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Maximum difference between node and its ancestor in a Directed Acyclic Graph ( DAG )

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  • Last Updated : 14 Sep, 2021
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Given a 2D array Edges[][], representing a directed edge between the pair of nodes in a Directed Acyclic Connected Graph consisting of N nodes valued from [1, N] and an array arr[] representing weights of each node, the task is to find the maximum absolute difference between the weights of any node and any of its ancestors.

Examples:

Input: N = 5, M = 4, Edges[][2] = {{1, 2}, {2, 3}, {4, 5}, {1, 3}}, arr[] = {13, 8, 3, 15, 18}
Output: 10
Explanation:

From the above graph, it can be observed that the maximum difference between the value of any node and any of its ancestors is 18 (Node 5) – 8 (Node 2) = 10.

Input: N = 4, M = 3, Edges[][2] = {{1, 2}, {2, 4}, {1, 3}}, arr[] = {2, 3, 1, 5}
Output: 3

 

Approach: The idea to solve the given problem is to perform DFS Traversal on the Graph and populate the maximum and minimum values from each node to its child node and find the maximum absolute difference.
Follow the steps below to solve the given problem:

  • Initialize a variable, say ans as INT_MIN to store the required maximum difference.
  • Perform DFS traversal on the given graph to find the maximum absolute difference between the weights of a node and any of its ancestors by performing the following operations:
    • For each source node, say src, update the value of ans to store the maximum of the absolute difference between the weight of src and currentMin and currentMax respectively.
    • Update the value of currentMin as the minimum of currentMin and the value of the source node src.
    • Update the value of currentMax as the maximum of currentMax and the value of the source node src.
    • Now, recursively traverse the child nodes of src and update values of currentMax and currentMin as DFS(child, Adj, ans, currentMin, currentMax).
  • After completing the above steps, print the value of ans as the resultant maximum difference.

Below is the implementation of the above approach:

C++




// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to perform DFS
// Traversal on the given graph
void DFS(int src, vector<int> Adj[],
         int& ans, int arr[],
         int currentMin, int currentMax)
{
 
    // Update the value of ans
    ans = max(
        ans, max(abs(
                     currentMax - arr[src - 1]),
                 abs(currentMin - arr[src - 1])));
 
    // Update the currentMin and currentMax
    currentMin = min(currentMin,
                     arr[src - 1]);
 
    currentMax = min(currentMax,
                     arr[src - 1]);
 
    // Traverse the adjacency
    // list of the node src
    for (auto& child : Adj[src]) {
 
        // Recursively call
        // for the child node
        DFS(child, Adj, ans, arr,
            currentMin, currentMax);
    }
}
 
// Function to calculate maximum absolute
// difference between a node and its ancestor
void getMaximumDifference(int Edges[][2],
                          int arr[], int N,
                          int M)
{
 
    // Stores the adjacency list of graph
    vector<int> Adj[N + 1];
 
    // Create Adjacency list
    for (int i = 0; i < M; i++) {
        int u = Edges[i][0];
        int v = Edges[i][1];
 
        // Add a directed edge
        Adj[u].push_back(v);
    }
 
    int ans = 0;
 
    // Perform DFS Traversal
    DFS(1, Adj, ans, arr,
        arr[0], arr[0]);
 
    // Print the maximum
    // absolute difference
    cout << ans;
}
 
// Driver Code
int main()
{
    int N = 5, M = 4;
    int Edges[][2]
        = { { 1, 2 }, { 2, 3 },
            { 4, 5 }, { 1, 3 } };
    int arr[] = { 13, 8, 3, 15, 18 };
 
    getMaximumDifference(Edges, arr, N, M);
 
    return 0;
}

Java




// Java program for the above approach
import java.util.*;
 
class GFG{
     
static int ans;
 
// Function to perform DFS
// Traversal on the given graph
static void DFS(int src,
                ArrayList<ArrayList<Integer> > Adj,
                int arr[], int currentMin,
                int currentMax)
{
     
    // Update the value of ans
    ans = Math.max(ans,
          Math.max(Math.abs(currentMax - arr[src - 1]),
                   Math.abs(currentMin - arr[src - 1])));
 
    // Update the currentMin and currentMax
    currentMin = Math.min(currentMin, arr[src - 1]);
 
    currentMax = Math.min(currentMax, arr[src - 1]);
 
    // Traverse the adjacency
    // list of the node src
    for(Integer child : Adj.get(src))
    {
         
        // Recursively call
        // for the child node
        DFS(child, Adj, arr, currentMin, currentMax);
    }
}
 
// Function to calculate maximum absolute
// difference between a node and its ancestor
static void getMaximumDifference(int Edges[][],
                                 int arr[], int N,
                                 int M)
{
    ans = 0;
     
    // Stores the adjacency list of graph
    ArrayList<ArrayList<Integer>> Adj = new ArrayList<>();
 
    for(int i = 0; i < N + 1; i++)
        Adj.add(new ArrayList<>());
 
    // Create Adjacency list
    for(int i = 0; i < M; i++)
    {
        int u = Edges[i][0];
        int v = Edges[i][1];
 
        // Add a directed edge
        Adj.get(u).add(v);
    }
 
    // Perform DFS Traversal
    DFS(1, Adj, arr, arr[0], arr[0]);
 
    // Print the maximum
    // absolute difference
    System.out.println(ans);
}
 
// Driver code
public static void main(String[] args)
{
    int N = 5, M = 4;
    int Edges[][] = { { 1, 2 }, { 2, 3 },
                      { 4, 5 }, { 1, 3 } };
    int arr[] = { 13, 8, 3, 15, 18 };
 
    getMaximumDifference(Edges, arr, N, M);
}
}
 
// This code is contributed by offbeat

Python3




# Python3 program for the above approach
ans = 0
 
# Function to perform DFS
# Traversal on the given graph
def DFS(src, Adj, arr, currentMin, currentMax):
     
    # Update the value of ans
    global ans
    ans = max(ans, max(abs(currentMax - arr[src - 1]),
                       abs(currentMin - arr[src - 1])))
 
    # Update the currentMin and currentMax
    currentMin = min(currentMin, arr[src - 1])
 
    currentMax = min(currentMax, arr[src - 1])
 
    # Traverse the adjacency
    # list of the node src
    for child in Adj[src]:
         
        # Recursively call
        # for the child node
        DFS(child, Adj, arr, currentMin, currentMax)
 
# Function to calculate maximum absolute
# difference between a node and its ancestor
def getMaximumDifference(Edges, arr, N, M):
     
    global ans
     
    # Stores the adjacency list of graph
    Adj = [[] for i in range(N + 1)]
 
    # Create Adjacency list
    for i in range(M):
        u = Edges[i][0]
        v = Edges[i][1]
 
        # Add a directed edge
        Adj[u].append(v)
 
    # Perform DFS Traversal
    DFS(1, Adj, arr, arr[0], arr[0])
 
    # Print the maximum
    # absolute difference
    print(ans)
 
# Driver Code
if __name__ == '__main__':
     
    N = 5
    M = 4
    Edges = [[1, 2], [2, 3], [4, 5], [1, 3]]
    arr =  [13, 8, 3, 15, 18]
     
    getMaximumDifference(Edges, arr, N, M)
 
# This code is contributed by ipg2016107

C#




// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG {
     
    static int ans;
  
    // Function to perform DFS
    // Traversal on the given graph
    static void DFS(int src, List<List<int>> Adj, int[] arr,
                    int currentMin, int currentMax)
    {
          
        // Update the value of ans
        ans = Math.Max(ans,
              Math.Max(Math.Abs(currentMax - arr[src - 1]),
                       Math.Abs(currentMin - arr[src - 1])));
      
        // Update the currentMin and currentMax
        currentMin = Math.Min(currentMin, arr[src - 1]);
      
        currentMax = Math.Min(currentMax, arr[src - 1]);
      
        // Traverse the adjacency
        // list of the node src
        foreach(int child in Adj[src])
        {
              
            // Recursively call
            // for the child node
            DFS(child, Adj, arr, currentMin, currentMax);
        }
    }
      
    // Function to calculate maximum absolute
    // difference between a node and its ancestor
    static void getMaximumDifference(int[,] Edges,
                                     int[] arr, int N, int M)
    {
        ans = 0;
          
        // Stores the adjacency list of graph
        List<List<int>> Adj = new List<List<int>>();
      
        for(int i = 0; i < N + 1; i++)
            Adj.Add(new List<int>());
      
        // Create Adjacency list
        for(int i = 0; i < M; i++)
        {
            int u = Edges[i,0];
            int v = Edges[i,1];
      
            // Add a directed edge
            Adj[u].Add(v);
        }
      
        // Perform DFS Traversal
        DFS(1, Adj, arr, arr[0], arr[0]);
      
        // Print the maximum
        // absolute difference
        Console.WriteLine(ans);
    }
     
  static void Main() {
    int N = 5, M = 4;
    int[,] Edges = { { 1, 2 }, { 2, 3 },
                      { 4, 5 }, { 1, 3 } };
    int[] arr = { 13, 8, 3, 15, 18 };
  
    getMaximumDifference(Edges, arr, N, M);
  }
}
 
// This code is contributed by rameshtravel07.

Javascript




<script>
 
// Javascript program for the above approach
 
var ans = 0;
 
// Function to perform DFS
// Traversal on the given graph
function DFS(src, Adj, arr, currentMin, currentMax)
{
 
    // Update the value of ans
    ans = Math.max(
        ans, Math.max(Math.abs(
                     currentMax - arr[src - 1]),
                 Math.abs(currentMin - arr[src - 1])));
 
    // Update the currentMin and currentMax
    currentMin = Math.min(currentMin,
                     arr[src - 1]);
 
    currentMax = Math.min(currentMax,
                     arr[src - 1]);
 
    // Traverse the adjacency
    // list of the node src
    Adj[src].forEach(child => {
       
        // Recursively call
        // for the child node
        DFS(child, Adj,arr,
            currentMin, currentMax);
    });
}
 
// Function to calculate maximum absolute
// difference between a node and its ancestor
function getMaximumDifference(Edges, arr, N, M)
{
 
    // Stores the adjacency list of graph
    var Adj = Array.from(Array(N+1), ()=> Array());
 
    // Create Adjacency list
    for (var i = 0; i < M; i++) {
        var u = Edges[i][0];
        var v = Edges[i][1];
 
        // Add a directed edge
        Adj[u].push(v);
    }
 
    // Perform DFS Traversal
    DFS(1, Adj, arr,
        arr[0], arr[0]);
 
    // Print the maximum
    // absolute difference
    document.write( ans);
}
 
// Driver Code
var N = 5, M = 4;
var Edges
    = [ [ 1, 2 ], [ 2, 3 ],
        [ 4, 5 ], [ 1, 3 ] ];
var arr = [13, 8, 3, 15, 18];
getMaximumDifference(Edges, arr, N, M);
 
</script>

Output: 

10

 

Time Complexity: O(N + M)
Auxiliary Space: O(N)

 


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