Longest Path in a Directed Acyclic Graph

Last Updated : 14 Mar, 2024

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.

LongestPath

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_MIN 
using 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  
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 DAG 
import 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  
    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 code 
if __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 DAG 
  
using 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 edge 
class 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 
// representation 
class 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 
    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.

Micro Focus

GeeksforGeeks

Improve

Maximum edges that can be added to DAG so that it remains DAG

Topological Sort of a graph using departure time of vertex

BFS and DFS on Graph

Cycle in a Graph

Shortest Paths in Graph

Minimum Spanning Tree in Graph

Topological Sorting in Graph

Connectivity of Graph

Maximum flow in a Graph

Some must do problems on Graph

Longest Path in a Directed Acyclic Graph

CPP

Java

Python3

C#

Javascript

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