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Maximum sum of nodes in Binary tree such that no two are adjacent | Dynamic Programming

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  • Difficulty Level : Hard
  • Last Updated : 21 Jun, 2022

Given a N-ary tree with a value associated with each node, the task is to choose a subset of these nodes such that sum of chosen nodes is maximum under a constraint that no two chosen node in subset should be directly connected that is, if we have taken a node in our sum then we can’t take its any children in consideration and vice versa.
Examples: 
 

The above diagram selects the nodes with a deep green color to get the maximum value 25. 
 

An approach to this problem has been discussed in the previous post using recursion
In this post, we will be discussing an approach using Dynamic Programming on Trees
While solving the problem, there arise two cases: 

  1. For a particular node, the maximum sum can be calculated by including the node itself along with nodes from its subtree.
  2. Or, the maximum sum is calculated by excluding the current node and including only the nodes from its subtree.

Let us assume

  • dp1[node] to be the maximum possible sum by choosing nodes from the subtree of this node and also including the node.
  • And, dp2[node] to be the maximum possible sum by choosing nodes from the subtree of the node and not including the node itself.

In the first case, if we include the current node, then its value is added and then we can not include any of its immediate children, hence the summation of dp2[] of all the children will be taken into the count to compute dp1[node]. That is, 
 

dp1[node] = tree[node] + sum(dp2[children1], dp2[children2], …) 

In the second case, if we do not include the current node, then its value is not added, but the children node can be taken or it cannot be taken, hence the summation of the maximum of both for all the children will be taken into count to compute dp2[node]. That is, 
 

dp2[node] = tree[node] + sum(max(dp1[children1], dp2[children1]), max(dp1[children2], dp2[children2])…) 

In the end, the final answer will be the maximum of dp1[root] and dp2[root]. 
Below is the implementation of the above approach: 
 

C++




// C++ program to find maximum sum
// of a subset of nodes that are not adjacent
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the diameter of the tree
// using Dynamic Programming
void dfs(int node, int parent, int dp1[], int dp2[],
                            list<int>* adj, int tree[])
{
 
    int sum1 = 0, sum2 = 0;
 
    // Traverse for all children of node
    for (auto i = adj[node].begin(); i != adj[node].end(); ++i) {
        if (*i == parent)
            continue;
 
        // Call DFS function again
        dfs(*i, node, dp1, dp2, adj, tree);
 
        // Include the current node
        // then donot include the children
        sum1 += dp2[*i];
 
        // Donot include current node,
        // then include children or not include them
        sum2 += max(dp1[*i], dp2[*i]);
    }
 
    // Recurrence value
    dp1[node] = tree[node] + sum1;
    dp2[node] = sum2;
}
 
/* Driver program to test above functions */
int main()
{
    int n = 5;
 
    /* Constructed tree is
        1
        / \
        2 3
       / \
       4 5 */
    list<int>* adj = new list<int>[n + 1];
 
    /* create undirected edges */
    adj[1].push_back(2);
    adj[2].push_back(1);
    adj[1].push_back(3);
    adj[3].push_back(1);
    adj[2].push_back(4);
    adj[4].push_back(2);
    adj[2].push_back(5);
    adj[5].push_back(2);
 
    // Numbers to node
    int tree[n + 1];
    tree[1] = 10;
    tree[2] = 5;
    tree[3] = 11;
    tree[4] = 6;
    tree[5] = 8;
 
    int dp1[n + 1], dp2[n + 1];
    memset(dp1, 0, sizeof dp1);
    memset(dp2, 0, sizeof dp2);
 
    dfs(1, 1, dp1, dp2, adj, tree);
 
    // Find maximum sum by calling function
    cout << "Maximum sum: "
         << max(dp1[1], dp2[1]) << endl;
    return 0;
}

Python3




# Python3 program to find
# maximum sum of a subset
# of nodes that are not
# adjacent
 
# Function to find the diameter
# of the tree using Dynamic
# Programming
def dfs(node, parent, dp1,
        dp2, adj, tree):
  
    sum1 = 0
    sum2 = 0
  
    # Traverse for all
    # children of node
    for i in adj[node]:   
        if (i == parent):
            continue;
  
        # Call DFS function
        # again
        dfs(i, node, dp1,
            dp2, adj, tree);
  
        # Include the current
        # node then donot include
        # the children
        sum1 += dp2[i];
  
        # Donot include current node,
        # then include children or not
        # include them
        sum2 += max(dp1[i],
                    dp2[i]);   
  
    # Recurrence value
    dp1[node] = tree[node] + sum1;
    dp2[node] = sum2;
 
# Driver code
if __name__=="__main__":
     
    n = 5;
     
    ''' Constructed tree is
        1
        / \
        2 3
       / \
       4 5 '''
        
    adj = [[] for i in range(n + 1)]
  
    # create undirected edges
    adj[1].append(2);
    adj[2].append(1);
    adj[1].append(3);
    adj[3].append(1);
    adj[2].append(4);
    adj[4].append(2);
    adj[2].append(5);
    adj[5].append(2);
  
    # Numbers to node
    tree = [0 for i in range(n + 1)];
    tree[1] = 10;
    tree[2] = 5;
    tree[3] = 11;
    tree[4] = 6;
    tree[5] = 8;
  
    dp1 = [0 for i in range(n + 1)];
    dp2 = [0 for i in range(n + 1)];
  
    dfs(1, 1, dp1, dp2, adj, tree);
  
    # Find maximum sum by calling
    # function
    print("Maximum sum:",
          max(dp1[1], dp2[1]))
     
# This code is contributed by Rutvik_56

C#




// C# program to find maximum sum
// of a subset of nodes that are not adjacent
using System;
using System.Collections;
 
class GFG
{
 
// Function to find the diameter of the tree
// using Dynamic Programming
public static void dfs(int node, int parent, int []dp1, int []dp2,
                            ArrayList []adj, int []tree)
{
  
    int sum1 = 0, sum2 = 0;
  
    // Traverse for all children of node
    foreach(int i in adj[node])
    {
        if (i == parent)
            continue;
  
        // Call DFS function again
        dfs(i, node, dp1, dp2, adj, tree);
  
        // Include the current node
        // then donot include the children
        sum1 += dp2[i];
  
        // Donot include current node,
        // then include children or not include them
        sum2 += Math.Max(dp1[i], dp2[i]);
    }
  
    // Recurrence value
    dp1[node] = tree[node] + sum1;
    dp2[node] = sum2;
}
  
/* Driver program to test above functions */
public static void Main(string []arg)
{
    int n = 5;
  
    /* Constructed tree is
        1
        / \
        2 3
       / \
       4 5 */
    ArrayList []adj = new ArrayList[n + 1];
     
    for(int i = 0; i < n + 1; i++)
    {
        adj[i] = new ArrayList();
    }
  
    /* create undirected edges */
    adj[1].Add(2);
    adj[2].Add(1);
    adj[1].Add(3);
    adj[3].Add(1);
    adj[2].Add(4);
    adj[4].Add(2);
    adj[2].Add(5);
    adj[5].Add(2);
  
    // Numbers to node
    int []tree = new int[n + 1];
    tree[1] = 10;
    tree[2] = 5;
    tree[3] = 11;
    tree[4] = 6;
    tree[5] = 8;
  
    int []dp1 = new int[n + 1];
    int []dp2 = new int[n + 1];
    Array.Fill(dp1, 0);
    Array.Fill(dp2, 0);
  
    dfs(1, 1, dp1, dp2, adj, tree);
  
    // Find maximum sum by calling function
    Console.Write("Maximum sum: "+ Math.Max(dp1[1], dp2[1]));
}
}
 
// This code is contributed by pratham76

Javascript




<script>
 
// Python3 program to find
// maximum sum of a subset
// of nodes that are not
// adjacent
 
// Function to find the diameter
// of the tree using Dynamic
// Programming
function dfs(node, parent, dp1,dp2, adj, tree){
  
    let sum1 = 0
    let sum2 = 0
  
    // Traverse for all
    // children of node
    for(let i of adj[node]){ 
        if (i == parent)
            continue;
  
        // Call DFS function
        // again
        dfs(i, node, dp1,
            dp2, adj, tree);
  
        // Include the current
        // node then donot include
        // the children
        sum1 += dp2[i];
  
        // Donot include current node,
        // then include children or not
        // include them
        sum2 += Math.max(dp1[i],
                    dp2[i]);   
    }
  
    // Recurrence value
    dp1[node] = tree[node] + sum1;
    dp2[node] = sum2;
}
 
// Driver code
     
    let n = 5;
     
    //  Constructed tree is
    //     1
    //     / \
    //     2 3
    //    / \
    //    4 5
        
    let adj = new Array(n+1);
    for(let i=0;i<n+1;i++){
      adj[i] = new Array();
    }
  
    // create undirected edges
    adj[1].push(2);
    adj[2].push(1);
    adj[1].push(3);
    adj[3].push(1);
    adj[2].push(4);
    adj[4].push(2);
    adj[2].push(5);
    adj[5].push(2);
  
    // Numbers to node
    let tree = new Array(n+1).fill(0);
    tree[1] = 10;
    tree[2] = 5;
    tree[3] = 11;
    tree[4] = 6;
    tree[5] = 8;
  
    let dp1 = new Array(n+1).fill(0);
    let dp2 = new Array(n+1).fill(0);
  
    dfs(1, 1, dp1, dp2, adj, tree);
  
    // Find maximum sum by calling
    // function
    document.write("Maximum sum: ", Math.max(dp1[1], dp2[1]),"</br>")
     
// This code is contributed by shinjanpatra
 
</script>

Output: 

Maximum sum: 25

 

Time Complexity: O(N), as we are using recursion to traverse the tree. Where N is the number of nodes in the tree.

Auxiliary Space: O(N), as we are using extra space for the array dp.


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