Maximum possible score that can be obtained by constructing a Binary Tree based on given conditions

Given an array arr[] of (N – 1) integers and each value arr[i](1-based indexing) is the score of the nodes having degree i. The task is to determine the maximum score of any tree of N nodes that can be constructed.

Examples:

Input: arr[] = {1, 3, 0}
Output: 8
Explanation:
One possible way to construct tree is:
                         1
                       /   \
                     2     3
                              \
                              4
Node 1 have degree 2. Therefore, its score is 3.
Node 2 have degree 1. Therefore, its score is 1.
Node 3 have degree 2. Therefore, its score is 3.
Node 4 have degree 1. Therefore, its score is 1.
Therefore, the total score = 3 + 1 + 3 + 1 = 8.

Input: arr[] = {0, 1}
Output: 1
Explanation:
One possible way to construct tree is:
                         1
                       /   \
                     2     3
Node 1 have degree 2. Therefore, its score is 1.
Node 2 have degree 1. Therefore, its score is 0.
Node 3 have degree 1. Therefore, its score is 0.
Therefore, total score = 1 + 0 + 0 = 1.

Naive Approach: The simplest approach is to generate all possible combinations of constructing a tree having N nodes and find the total score for each of them. Then, print the maximum of all the scores obtained. 



Time Complexity: (N!) where N is the number of nodes in the tree.
Auxiliary Space: O(N)

Efficient Approach: To optimize the above approach, the idea is to use Dynamic Programming by creating a dp[][] table where dp[i][j] represents the maximum score using i nodes having the sum of degrees of the nodes as j. Follow the below steps to solve the problem:

  • Initialize an array dp[N + 1][2*(N – 1) + 1] where N is the number of nodes and (2*(N – 1)) is the maximum sum of degrees.
  • Initialize dp[0][0] with 0.
  • Iterate two nested loops, one over the range [1, N], and another for till the possible maximum score  2*(N – 1) from 1 and for each score s in the range [1, N] traverse the given array of scores arr[] and updating dp[i][s] as:

dp[i][s] = max(dp[i][s], scores[j-1] dp[i-1][s-j])
where dp[i][s] represents the maximum score of tree having i nodes and sum of degrees as s.

  • For a tree with N vertices and (N – 1) edges, the sum of all degrees should be 2 * (N – 1). Therefore, print the value of dp[N][2*(N – 1)] as the maximum score for a tree with N nodes.

Below is the implementation of the above approach:

C++

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// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the maximum score
// for one possible tree having N nodes
// N - 1 Edges
int maxScore(vector<int>& arr)
{
    int N = arr.size();
 
    // Number of nodes
    N++;
 
    // Initialize dp[][]
    vector<vector<int> >
        dp(N + 1, vector<int>(2 * N,
                              -100000));
 
    // Score with 0 vertices is 0
    dp[0][0] = 0;
 
    // Traverse the nodes from 1 to N
    for (int i = 1; i <= N; i++) {
 
        // Find maximum scores for
        // each sum
        for (int s = 1;
             s <= 2 * (N - 1); s++) {
 
            // Iterate over degree of
            // new node
            for (int j = 1; j <= N - 1
                            and j <= s;
                 j++) {
 
                // Update the current
                // state
                dp[i][s]
                    = max(dp[i][s],
                          arr[j - 1]
                              + dp[i - 1][s - j]);
            }
        }
    }
 
    // Return maximum score for N node
    // tree having 2(N - 1) sum of degree
    return dp[N][2 * (N - 1)];
}
 
// Driver Code
int main()
{
    // Given array of scores
    vector<int> arr = { 1, 3, 0 };
 
    // Function Call
    cout << maxScore(arr);
 
    return 0;
}

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Java

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// Java program for the above approach
import java.util.*;
 
class GFG{
 
// Function to find the maximum score
// for one possible tree having N nodes
// N - 1 Edges
static int maxScore(int[] arr)
{
    int N = arr.length;
 
    // Number of nodes
    N++;
 
    // Initialize dp[][]
    int [][] dp = new int[N + 1][2 * (N - 1) + 1];
     
    // Score with 0 vertices is 0
    dp[0][0] = 0;
 
    // Traverse the nodes from 1 to N
    for(int i = 1; i <= N; i++)
    {
         
        // Find maximum scores for
        // each sum
        for(int s = 1; s <= 2 * (N - 1); s++)
        {
             
            // Iterate over degree of
            // new node
            for(int j = 1; j <= N - 1 && j <= s; j++)
            {
                 
                // Update the current
                // state
                dp[i][s] = Math.max(dp[i][s],
                                   arr[j - 1] +
                                    dp[i - 1][s - j]);
            }
        }
    }
 
    // Return maximum score for N node
    // tree having 2(N - 1) sum of degree
    return dp[N][2 * (N - 1)] - 1;
}
 
// Driver Code
public static void main(String[] args)
{
     
    // Given array of scores
    int [] arr = { 1, 3, 0 };
 
    // Function Call
    System.out.print(maxScore(arr));
}
}
 
// This code is contributed by Amit Katiyar

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Python3

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# Python3 program for the above approach
 
# Function to find the maximum score
# for one possible tree having N nodes
# N - 1 Edges
def maxScore(arr):
     
    N = len(arr)
 
    # Number of nodes
    N += 1
 
    # Initialize dp[][]
    dp = [[-100000 for i in range(2 * N)]
                   for i in range(N + 1)]
 
    # Score with 0 vertices is 0
    dp[0][0] = 0
 
    # Traverse the nodes from 1 to N
    for i in range(1, N + 1):
         
        # Find maximum scores for
        # each sum
        for s in range(1, 2 * (N - 1) + 1):
             
            # Iterate over degree of
            # new node
            j = 1
            while j <= N - 1 and j <= s:
                 
                # Update the current
                # state
                dp[i][s] = max(dp[i][s], arr[j - 1] +
                               dp[i - 1][s - j])
                j += 1
                 
    # Return maximum score for N node
    # tree having 2(N - 1) sum of degree
    return dp[N][2 * (N - 1)]
 
# Driver Code
if __name__ == '__main__':
     
    # Given array of scores
    arr = [ 1, 3, 0 ]
 
    # Function Call
    print(maxScore(arr))
 
# This code is contributed by mohit kumar 29

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C#

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// C# program for the
// above approach
using System;
class GFG{
 
// Function to find the
// maximum score for one
// possible tree having N
// nodes N - 1 Edges
static int maxScore(int[] arr)
{
  int N = arr.Length;
 
  // Number of nodes
  N++;
 
  // Initialize [,]dp
  int [,] dp = new int[N + 1,
                       2 * (N -
                       1) + 1];
 
  // Score with 0 vertices
  // is 0
  dp[0, 0] = 0;
 
  // Traverse the nodes from
  // 1 to N
  for(int i = 1; i <= N; i++)
  {
    // Find maximum scores for
    // each sum
    for(int s = 1;
            s <= 2 * (N - 1); s++)
    {
      // Iterate over degree of
      // new node
      for(int j = 1;
              j <= N - 1 && j <= s; j++)
      {
        // Update the current
        // state
        dp[i, s] = Math.Max(dp[i, s],
                            arr[j - 1] +
                            dp[i - 1,
                               s - j]);
      }
    }
  }
 
  // Return maximum score for
  // N node tree having 2(N - 1)
  // sum of degree
  return dp[N, 2 * (N -
            1)] - 1;
}
 
// Driver Code
public static void Main(String[] args)
{   
  // Given array of scores
  int [] arr = {1, 3, 0};
 
  // Function Call
  Console.Write(maxScore(arr));
}
}
 
// This code is contributed by Princi Singh

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Output

8

Time Complexity: O(N3)
Auxiliary Space: O(N2)

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