Count of Array elements greater than all elements on its left and at least K elements on its right

Given an array A[ ] consisting of N distinct integers, the task is to find the number of elements which are strictly greater than all the elements preceding it and strictly greater than at least K elements on its right.

Examples:

Input: A[] = {2, 5, 1, 7, 3, 4, 0}, K = 3
Output: 2
Explanation:
The only array elements satisfying the given conditions are:

  • 5: Greater than all elements on its left {2} and at least K(= 3) elements on its right {1, 3, 4, 0}
  • 7: Greater than all elements on its left {2, 5, 1} and at least K(= 3) elements on its right {3, 4, 0}

Therefore, the count is 2.

Input: A[] = {11, 2, 4, 7, 5, 9, 6, 3}, K = 2
Output: 1



Naive Approach:
The simplest approach to solve the problem is to traverse the array and for each element, traverse all the elements on its left and check if all of them are smaller than it or not and traverse all elements on its right to check if at least K elements are smaller than it or not. For every element satisfying the conditions, increase count. Finally, print the value of count.
Time Complexity: O(N2)
Auxiliary Space: O(1)

Efficient Approach:
The above approach can be further optimized by using Self-Balancing BST. Follow the steps below:

  • Traverse the array from right to left and insert all elements one by one in an AVL Tree
  • Using the AVL Tree generate an array countSmaller[] which contains the count of smaller elements on the right of every array element.
  • Traverse the array and for every ith element, check if it is the maximum obtained so far and countSmaller[i] is greater than or equal to K.
  • If so, increase count.
  • Print the final value of count as the answer.

Below is the implementation of the above approach:

C++

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// C++ Program to implement
// the above appraoch
#include <bits/stdc++.h>
using namespace std;
  
// Structure of an AVL Tree Node
struct node {
    int key;
    struct node* left;
    struct node* right;
    int height;
    // Size of the tree rooted
    // with this node
    int size;
};
  
// Utility function to get maximum
// of two integers
int max(int a, int b);
  
// Utility function to get height
// of the tree rooted with N
int height(struct node* N)
{
    if (N == NULL)
        return 0;
    return N->height;
}
  
// Utility function to find size of
// the tree rooted with N
int size(struct node* N)
{
    if (N == NULL)
        return 0;
    return N->size;
}
  
// Utility function to get maximum
// of two integers
int max(int a, int b)
{
    return (a > b) ? a : b;
}
  
// Helper function to allocates a
// new node with the given key
struct node* newNode(int key)
{
    struct node* node
        = (struct node*)
            malloc(sizeof(struct node));
    node->key = key;
    node->left = NULL;
    node->right = NULL;
    node->height = 1;
    node->size = 1;
    return (node);
}
  
// Utility function to right rotate
// subtree rooted with y
struct node* rightRotate(struct node* y)
{
    struct node* x = y->left;
    struct node* T2 = x->right;
  
    // Perform rotation
    x->right = y;
    y->left = T2;
  
    // Update heights
    y->height = max(height(y->left),
                    height(y->right))
                + 1;
    x->height = max(height(x->left),
                    height(x->right))
                + 1;
  
    // Update sizes
    y->size = size(y->left)
              + size(y->right) + 1;
    x->size = size(x->left)
              + size(x->right) + 1;
  
    // Return new root
    return x;
}
  
// Utility function to left rotate
// subtree rooted with x
struct node* leftRotate(struct node* x)
{
    struct node* y = x->right;
    struct node* T2 = y->left;
  
    // Perform rotation
    y->left = x;
    x->right = T2;
  
    // Update heights
    x->height = max(height(x->left),
                    height(x->right))
                + 1;
    y->height = max(height(y->left),
                    height(y->right))
                + 1;
  
    // Update sizes
    x->size = size(x->left)
              + size(x->right) + 1;
    y->size = size(y->left)
              + size(y->right) + 1;
  
    // Return new root
    return y;
}
  
// Function to obtain Balance factor
// of node N
int getBalance(struct node* N)
{
    if (N == NULL)
        return 0;
  
    return height(N->left)
           - height(N->right);
}
  
// Function to insert a new key to the
// tree rooted with node
struct node* insert(struct node* node, int key,
                    int* count)
{
    // Perform the normal BST rotation
    if (node == NULL)
        return (newNode(key));
  
    if (key < node->key)
        node->left
            = insert(node->left, key, count);
    else {
        node->right
            = insert(node->right, key, count);
  
        // Update count of smaller elements
        *count = *count + size(node->left) + 1;
    }
  
    // Update height and size of the ancestor
    node->height = max(height(node->left),
                       height(node->right))
                   + 1;
    node->size = size(node->left)
                 + size(node->right) + 1;
  
    // Get the balance factor of the ancestor
    int balance = getBalance(node);
  
    // Left Left Case
    if (balance > 1 && key < node->left->key)
        return rightRotate(node);
  
    // Right Right Case
    if (balance < -1 && key > node->right->key)
        return leftRotate(node);
  
    // Left Right Case
    if (balance > 1 && key > node->left->key) {
        node->left = leftRotate(node->left);
        return rightRotate(node);
    }
  
    // Right Left Case
    if (balance < -1 && key < node->right->key) {
        node->right = rightRotate(node->right);
        return leftRotate(node);
    }
  
    return node;
}
  
// Function to generate an array which contains
// count of smaller elements on the right
void constructLowerArray(int arr[],
                         int countSmaller[],
                         int n)
{
    int i, j;
    struct node* root = NULL;
  
    for (i = 0; i < n; i++)
        countSmaller[i] = 0;
  
    // Insert all elements in the AVL Tree
    // and get the count of smaller elements
    for (i = n - 1; i >= 0; i--) {
        root = insert(root, arr[i],
                      &countSmaller[i]);
    }
}
  
// Function to find the number
// of elements which are greater
// than all elements on its left
// and K elements on its right
int countElements(int A[], int n, int K)
{
  
    int count = 0;
  
    // Stores the count of smaller
    // elements on its right
    int* countSmaller
        = (int*)malloc(sizeof(int) * n);
    constructLowerArray(A, countSmaller, n);
  
    int maxi = INT_MIN;
    for (int i = 0; i <= (n - K - 1); i++) {
        if (A[i] > maxi && countSmaller[i] >= K) {
            count++;
            maxi = A[i];
        }
    }
  
    return count;
}
  
// Driver Code
int main()
{
  
    int A[] = { 2, 5, 1, 7, 3, 4, 0 };
    int n = sizeof(A) / sizeof(int);
    int K = 3;
  
    cout << countElements(A, n, K);
  
    return 0;
}

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Output:

2

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

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