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:
C++
// C++ program to implement// the above approach#include<bits/stdc++.h>using namespace std;static long sum = 0, total_sum = 0;static long mod = 1000000007;// Structure of a nodestruct Node{ Node *left, *right; int data; Node(int x) { data = x; left = NULL; right = NULL; }};// Function to calculate and return// the Alpha Score of the journeylong 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[]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 Codeint main(){ int arr[] = { 10, 11, 12 }; int length = 3; // Sort the array sort(arr, arr + length); Node *root = NULL; // Construct BST from the sorted array root = constructBST(arr, 0, length - 1, root); cout << (getAlphaScore(root));}// This code is contributed by mohit kumar 29 |
Java
// Java Program to implement// the above approachimport java.lang.*;import java.util.*;// Structure of a nodeclass 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 nodeclass Node: def __init__(self, data): self.data = data self.left = None self.right = None sum = 0total_sum = 0mod = 1000000007 # Function to calculate and return# the Alpha Score of the journeydef 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 codearr = [ 10, 11, 12 ]length = len(arr)# Sort the arrayarr.sort()root = None# Construct BST from the sorted arrayroot = constructBST(arr, 0, length - 1, root)print(getAlphaScore(root))# This code is contributed by Shivam Singh |
C#
// C# program to implement// the above approachusing System;// Structure of a nodeclass 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 journeystatic 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 []arrstatic 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 Codepublic 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 |
64
Time Complexity: O(NlogN)
Auxiliary Space: O(N)
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