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Maximize subarrays count containing the maximum and minimum Array element after deleting at most one element

  • Difficulty Level : Hard
  • Last Updated : 29 Sep, 2021
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Given an array arr[] of size N. The task is to maximize the count of subarrays that contain both the minimum and maximum elements of the array by deleting at most one element from the array.

Examples: 
 

Input: arr[] = {7, 2, 5, 4, 3, 1} 
Output:
Explanation: 
Delete 1 from the array then resultant array will be {7, 2, 5, 4, 3}. So the number of subarrays which contain maximum element 7 and minimum element 2 will be 4 {[7, 2], [7, 2, 5], [7, 2, 5, 4], [7, 2, 5, 4, 3]} 
 

Input: arr[] = {9, 9, 8, 9, 8, 9, 9, 8, 9, 8} 
Output: 43

 



 

Naive Approach: The simplest approach is to delete every element and then count the number of subarrays having the minimum and maximum element of the resultant array.

Time Complexity: O(N3
Auxiliary Space: O(1)
 

Efficient Approach: This approach is based on the observation that deletion of elements other than the maximum or minimum element never maximizes the overall result. Below are the steps: 
 

  1. Initialize the overall result with INT_MIN.
  2. Create a function say proc which returns the number of subarrays containing the smallest and the largest element of the array.
  3. To calculate the number of subarrays find the starting and ending index of the subarray using two pointers approach:
    • Initialize the smallest and the largest element say low and high with the last element of the array.
    • Initialize two pointers p1 and p2 with the last index of array which stores the location of low and high.
    • Now, iterate over the array and check if the current element is less than low, then update p1.
    • If the current element is more than high, then update p2.
    • At each step, update the maximum number of subarrays.
  4. Now, calculate the number of subarrays in the following three cases: 
    • Without removing any element 
       
    • After removing the largest element
    • After removing the smallest element.
  5. Take the maximum of all three cases.

Below is the implementation of the above approach:
 

C++




// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
// Returns the count of subarrays
// which contains both the maximum and
// minimum elements in the given vector
long long proc(vector<int> &v)
    {
   
        long long int n = v.size();
   
        // Initialize the low and
        // high of array
        int low = v[n - 1], high = v[n - 1];
        long long int p1 = n, p2 = n;
        long long ans = 0;
   
        for (int i = n - 1; i >= 0; i--) {
            int x = v[i];
   
            // If current element is
            // less than least element
            if (x < low) {
                low = x;
                ans = 0;
            }
   
            // If current element is
            // more than highest element
            else if (x > high) {
                high = x;
                ans = 0;
            }
   
            // If current element is
            // equal to low or high
            // then update the pointers
            if (x == low)
                p1 = i;
            if (x == high)
                p2 = i;
   
            // Update number of subarrays
            ans += n - max(p1, p2);
        }
   
        // Return the result
        return ans;
    }
  
// Function to find the maximum
    // count of subarrays
long long subarray(vector<int>& v)
{
    long long int n=v.size();
    if(n<=1)
    return n;
    long long ans=proc(v);
    int low=v[0],pos_low=0,high=v[0],pos_high=0;
    // Iterate the array to find
    // the maximum and minimum element
    for (int i = 1; i < n; i++) {
            int x = v[i];
            if (x < low) {
                low = x;
                pos_low = i;
            }
            else if (x > high) {
                high = x;
                pos_high = i;
            }
        }
        // Vector after removing the
        // minimum element
        vector<int>u;
         // Using assignment operator to copy one
         //  vector to other
         u=v;
         u.erase(u.begin()+pos_low);
         ans=max(ans,proc(u));
         // Vector after removing the
        // maximum element
        vector<int>w;
        w=v;
        w.erase(w.begin()+pos_high);
        return max(ans,proc(w));
     
}
 
// Driver Code
int main()
{
   
    // Given array
    vector<int>v;
    v.push_back(7);
    v.push_back(2);
    v.push_back(5);
    v.push_back(4);
    v.push_back(3);
    v.push_back(1);
   
  // Function Call
  cout<<subarray(v)<<endl;
  
    return 0;
}
// This code is contributed by dwivediyash

Java




// Java implementation of the above approach
import java.util.*;
import java.lang.*;
 
class GFG {
 
    // Function to find the maximum
    // count of subarrays
    static long subarray(List<Integer> v)
    {
        int n = v.size();
        if (n <= 1)
            return n;
 
        long ans = proc(v);
        int low = v.get(0), pos_low = 0;
        int high = v.get(0), pos_high = 0;
 
        // Iterate the array to find
        // the maximum and minimum element
        for (int i = 1; i < n; i++) {
            int x = v.get(i);
            if (x < low) {
                low = x;
                pos_low = i;
            }
            else if (x > high) {
                high = x;
                pos_high = i;
            }
        }
 
        // List after removing the
        // minimum element
        List<Integer> u
            = new ArrayList<>(
                Collections.nCopies(n, 0));
        Collections.copy(u, v);
        u.remove(pos_low);
        ans = Math.max(ans, proc(u));
 
        // List after removing the
        // maximum element
        List<Integer> w
            = new ArrayList<>(
                Collections.nCopies(n, 0));
        Collections.copy(w, v);
        w.remove(pos_high);
 
        return Math.max(ans, proc(w));
    }
 
    // Returns the count of subarrays
    // which contains both the maximum and
    // minimum elements in the given list
    static long proc(List<Integer> v)
    {
 
        int n = v.size();
 
        // Initialize the low and
        // high of array
        int low = v.get(n - 1), high = v.get(n - 1);
        int p1 = n, p2 = n;
        long ans = 0;
 
        for (int i = n - 1; i >= 0; i--) {
            int x = v.get(i);
 
            // If current element is
            // less than least element
            if (x < low) {
                low = x;
                ans = 0;
            }
 
            // If current element is
            // more than highest element
            else if (x > high) {
                high = x;
                ans = 0;
            }
 
            // If current element is
            // equal to low or high
            // then update the pointers
            if (x == low)
                p1 = i;
            if (x == high)
                p2 = i;
 
            // Update number of subarrays
            ans += n - Math.max(p1, p2);
        }
 
        // Return the result
        return ans;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        // Given array
        List<Integer> arr = Arrays.asList(7, 2, 5, 4, 3, 1);
 
        // Function Call
        System.out.println(subarray(arr));
    }
}

Python3




# Python program for the above approach
 
# Returns the count of subarrays
# which contains both the maximum and
# minimum elements in the given vector
def proc(v):
  n = len(v);
 
  # Initialize the low and
  # high of array
  low = v[n - 1]
  high = v[n - 1]
  p1 = n
  p2 = n;
  ans = 0;
 
  for i in range(n - 1, -1, -1):
    x = v[i];
 
    # If current element is
    # less than least element
    if (x < low):
      low = x;
      ans = 0;
     
    # If current element is
    # more than highest element
    elif (x > high):
      high = x;
      ans = 0;
     
    # If current element is
    # equal to low or high
    # then update the pointers
    if (x == low): p1 = i;
    if (x == high): p2 = i;
 
    # Update number of subarrays
    ans += n - max(p1, p2);
   
  # Return the result
  return ans;
 
# Function to find the maximum
# count of subarrays
def subarray(v):
  n = len(v);
 
  if (n <= 1):
    return n;
   
  ans = proc(v);
  low = v[0]
  pos_low = 0
  high = v[0]
  pos_high = 0
     
  # Iterate the array to find
  # the maximum and minimum element
  for i in range(1, n):
    x = v[i];
    if (x < low):
      low = x;
      pos_low = i;
    elif (x > high):
      high = x;
      pos_high = i;
     
  # Vector after removing the
  # minimum element
  u = v[:];
   
  # Using assignment operator to copy one
  #  vector to other
  del u[pos_low];
 
  ans = max(ans, proc(u));
 
  # Vector after removing the
  # maximum element
  w = v[:];
 
  del w[pos_high];
  return max(ans, proc(w));
 
# Driver Code
 
# Given array
v = [];
v.append(7);
v.append(2);
v.append(5);
v.append(4);
v.append(3);
v.append(1);
 
# Function Call
print(subarray(v));
 
# This code is contributed by gfgking

Javascript




<script>
// Javascript program for the above approach
 
// Returns the count of subarrays
// which contains both the maximum and
// minimum elements in the given vector
function proc(v)
{
  let n = v.length;
 
  // Initialize the low and
  // high of array
  let low = v[n - 1],
    high = v[n - 1];
  let p1 = n,
    p2 = n;
  let ans = 0;
 
  for (let i = n - 1; i >= 0; i--) {
    let x = v[i];
 
    // If current element is
    // less than least element
    if (x < low) {
      low = x;
      ans = 0;
    }
 
    // If current element is
    // more than highest element
    else if (x > high) {
      high = x;
      ans = 0;
    }
 
    // If current element is
    // equal to low or high
    // then update the pointers
    if (x == low) p1 = i;
    if (x == high) p2 = i;
 
    // Update number of subarrays
    ans += n - Math.max(p1, p2);
  }
 
  // Return the result
  return ans;
}
 
// Function to find the maximum
// count of subarrays
function subarray(v) {
  let n = v.length;
 
  if (n <= 1) {
    return n;
  }
 
  let ans = proc(v);
  let low = v[0],
    pos_low = 0,
    high = v[0],
    pos_high = 0;
     
  // Iterate the array to find
  // the maximum and minimum element
  for (let i = 1; i < n; i++) {
    let x = v[i];
    if (x < low) {
      low = x;
      pos_low = i;
    } else if (x > high) {
      high = x;
      pos_high = i;
    }
  }
   
  // Vector after removing the
  // minimum element
  let u = [...v];
   
  // Using assignment operator to copy one
  //  vector to other
  u.splice(pos_low, 1);
 
  ans = Math.max(ans, proc(u));
 
  // Vector after removing the
  // maximum element
  let w = [...v];
 
  w.splice(pos_high, 1);
 
  return Math.max(ans, proc(w));
}
 
// Driver Code
 
// Given array
let v = [];
v.push(7);
v.push(2);
v.push(5);
v.push(4);
v.push(3);
v.push(1);
 
// Function Call
document.write(subarray(v));
 
// This code is contributed by gfgking
 
</script>
Output 
4

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

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