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

  • Difficulty Level : Hard
  • Last Updated : 12 Jan, 2022

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:
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:

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++




// C++ Program to implement
// the above approach
#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;
}

Java




// Java program to implement
// the above approach
class GFG{
 
// Structure of an AVL Tree Node
static class Node
{
    int key;
    Node left;
    Node right;
    int height;
     
    // Size of the tree rooted
    // with this Node
    int size;
 
    public Node(int key)
    {
        this.key = key;
        this.left = this.right = null;
        this.size = this.height = 1;
    }
};
 
// Helper class to pass Integer
// as referencee
static class RefInteger
{
    Integer value;
     
    public RefInteger(Integer value)
    {
        this.value = value;
    }
}
 
// Utility function to get height
// of the tree rooted with N
static int height(Node N)
{
    if (N == null)
        return 0;
         
    return N.height;
}
 
// Utility function to find size of
// the tree rooted with N
static int size(Node N)
{
    if (N == null)
        return 0;
         
    return N.size;
}
 
// Utility function to get maximum
// of two integers
static int max(int a, int b)
{
    return (a > b) ? a : b;
}
 
// Utility function to right rotate
// subtree rooted with y
static Node rightRotate(Node y)
{
    Node x = y.left;
    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
static Node leftRotate(Node x)
{
    Node y = x.right;
    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
static int getBalance(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
static Node insert(Node Node, int key,
                   RefInteger count)
{
     
    // Perform the normal BST rotation
    if (Node == null)
        return (new Node(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.value = count.value +
                  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
static void constructLowerArray(int arr[],
     RefInteger[] countSmaller, int n)
{
    int i, j;
    Node root = null;
 
    for(i = 0; i < n; i++)
        countSmaller[i] = new RefInteger(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
static int countElements(int A[], int n,
                         int K)
{
    int count = 0;
 
    // Stores the count of smaller
    // elements on its right
    RefInteger[] countSmaller = new RefInteger[n];
    constructLowerArray(A, countSmaller, n);
 
    int maxi = Integer.MIN_VALUE;
    for(int i = 0; i <= (n - K - 1); i++)
    {
        if (A[i] > maxi &&
            countSmaller[i].value >= K)
        {
            count++;
            maxi = A[i];
        }
    }
    return count;
}
 
// Driver Code
public static void main(String[] args)
{
    int A[] = { 2, 5, 1, 7, 3, 4, 0 };
    int n = A.length;
    int K = 3;
 
    System.out.println(countElements(A, n, K));
}
}
 
// This code is contributed by sanjeev2552

Python3




# Python program to implement
# the above approach
import sys
 
# Structure of an AVL Tree Node
class Node:
    def __init__(self, key):
        self.key = key;
        self.left = None;
        self.right = None;
        self.height = 1;
        self.size = 1;
 
# Helper class to pass Integer
# as referencee
class RefInteger:
    def __init__(self, data):
        self.value = data;
     
# Utility function to get height
# of the tree rooted with N
def height(N):
    if (N == None):
        return 0;
 
    return N.height;
 
# Utility function to find size of
# the tree rooted with N
def size(N):
    if (N == None):
        return 0;
 
    return N.size;
 
# Utility function to get maximum
# of two integers
def max(a, b):
    if(a>b):
        return a;
    else:
        return b;
 
# Utility function to right rotate
# subtree rooted with y
def rightRotate(y):
    x = y.left;
    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
def leftRotate(x):
    y = x.right;
    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
def getBalance(N):
    if (N == None):
        return 0;
 
    return height(N.left) - height(N.right);
 
# Function to insert a new key to the
# tree rooted with Node
def insert(node, key, count):
 
    # Perform the normal BST rotation
    if (node == None):
        return  Node(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.value = count.value + 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
    balance = getBalance(node);
 
    # Left Left Case
    if (balance > 1 and key < node.left.key):
        return rightRotate(node);
 
    # Right Right Case
    if (balance < -1 and key > node.right.key):
        return leftRotate(node);
 
    # Left Right Case
    if (balance > 1 and key > node.left.key):
        node.left = leftRotate(node.left);
        return rightRotate(node);
     
 
    # Right Left Case
    if (balance < -1 and 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
def constructLowerArray(arr, countSmaller, n):
    root = None;
 
    for i in range(n):
        countSmaller[i] =  RefInteger(0);
 
    # Insert all elements in the AVL Tree
    # and get the count of smaller elements
    for i in range(n - 1, -1,-1):
        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
def countElements(A, n, K):
    count = 0;
 
    # Stores the count of smaller
    # elements on its right
    countSmaller = [0 for i in range(n)];
    constructLowerArray(A, countSmaller, n);
 
    maxi = -sys.maxsize;
    for i in range(n - K):
        if (A[i] > maxi and countSmaller[i].value >= K):
            count += 1;
            maxi = A[i];
         
    return count;
 
# Driver Code
if __name__ == '__main__':
    A = [ 2, 5, 1, 7, 3, 4, 0 ];
    n = len(A);
    K = 3;
 
    print(countElements(A, n, K));
 
# This code is contributed by Rajput-Ji

C#




// C# program to implement
// the above approach
using System;
using System.Collections.Generic;
 
class GFG{
 
// Structure of an AVL Tree Node
public class Node
{
    public int key;
    public Node left;
    public Node right;
    public int height;
     
    // Size of the tree rooted
    // with this Node
    public int size;
 
    public Node(int key)
    {
        this.key = key;
        this.left = this.right = null;
        this.size = this.height = 1;
    }
};
 
// Helper class to pass int
// as referencee
public class Refint
{
    public int value;
     
    public Refint(int value)
    {
        this.value = value;
    }
}
 
// Utility function to get height
// of the tree rooted with N
static int height(Node N)
{
    if (N == null)
        return 0;
         
    return N.height;
}
 
// Utility function to find size of
// the tree rooted with N
static int size(Node N)
{
    if (N == null)
        return 0;
         
    return N.size;
}
 
// Utility function to get maximum
// of two integers
static int max(int a, int b)
{
    return (a > b) ? a : b;
}
 
// Utility function to right rotate
// subtree rooted with y
static Node rightRotate(Node y)
{
    Node x = y.left;
    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
static Node leftRotate(Node x)
{
    Node y = x.right;
    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
static int getBalance(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
static Node insert(Node Node, int key,
                   Refint count)
{
     
    // Perform the normal BST rotation
    if (Node == null)
        return (new Node(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.value = count.value +
                  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
static void constructLowerArray(int []arr,
         Refint[] countSmaller, int n)
{
    int i;
    //int j;
    Node root = null;
 
    for(i = 0; i < n; i++)
        countSmaller[i] = new Refint(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
static int countElements(int []A, int n,
                         int K)
{
    int count = 0;
 
    // Stores the count of smaller
    // elements on its right
    Refint[] countSmaller = new Refint[n];
    constructLowerArray(A, countSmaller, n);
 
    int maxi = int.MinValue;
    for(int i = 0; i <= (n - K - 1); i++)
    {
        if (A[i] > maxi &&
            countSmaller[i].value >= K)
        {
            count++;
            maxi = A[i];
        }
    }
    return count;
}
 
// Driver Code
public static void Main(String[] args)
{
    int []A = { 2, 5, 1, 7, 3, 4, 0 };
    int n = A.Length;
    int K = 3;
 
    Console.WriteLine(countElements(A, n, K));
}
}
 
// This code is contributed by Princi Singh

Javascript




<script>
// javascript program to implement
// the above approach   
 
// Structure of an AVL Tree Node
     class Node
     {
 
        // Size of the tree rooted
        // with this Node
        constructor(key) {
            this.key = key;
            this.left = this.right = null;
            this.size = this.height = 1;
        }
    };
 
    // Helper class to pass Integer
    // as referencee
     class RefInteger {
         constructor(value) {
            this.value = value;
        }
    }
 
    // Utility function to get height
    // of the tree rooted with N
    function height(N) {
        if (N == null)
            return 0;
 
        return N.height;
    }
 
    // Utility function to find size of
    // the tree rooted with N
    function size(N) {
        if (N == null)
            return 0;
 
        return N.size;
    }
 
    // Utility function to get maximum
    // of two integers
    function max(a , b) {
        return (a > b) ? a : b;
    }
 
    // Utility function to right rotate
    // subtree rooted with y
    function rightRotate(y)
    {
        var x = y.left;
        var 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
    function leftRotate(x)
    {
        var y = x.right;
        var 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
    function getBalance(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
    function insert(node , key,  count) {
 
        // Perform the normal BST rotation
        if (node == null)
            return (new Node(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.value = count.value + 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
        var 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
    function constructLowerArray(arr,  countSmaller , n)
    {
        var i, j;
        var root = null;
 
        for (i = 0; i < n; i++)
            countSmaller[i] = new RefInteger(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
    function countElements(A , n , K) {
        var count = 0;
 
        // Stores the count of smaller
        // elements on its right
        var countSmaller = [];
        constructLowerArray(A, countSmaller, n);
 
        var maxi = Number.MIN_VALUE;
        for (i = 0; i <= (n - K - 1); i++) {
            if (A[i] > maxi && countSmaller[i].value >= K) {
                count++;
                maxi = A[i];
            }
        }
        return count;
    }
 
    // Driver Code
     
        var A = [ 2, 5, 1, 7, 3, 4, 0 ];
        var n = A.length;
        var K = 3;
 
        document.write(countElements(A, n, K));
 
// This code is contributed by Rajput-Ji
</script>
Output: 
2

 

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


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