process all vertices and for every vertex, it runs a loop for all adjacent vertices. Total adjacent vertices in a graph is O(E). So the inner loop runs O(V+E) times. Therefore, overall time complexity of this algorithm is O(V+E).<\/p>\n[ad type=”banner”]\n<\/div>\n","protected":false},"excerpt":{"rendered":"
Longest Path in a Directed Acyclic Graph – Graph Algorithms – Given a Weighted Directed Acyclic Graph (DAG) and a source vertex s in it.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[74168,83583,73906],"tags":[76584,76585,76586,76580,76587,76582,76581,76583],"class_list":["post-26017","post","type-post","status-publish","format-standard","hentry","category-dfs-and-bfs","category-directed-acyclic-graph","category-graph-algorithms","tag-longest-path-algorithm-dijkstra","tag-longest-path-between-two-nodes-in-a-graph","tag-longest-path-in-a-tree","tag-longest-path-in-a-undirected-graph","tag-longest-path-in-directed-cyclic-graph","tag-longest-path-in-undirected-acyclic-graph","tag-longest-path-in-unweighted-graph","tag-longest-path-in-weighted-undirected-graph"],"_links":{"self":[{"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/posts\/26017","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=26017"}],"version-history":[{"count":0,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/posts\/26017\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/media?parent=26017"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/categories?post=26017"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/tags?post=26017"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}