# Find the Alpha Score of the Given Steps (Using BST)

Given an array A[] consisting of N numbers denoting the values written on N steps, the task is to find the alpha score of the total journey of climbing up all the stairs. Since the answer can be very large, print the answer modulo 109 + 7.

Alpha Score: The alpha score at each step is the sum of all the numbers previously seen on the stairs climbed which are smaller than the number on the current stair.
The alpha score of the total journey is the sum of the alpha scores of each step.

Examples:

Input: A[] = {13, 14, 20}
Output: 87
Explanation:
Alpha Score at the first stair = 13
Alpha Score at the second stair = 13 + 14 = 27
Alpha Score of the third stair = 13 + 14 + 20 = 47
Sum of all the Alpha Scores=13 + 27 + 47 = 87
Therefore, the Alpha Score of the total journey is 87.

Input: A[] = {10, 11, 12}
Output: 64
Explanation:
Alpha Score at the first stair = 10
Alpha Score at the second stair = 10 + 11 = 21
Alpha Score of the third stair = 10+11 + 12 = 33
Sum of all the Alpha Scores =10 + 21 + 33
Therefore, the Alpha Score of the total journey is 64.

Naive Approach:
The simplest approach to solve the problem is to traverse each element in the array and calculate the sum of all the elements smaller than the current element present at previous indices to calculate the Alpha Score of the current stair. For each sum calculated, update the total_sum(Alpha Score of the total journey). Finally, print the total_sum as the Alpha Score of the complete journey.
Time Complexity: O(N2)
Auxiliary Space: O(1)

Efficient Approach:
The above approach can be optimized using Binary Search Trees. Follow the steps below:

• Sort the given array.
• Construct a BST from the sorted array.
• Recursively traverse the BST and follow the steps below:
• Traverse the left subtree.
• Add the value of the current node to sum (Alpha score for current stair) and update the total_sum (Alpha score of the journey till now).
• Traverse the right subtree.
• After complete traversal of the BST, print the total_sum.

Below is the implementation of the above approach:

## Java

 // Java Program to implement // the above approach import java.lang.*; import java.util.*;    // Structure of a node class Node {     Node left, right;     int data;     public Node(int data)     {         this.data = data;         left = null;         right = null;     } }    class AlphaScore {        Node root;        AlphaScore() { root = null; }        static long sum = 0, total_sum = 0;     static long mod = 1000000007;        // Function to calculate and return     // the Alpha Score of the journey     public static long getAlphaScore(Node node)     {         // Traverse left subtree         if (node.left != null)             getAlphaScore(node.left);            // Calculate the alpha score         // of the current step         sum = (sum + node.data) % mod;            // Update alpha score of         // the journey         total_sum = (total_sum + sum) % mod;            // Traverse right subtree         if (node.right != null)             getAlphaScore(node.right);            // Return         return total_sum;     }        // Function to construct a BST     // from the sorted array arr[]     public static Node constructBST(int[] arr, int start,                                     int end, Node root)     {         if (start > end)             return null;            int mid = (start + end) / 2;            // Insert root         if (root == null)             root = new Node(arr[mid]);            // Construct left subtree         root.left             = constructBST(arr, start, mid - 1, root.left);            // Construct right subtree         root.right             = constructBST(arr, mid + 1, end, root.right);            // Return root         return root;     }        // Driver Code     public static void main(String args[])     {            int arr[] = { 10, 11, 12 };         int length = arr.length;            // Sort the array         Arrays.sort(arr);            Node root = null;            // Construct BST from the sorted array         root = constructBST(arr, 0, length - 1, root);            System.out.println(getAlphaScore(root));     } }

## Python3

 # Python3 program to implement  # the above approach     # Structure of a node class Node:     def __init__(self, data):                    self.data = data         self.left = None         self.right = None            sum = 0 total_sum = 0 mod = 1000000007        # Function to calculate and return  # the Alpha Score of the journey  def getAlphaScore(node):            global sum     global total_sum            # Traverse left subtree      if node.left != None:         getAlphaScore(node.left)                # Calculate the alpha score      # of the current step      sum = (sum + node.data) % mod            # Update alpha score of      # the journey      total_sum = (total_sum + sum) % mod            # Traverse right subtree     if node.right != None:         getAlphaScore(node.right)                # Return     return total_sum    # Function to construct a BST  # from the sorted array arr[] def constructBST(arr, start, end, root):            if start > end:         return None            mid = (start + end) // 2            # Insert root     if root == None:         root = Node(arr[mid])                # Construct left subtree     root.left = constructBST(arr, start,                              mid - 1,                               root.left)            # Construct right subtree     root.right = constructBST(arr, mid + 1,                               end, root.right)            # Return root     return root    # Driver code arr = [ 10, 11, 12 ] length = len(arr)    # Sort the array arr.sort()    root = None    # Construct BST from the sorted array root = constructBST(arr, 0, length - 1, root)    print(getAlphaScore(root))    # This code is contributed by Shivam Singh

## C#

 // C# program to implement  // the above approach  using System;    // Structure of a node  class Node  {      public Node left, right;      public int data;      public Node(int data)      {          this.data = data;          left = null;          right = null;      }  }     class AlphaScore{     Node root;     AlphaScore(){root = null;}     static long sum = 0, total_sum = 0;  static long mod = 1000000007;     // Function to calculate and return  // the Alpha Score of the journey  static long getAlphaScore(Node node)  {             // Traverse left subtree      if (node.left != null)          getAlphaScore(node.left);         // Calculate the alpha score      // of the current step      sum = (sum + node.data) % mod;         // Update alpha score of      // the journey      total_sum = (total_sum + sum) % mod;         // Traverse right subtree      if (node.right != null)          getAlphaScore(node.right);         // Return      return total_sum;  }     // Function to construct a BST  // from the sorted array []arr  static Node constructBST(int[] arr, int start,                           int end, Node root)  {      if (start > end)          return null;         int mid = (start + end) / 2;         // Insert root      if (root == null)          root = new Node(arr[mid]);         // Construct left subtree      root.left = constructBST(arr, start,                               mid - 1, root.left);         // Construct right subtree      root.right = constructBST(arr, mid + 1,                                end, root.right);         // Return root      return root;  }     // Driver Code  public static void Main(String []args)  {         int []arr = { 10, 11, 12 };      int length = arr.Length;         // Sort the array      Array.Sort(arr);         Node root = null;         // Construct BST from the sorted array      root = constructBST(arr, 0, length - 1, root);         Console.WriteLine(getAlphaScore(root));  }  }    // This is code contributed by PrinciRaj1992

Output:

64

Time Complexity: O(NlogN)
Auxiliary Space: O(N)

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