Given a Weighted Directed Acyclic Graph (DAG) and a source vertex s in it, find the longest distances from s to all other vertices in the given graph.
The longest path problem for a general graph is not as easy as the shortest path problem because the longest path problem doesn’t 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 Topological Sorting.
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.
Following figure shows step by step process of finding longest paths.
Following is complete algorithm for finding longest distances.
- Initialize dist[] = {NINF, NINF, ….} and dist[s] = 0 where s is the source vertex. Here NINF means negative infinite.
- Create a topological order of all vertices.
- Do following for every vertex u in topological order.
- ..Do following for every adjacent vertex v of u
- ……if (dist[v] < dist[u] + weight(u, v))
- ………dist[v] = dist[u] + weight(u, v)
Following is C++ implementation of the above algorithm.
CPP
// A C++ program to find single source longest distances// in a DAG#include <iostream>#include <limits.h>#include <list>#include <stack>#define NINF INT_MINusing namespace std;// 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 edge class 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 // representation class 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 ~Graph(); // Destructor // 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]; } Graph::~Graph() // Destructor { delete [] adj; } 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 // https:// 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] << " "; delete [] visited;} // Driver program to test above functions int 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; } |
Java
// A Java program to find single source longest distances// in a DAGimport java.util.*;class GFG{ // 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 edge static class AdjListNode { int v; int weight; 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 // representation static class Graph { int V; // No. of vertices' // Pointer to an array containing adjacency lists ArrayList<ArrayList<AdjListNode>> adj; Graph(int V) // Constructor { this.V = V; adj = new ArrayList<ArrayList<AdjListNode>>(V); for(int i = 0; i < V; i++){ adj.add(new ArrayList<AdjListNode>()); } } void addEdge(int u, int v, int weight) { AdjListNode node = new AdjListNode(v, weight); adj.get(u).add(node); // Add v to u's list } // A recursive function used by longestPath. See below // link for details // https:// www.geeksforgeeks.org/topological-sorting/ void topologicalSortUtil(int v, boolean visited[], Stack<Integer> stack) { // Mark the current node as visited visited[v] = true; // Recur for all the vertices adjacent to this vertex for (int i = 0; i<adj.get(v).size(); i++) { AdjListNode node = adj.get(v).get(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 longestPath(int s) { Stack<Integer> stack = new Stack<Integer>(); int dist[] = new int[V]; // Mark all the vertices as not visited boolean visited[] = new boolean[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] = Integer.MIN_VALUE; dist[s] = 0; // Process vertices in topological order while (stack.isEmpty() == false) { // Get the next vertex from topological order int u = stack.peek(); stack.pop(); // Update distances of all adjacent vertices ; if (dist[u] != Integer.MIN_VALUE) { for (int i = 0; i<adj.get(u).size(); i++) { AdjListNode node = adj.get(u).get(i); if (dist[node.getV()] < dist[u] + node.getWeight()) dist[node.getV()] = dist[u] + node.getWeight(); } } } // Print the calculated longest distances for (int i = 0; i < V; i++) if(dist[i] == Integer.MIN_VALUE) System.out.print("INF "); else System.out.print(dist[i] + " "); } } // Driver program to test above functions public static void main(String args[]) { // 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 = new Graph(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; System.out.print("Following are longest distances from source vertex "+ s + " \n" ); g.longestPath(s); }}// This code is contribute by adityapande88. |
Python3
# A recursive function used by longestPath. See below# link for details# https:#www.geeksforgeeks.org/topological-sorting/def topologicalSortUtil(v): global Stack, visited, adj visited[v] = True # Recur for all the vertices adjacent to this vertex # list<AdjListNode>::iterator i for i in adj[v]: if (not visited[i[0]]): topologicalSortUtil(i[0]) # Push current vertex to stack which stores topological # sort Stack.append(v)# The function to find longest distances from a given vertex.# It uses recursive topologicalSortUtil() to get topological# sorting.def longestPath(s): global Stack, visited, adj, V dist = [-10**9 for i in range(V)] # Call the recursive helper function to store Topological # Sort starting from all vertices one by one for i in range(V): if (visited[i] == False): topologicalSortUtil(i) # print(Stack) # Initialize distances to all vertices as infinite and # distance to source as 0 dist[s] = 0 # Stack.append(1) # Process vertices in topological order while (len(Stack) > 0): # Get the next vertex from topological order u = Stack[-1] del Stack[-1] #print(u) # Update distances of all adjacent vertices # list<AdjListNode>::iterator i if (dist[u] != 10**9): for i in adj[u]: # print(u, i) if (dist[i[0]] < dist[u] + i[1]): dist[i[0]] = dist[u] + i[1] # Print calculated longest distances # print(dist) for i in range(V): print("INF ",end="") if (dist[i] == -10**9) else print(dist[i],end=" ")# Driver codeif __name__ == '__main__': V, Stack, visited = 6, [], [False for i in range(7)] adj = [[] for i in range(7)] # 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 adj[0].append([1, 5]) adj[0].append([2, 3]) adj[1].append([3, 6]) adj[1].append([2, 2]) adj[2].append([4, 4]) adj[2].append([5, 2]) adj[2].append([3, 7]) adj[3].append([5, 1]) adj[3].append([4, -1]) adj[4].append([5, -2]) s = 1 print("Following are longest distances from source vertex ",s) longestPath(s) # This code is contributed by mohit kumar 29. |
C#
// C# program to find single source longest distances// in a DAGusing System;using System.Collections.Generic;// 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 { private int v; private int weight; public AdjListNode(int _v, int _w) { v = _v; weight = _w; } public int getV() { return v; } public int getWeight() { return weight; }}// Class to represent a graph using adjacency list// representationclass Graph { private int V; // No. of vertices' // Pointer to an array containing adjacency lists private List<AdjListNode>[] adj; public Graph(int V) // Constructor { this.V = V; adj = new List<AdjListNode>[ V ]; for (int i = 0; i < V; i++) { adj[i] = new List<AdjListNode>(); } } public void AddEdge(int u, int v, int weight) { AdjListNode node = new AdjListNode(v, weight); adj[u].Add(node); // Add v to u's list } // A recursive function used by longestPath. See below // link for details // https:// www.geeksforgeeks.org/topological-sorting/ private void 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 for (int i = 0; i < adj[v].Count; i++) { AdjListNode node = adj[v][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. public void LongestPath(int s) { Stack<int> stack = new Stack<int>(); int[] dist = new int[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] = Int32.MinValue; dist[s] = 0; // Process vertices in topological order while (stack.Count > 0) { // Get the next vertex from topological order int u = stack.Pop(); // Update distances of all adjacent vertices ; if (dist[u] != Int32.MinValue) { for (int i = 0; i < adj[u].Count; i++) { AdjListNode node = adj[u][i]; if (dist[node.getV()] < dist[u] + node.getWeight()) dist[node.getV()] = dist[u] + node.getWeight(); } } } // Print the calculated longest distances for (int i = 0; i < V; i++) { if (dist[i] == Int32.MinValue) Console.Write("INF "); else Console.Write(dist[i] + " "); } Console.WriteLine(); }}public class GFG { // Driver program to test above functions static void Main(string[] args) { // 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 = new Graph(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; Console.WriteLine( "Following are longest distances from source vertex {0}", s); g.LongestPath(s); }}// This code is contributed by cavi4762 |
Javascript
// A Javascript program to find single source longest distances in a DAG // program to implement stack data structure class stack { constructor() { this.items = []; } // add element to the stack push(element) { return this.items.push(element); } // remove element from the stack pop() { if (this.items.length > 0) { return this.items.pop(); } } // view the last element top() { return this.items[this.items.length - 1]; } // check if the stack is empty empty() { return this.items.length == 0; } // the size of the stack size() { return this.items.length; } // empty the stack clear() { this.items = []; } } let NINF = Number.MIN_VALUE; // 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 edge class AdjListNode { constructor(_v, _w) { this.v = _v; this.weight = _w; } getV() { return this.v; } getWeight() { return this.weight; } } // Class to represent a graph using adjacency list // representation class Graph { // Constructor constructor(V) { this.V = V; // No. of vertices' // Pointer to an array containing adjacency lists this.adj = Array.from(Array(V), () => new Array()); } // A function used by longestPath topologicalSortUtil(v, visited, Stack) { // Mark the current node as visited visited[v] = true; // Recur for all the vertices adjacent to this vertex for (let j in this.adj[v]) { let i = this.adj[v][j]; let node = i; if (!visited[node.getV()]) this.topologicalSortUtil(node.getV(), visited, Stack); } // Push current vertex to stack which stores topological // sort Stack.push(v); } // function to add an edge to graph addEdge(u, v, weight) { let node = new AdjListNode(v, weight); this.adj[u].push(node); // Add v to u's list } // The function to find longest distances from a given vertex. // It uses recursive topologicalSortUtil() to get topological // sorting. longestPath(s) { let Stack = new stack(); let dist = new Array(this.V); // Mark all the vertices as not visited let visited = new Array(this.V); for (let i = 0; i < this.V; i++) { visited[i] = false; } // Call the recursive helper function to store Topological // Sort starting from all vertices one by one for (let i = 0; i < this.V; i++) if (visited[i] == false) this.topologicalSortUtil(i, visited, Stack); // Initialize distances to all vertices as infinite and // distance to source as 0 for (let i = 0; i < this.V; i++) dist[i] = NINF; dist[s] = 0; // Process vertices in topological order while (Stack.empty() == false) { // Get the next vertex from topological order let u = Stack.top(); Stack.pop(); // Update distances of all adjacent vertices if (dist[u] != NINF) { for (let j in this.adj[u]) { let i = this.adj[u][j]; if (dist[i.getV()] < dist[u] + i.getWeight()) dist[i.getV()] = dist[u] + i.getWeight(); } } } // Print the calculated longest distances for (let i = 0; i < this.V; i++) dist[i] == NINF ? console.log("INF ") : console.log(dist[i] + " "); } } // Driver program to test above functions // 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 let g = new Graph(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); let s = 1; console.log("Following are longest distances from source vertex " + s); g.longestPath(s); // This code is contributed by satwiksuman. |
Output
Following are longest distances from source vertex 1 INF 0 2 9 8 10
Time Complexity: 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).
Space complexity : O(V + E), where V is the number of vertices and E is the number of edges in the graph.
Exercise: The above solution print longest distances, extend the code to print paths also.
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