给定一个由N个数字组成的数组A [] ，表示在N个台阶上写入的值，任务是找到爬上所有楼梯的总行程的alpha分数。由于答案可能非常大，请以10 9 + 7模输出答案。
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.
例子：
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.
天真的方法：
解决此问题的最简单方法是遍历数组中的每个元素，并计算所有小于先前索引处当前元素的元素的总和，以计算当前楼梯的Alpha得分。对于每个计算出的总和，更新total_sum (总旅程的Alpha分数)。最后，将total_sum打印为整个旅程的Alpha分数。
时间复杂度：O(N 2 )
辅助空间：O(1)
高效方法：
可以使用二进制搜索树来优化上述方法。请按照以下步骤操作：
- 对给定的数组进行排序。
- 从排序后的数组构造一个BST。
- 递归地遍历BST并遵循以下步骤：
- 遍历左子树。
- 将当前节点的值加到总和(当前楼梯的Alpha分数)并更新total_sum (到目前为止的旅程的Alpha分数)。
- 遍历右子树。
- 完成BST遍历后，打印total_sum 。
下面是上述方法的实现：
C++
// C++ program to implement
// the above approach
#include
using namespace std;
static long sum = 0, total_sum = 0;
static long mod = 1000000007;
// Structure of a node
struct 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 journey
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[]
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
int 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 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 SinghC#
// 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输出：
64
时间复杂度： O(NlogN)
辅助空间：O(N)