Queries to calculate average of an array after removing K smallest and largest elements with updates

Given two positive integers N and K, initialize an empty array arr[] and Q number of queries of the following two types:

• addInteger(x): Insert element X in the array arr[]. If the size of the array becomes greater than N, then remove the element from the beginning of the array.
• calculateSpecialAverage(): Find the average of array elements after removing the first K smallest and the largest elements. If the size of the array is less than N, then print -1.

The task is to process the given queries and print the results accordingly.

Examples:

Input: N = 3, K = 1, Q = 5, Queries[] = { addInteger(4), addInteger(2), calculateSpecialAverage(), addInteger(10), calculateSpecialAverage() }
Output: -1, 4
Explanation:
Below are the queries performed:
Query 1 is to insert element 4 in the array, arr[] becomes {4}.
Query 2 is to insert the element 2 in the array, arr[] becomes {4, 2}.
Query 3 is to find the average. Since the size of the array is less than N(= 3), print -1.
Query 4 is to insert the element 10 in the array, arr[] becomes {4, 2, 10}.
Query 5 is to find the average. Since the size of the array =3, remove K(= 1) the smallest element i.e., 2 and then remove the largest element i.e., 10. Now the average will be 4/1 = 4.

Input: N = 3, K = 1, Q = 5, Queries[] = { addInteger(4), addInteger(21), calculateSpecialAverage() }
Output: -1

Approach: The given problem can be solved by using the multiset. Follow the steps below to solve the problem:

• Initialize three multisets left, right and mid to store K smallest, K largest, and the remaining (N – 2*K) integers.
• Declare a vector v of size N to store the last N integers.
• Declare an integer variable pos to keep track of the current index and an integer sum to store the sum of values in the mid multiset, which is the desired sum for calculating the average of (N – 2*K) integers.
• Define a function add to insert an integer in left, mid, or right multiset based on its value. To insert integer in the correct multiset follow the following steps:
  • Initially, insert the integer in the left multiset.
  • If the size of the left multiset exceeds k, delete the largest integer from the left multiset and insert it in the mid multiset.
  • If the size of mid multiset exceeds (n – 2*k), delete the largest integer from mid multiset and insert it in the right multiset.
  • While removing and adding integers in mid multiset maintain the correct value of sum i.e. if an integer is removed from mid multiset subtract its value from the sum and similarly if an integer is added in the mid multiset, add its value to sum, to keep the value of sum updated.
• Declare a function remove to erase the integers from left, right, or mid multiset.
  • If the value of the integer num is less than or equal to the largest integer in the left multiset, then find and erase the integer from the left multiset, otherwise search for it in mid or right multiset, based on its value.
  • If after the removal of num, the size of the left multiset decreases, delete the smallest integer from the mid multiset and insert it in the left multiset.
  • Similarly, for mid multiset, if the size decreases, delete the smallest integer from the right multiset and insert it in the mid multiset.
• Define a function addInteger to add an integer to the current stream:
  • If the number of integers in the current stream(sum of the size of left, mid, right multiset) exceeds n, remove the integer at index pos%n.
  • Call the function add to insert the integer in the left, mid, or right multiset, based on its value.
• Define the function calculateSpecialAverage to return the special average:
  • If the number of integers in the stream is less than N, then print -1.
  • Otherwise, print the average as (sum/(N – 2*K)).

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Special Average class
class SpecialAverage {
public:
    // Three multisets to store the
    // smaller K larger K and the
    // remaining (n-2*k) integers
    multiset<int> left, mid, right;

    int n, k;

    // Stores the index of current
    // integer in running stream of
    // integers and the sum of integers
    // in mid multiset
    long pos, sum;

    // Array to store last n integers
    vector<int> v;

    // Constructor to initialize the
    // values of n and k and initialize
    // default values
    SpecialAverage(int nvalue, int kvalue)
    {
        n = nvalue;
        k = kvalue;
        pos = 0;
        sum = 0;
        for (int i = 0; i < n; i++)
            v.push_back(0);
    }

    // Add current integer in the
    // multiset left, mid or right
    // based on its value
    void add(int num)
    {
        // Firstly, insert num in the
        // left multiset
        left.insert(num);

        // Check if size of the left
        // multiset is greater than k
        if (left.size() > k) {

            // Stores the value of the
            // largest integer in the
            // left multiset
            int temp = *(prev(end(left)));

            // Insert temp in the
            // mid multiset
            mid.insert(temp);

            // Remove temp from the
            // left multiset
            left.erase(prev(end(left)));

            // Add the value of temp
            // to the current sum
            sum += temp;
        }

        // Check if the size of mid
        // multiset exceeds (n-2*k)
        if (mid.size() > (n - 2 * k)) {

            // Stores the value of the
            // largest integer in the
            // mid multiset
            int temp = *(prev(end(mid)));

            // Insert temp in the
            // right multiset
            right.insert(temp);

            // Remove temp from the
            // mid multiset
            mid.erase(prev(end(mid)));

            // Subtract the value of
            // temp from current sum
            sum -= temp;
        }
    }

    // Remove integer from the
    // multiset left, mid or right
    // based on its value
    void remove(int ele)
    {

        // If integer exists in the
        // left multiset
        if (ele <= *(left.rbegin()))
            left.erase(left.find(ele));

        // If integer exists in the
        // mid multiset
        else if (ele <= *(mid.rbegin())) {

            // Subtract the value of
            // integer from current sum
            sum -= ele;
            mid.erase(mid.find(ele));
        }

        // If integer exists in the
        // right multiset
        else
            right.erase(right.find(ele));

        // Balance all the multisets
        // left, right and mid check
        // if size of the left multiset
        // is less than k
        if (left.size() < k) {

            // Stores the value of
            // smallest integer
            // in mid multiset
            int temp = *(mid.begin());

            // Insert temp in the
            // left multiset
            left.insert(temp);

            // Remove temp from the
            // mid multiset
            mid.erase(mid.begin());

            // Subtract the value of
            // temp from current sum
            sum -= temp;
        }

        // Check if the size of mid
        // multiset becomes lesser
        // than (n-2*k)
        if (mid.size() < (n - 2 * k)) {

            // Stores the value of
            // smallest integer in
            // right multiset
            int temp = *(right.begin());

            // Insert temp in the
            // mid multiset
            mid.insert(temp);

            // Remove temp from the
            // right multiset
            right.erase(right.begin());

            // Add the value of temp
            // to the current sum
            sum += temp;
        }
    }

    // Function to add integer to
    // the current stream
    void addInteger(int num)
    {

        // Check if the total number
        // of elements in stream > n
        if (pos >= n)

            // Call the function to
            // remove  extra integer
            remove(v[pos % n]);

        // Insert the current integer
        // in the array v
        v[(pos++) % n] = num;

        // Call the function to add
        // the current integer in left,
        // mid or right multiset
        // based on its value
        add(num);
    }

    // Function to calculate the
    // special average
    int calculateSpecialAverage()
    {
        // Check if the total number
        // of elements is greater than
        // or equal to n
        if (pos >= n)

            // Return desired average
            return ((double)sum) / (n - 2 * k);
        return -1;
    }
};

// Function to process all the queries
void processQueries(int n, int k)
{
    // Create an object of the class
    SpecialAverage* avg
        = new SpecialAverage(n, k);

    // Add integer Query
    avg->addInteger(4);
    avg->addInteger(2);

    // Find average Query
    cout << avg->calculateSpecialAverage()
         << endl;

    // Add integer Query
    avg->addInteger(10);

    // Find average Query
    cout << avg->calculateSpecialAverage()
         << endl;
}

// Driver Code
int main()
{
    int N = 3, K = 1;
    processQueries(N, K);

    return 0;
}

Java

// Java program for the above approach
import java.util.*;

class SpecialAverage {
    // Three TreeSets to store the
    // smaller K larger K and the
    // remaining (n-2*k) integers
    TreeSet<Integer> left, mid, right;

    int n, k;

    // Stores the index of current
    // integer in running stream of
    // integers and the sum of integers
    // in mid TreeSet
    long pos, sum;

    // Array to store last n integers
    ArrayList<Integer> v;

    // Constructor to initialize the
    // values of n and k and initialize
    // default values
    SpecialAverage(int nvalue, int kvalue) {
        n = nvalue;
        k = kvalue;
        pos = 0;
        sum = 0;
        v = new ArrayList<>();
        for (int i = 0; i < n; i++)
            v.add(0);
        left = new TreeSet<>();
        mid = new TreeSet<>();
        right = new TreeSet<>();
    }

    // Add current integer in the
    // TreeSet left, mid or right
    // based on its value
    void add(int num) {
        // Firstly, insert num in the
        // left TreeSet
        left.add(num);

        // Check if size of the left
        // TreeSet is greater than k
        if (left.size() > k) {
            // Stores the value of the
            // largest integer in the
            // left TreeSet
            int temp = left.pollLast();

            // Insert temp in the
            // mid TreeSet
            mid.add(temp);

            // Add the value of temp
            // to the current sum
            sum += temp;
        }

        // Check if the size of mid
        // TreeSet exceeds (n-2*k)
        if (mid.size() > (n - 2 * k)) {
            // Stores the value of the
            // largest integer in the
            // mid TreeSet
            int temp = mid.pollLast();

            // Insert temp in the
            // right TreeSet
            right.add(temp);

            // Subtract the value of
            // temp from current sum
            sum -= temp;
        }
    }

    // Remove integer from the
    // TreeSet left, mid or right
    // based on its value
    void remove(int ele) {
        // If integer exists in the
        // left TreeSet
        if (ele <= left.last())
            left.remove(ele);

        // If integer exists in the
        // mid TreeSet
        else if (ele <= mid.last()) {
            // Subtract the value of
            // integer from current sum
            sum -= ele;
            mid.remove(ele);
        }

        // If integer exists in the
        // right TreeSet
        else
            right.remove(ele);

        // Balance all the TreeSets
        // left, right and mid check
        // if size of the left TreeSet
        // is less than k
        if (left.size() < k) {
            // Stores the value of
            // smallest integer
            // in mid TreeSet
            int temp = mid.first();

            // Insert temp in the
            // left TreeSet
            left.add(temp);

            // Subtract the value of
            // temp from current sum
            sum -= temp;

            // Remove temp from the
            // mid TreeSet
            mid.remove(temp);
        }

        // Check if the size of mid
        // TreeSet becomes lesser
        // than (n-2*k)
        if (mid.size() < (n - 2 * k)) {
            // Stores the value of
            // smallest integer in
            // right TreeSet
            int temp = right.first();

            // Insert temp in the
            // mid TreeSet
            mid.add(temp);

            // Add the value of temp
            // to the current sum
            sum += temp;

            // Remove temp from the
            // right TreeSet
            right.remove(temp);
        }
    }

    // Function to add integer to
    // the current stream
    void addInteger(int num) {
        // Check if the total number
        // of elements in stream > n
        if (pos >= n)
            // Call the function to
            // remove extra integer
            remove(v.get((int) (pos % n)));

        // Insert the current integer
        // in the array v
        v.set((int) (pos++ % n), num);

        // Call the function to add
        // the current integer in left,
        // mid or right TreeSet
        // based on its value
        add(num);
    }

    // Function to calculate the
    // special average
    int calculateSpecialAverage() {
        // Check if the total number
        // of elements is greater than
        // or equal to n
        if (pos >= n)
            // Return desired average
            return ((int) sum) / (n - 2 * k);
        return -1;
    }

    // Function to process all the queries
    static void processQueries(int n, int k) {
        // Create an object of the class
        SpecialAverage avg = new SpecialAverage(n, k);

        // Add integer Query
        avg.addInteger(4);
        avg.addInteger(2);

        // Find average Query
        System.out.println(avg.calculateSpecialAverage());

        // Add integer Query
        avg.addInteger(10);

        // Find average Query
        System.out.println(avg.calculateSpecialAverage());
    }

    // Driver Code
    public static void main(String[] args) {
        int N = 3, K = 1;
        processQueries(N, K);
    }
}

Python3

class SpecialAverage:
    def __init__(self, nvalue, kvalue):
        # Three sets to store the smaller K, larger K, and the remaining (n-2*k) integers
        self.left = set()
        self.mid = set()
        self.right = set()
        self.n = nvalue
        self.k = kvalue
        self.pos = 0
        self.sum = 0
        self.v = [0] * nvalue

    # Add the current integer to the SortedSets left, mid, or right based on its value
    def add(self, num):
        # Firstly, insert num into the left SortedSet
        self.left.add(num)
        # Check if the size of the left SortedSet is greater than k
        if len(self.left) > self.k:
            # Stores the value of the largest integer in the left SortedSet

            temp = max(self.left)
            # Insert temp into the mid SortedSet

            self.mid.add(temp)
            # Add the value of temp to the current sum

            self.sum += temp
            # Remove temp from the left SortedSet

            self.left.remove(temp)
        # Check if the size of the mid SortedSet exceeds (n-2*k)

        if len(self.mid) > (self.n - 2 * self.k):
            # Stores the value of the largest integer in the mid SortedSet

            temp = max(self.mid)
            # Insert temp into the right SortedSet

            self.right.add(temp)
            # Subtract the value of temp from the current sum

            self.sum -= temp

            # Remove temp from the mid SortedSet

            self.mid.remove(temp)
    # Remove the integer from the SortedSets left, mid, or right based on its value

    def remove(self, ele):
        # If the integer exists in the left SortedSet
        if ele <= max(self.left, default=float('-inf')):
            self.left.remove(ele)
        # If the integer exists in the mid SortedSet
        elif ele <= max(self.mid, default=float('-inf')):
            # Subtract the value of the integer from the current sum
            self.sum -= ele
            self.mid.remove(ele)
        # If the integer exists in the right SortedSet
        else:
            self.right.remove(ele)

        # Balance all the SortedSets left, right, and mid
        # Check if the size of the left SortedSet is less than k
        if len(self.left) < self.k:
            # Stores the value of the smallest integer in the mid SortedSet
            temp = min(self.mid, default=float('inf'))
            # Insert temp into the left SortedSet
            self.left.add(temp)
            # Subtract the value of temp from the current sum
            self.sum -= temp
            # Remove temp from the mid SortedSet
            self.mid.remove(temp)

        # Check if the size of the mid SortedSet becomes less than (n-2*k)
        if len(self.mid) < (self.n - 2 * self.k):
            # Stores the value of the smallest integer in the right SortedSet
            temp = min(self.right, default=float('inf'))
            # Insert temp into the mid SortedSet
            self.mid.add(temp)
            # Add the value of temp to the current sum
            self.sum += temp
            # Remove temp from the right SortedSet
            self.right.remove(temp)

    # Function to add an integer to the current stream
    def add_integer(self, num):
        # Check if the total number of elements in the stream is greater than n
        if self.pos >= self.n:
            # Call the function to remove the extra integer
            self.remove(self.v[int(self.pos % self.n)])

        # Insert the current integer into the array v
        self.v[int(self.pos % self.n)] = num

        # Call the function to add the current integer into left, mid, or right SortedSets
        # based on its value
        self.add(num)
        self.pos += 1
    # Function to calculate the special average

    def calculate_special_average(self):
        # Check if the total number of elements is greater than or equal to n
        if self.pos >= self.n:
            # Return the desired average
            return int(self.sum / (self.n - 2 * self.k))
        return -1

# Function to process all the queries


def process_queries(n, k):
    # Create an object of the SpecialAverage class
    avg = SpecialAverage(n, k)

    # Add integer Query
    avg.add_integer(4)
    avg.add_integer(2)

    # Find average Query
    print(avg.calculate_special_average())

    # Add integer Query
    avg.add_integer(10)

    # Find average Query
    print(avg.calculate_special_average())


if __name__ == "__main__":
    N, K = 3, 1
    process_queries(N, K)

C#

using System;
using System.Collections.Generic;

class SpecialAverage
{
    // Three SortedSets to store the smaller K, larger K, and the remaining (n-2*k) integers
    SortedSet<int> left, mid, right;

    int n, k;

    // Stores the index of the current integer in the running stream of integers
    // and the sum of integers in the mid SortedSet
    long pos, sum;

    // Array to store the last n integers
    List<int> v;

    // Constructor to initialize the values of n and k and initialize default values
    public SpecialAverage(int nvalue, int kvalue)
    {
        n = nvalue;
        k = kvalue;
        pos = 0;
        sum = 0;
        v = new List<int>();
        for (int i = 0; i < n; i++)
            v.Add(0);
        left = new SortedSet<int>();
        mid = new SortedSet<int>();
        right = new SortedSet<int>();
    }

    // Add the current integer to the SortedSets left, mid, or right based on its value
    public void Add(int num)
    {
        // Firstly, insert num into the left SortedSet
        left.Add(num);

        // Check if the size of the left SortedSet is greater than k
        if (left.Count > k)
        {
            // Stores the value of the largest integer in the left SortedSet
            int temp = left.Max;

            // Insert temp into the mid SortedSet
            mid.Add(temp);

            // Add the value of temp to the current sum
            sum += temp;

            // Remove temp from the left SortedSet
            left.Remove(temp);
        }

        // Check if the size of the mid SortedSet exceeds (n-2*k)
        if (mid.Count > (n - 2 * k))
        {
            // Stores the value of the largest integer in the mid SortedSet
            int temp = mid.Max;

            // Insert temp into the right SortedSet
            right.Add(temp);

            // Subtract the value of temp from the current sum
            sum -= temp;

            // Remove temp from the mid SortedSet
            mid.Remove(temp);
        }
    }

    // Remove the integer from the SortedSets left, mid, or right based on its value
    public void Remove(int ele)
    {
        // If the integer exists in the left SortedSet
        if (ele <= left.Max)
            left.Remove(ele);

        // If the integer exists in the mid SortedSet
        else if (ele <= mid.Max)
        {
            // Subtract the value of the integer from the current sum
            sum -= ele;
            mid.Remove(ele);
        }

        // If the integer exists in the right SortedSet
        else
            right.Remove(ele);

        // Balance all the SortedSets left, right, and mid
        // Check if the size of the left SortedSet is less than k
        if (left.Count < k)
        {
            // Stores the value of the smallest integer in the mid SortedSet
            int temp = mid.Min;

            // Insert temp into the left SortedSet
            left.Add(temp);

            // Subtract the value of temp from the current sum
            sum -= temp;

            // Remove temp from the mid SortedSet
            mid.Remove(temp);
        }

        // Check if the size of the mid SortedSet becomes less than (n-2*k)
        if (mid.Count < (n - 2 * k))
        {
            // Stores the value of the smallest integer in the right SortedSet
            int temp = right.Min;

            // Insert temp into the mid SortedSet
            mid.Add(temp);

            // Add the value of temp to the current sum
            sum += temp;

            // Remove temp from the right SortedSet
            right.Remove(temp);
        }
    }

    // Function to add an integer to the current stream
    public void AddInteger(int num)
    {
        // Check if the total number of elements in the stream is greater than n
        if (pos >= n)
            // Call the function to remove the extra integer
            Remove(v[(int)(pos % n)]);

        // Insert the current integer into the array v
        v[(int)(pos++ % n)] = num;

        // Call the function to add the current integer into left, mid, or right SortedSets
        // based on its value
        Add(num);
    }

    // Function to calculate the special average
    public int CalculateSpecialAverage()
    {
        // Check if the total number of elements is greater than or equal to n
        if (pos >= n)
            // Return the desired average
            return (int)sum / (n - 2 * k);
        return -1;
    }

    // Function to process all the queries
    public static void ProcessQueries(int n, int k)
    {
        // Create an object of the SpecialAverage class
        SpecialAverage avg = new SpecialAverage(n, k);

        // Add integer Query
        avg.AddInteger(4);
        avg.AddInteger(2);

        // Find average Query
        Console.WriteLine(avg.CalculateSpecialAverage());

        // Add integer Query
        avg.AddInteger(10);

        // Find average Query
        Console.WriteLine(avg.CalculateSpecialAverage());
    }

    // Driver Code
    public static void Main(string[] args)
    {
        int N = 3, K = 1;
        ProcessQueries(N, K);
    }
}

Javascript

class SpecialAverage {
  constructor(nvalue, kvalue) {
    // Three sets to store smaller K, larger K, and remaining integers
    this.left = new Set();
    this.mid = new Set();
    this.right = new Set();
    this.n = nvalue; // Total number of integers to consider
    this.k = kvalue; // Parameter value K
    this.pos = 0; // Position tracker
    this.sum = 0; // Sum of selected integers
    this.v = new Array(nvalue).fill(0); // Array to store the integers
  }

  // Add the current integer to the appropriate set
  add(num) {
    this.left.add(num);

    // Check and move integers from left to mid set if left set size exceeds K
    if (this.left.size > this.k) {
      let temp = Math.max(...this.left);
      this.mid.add(temp);
      this.sum += temp;
      this.left.delete(temp);
    }

    // Check and move integers from mid to right set if mid set size exceeds (n - 2 * K)
    if (this.mid.size > this.n - 2 * this.k) {
      let temp = Math.max(...this.mid);
      this.right.add(temp);
      this.sum -= temp;
      this.mid.delete(temp);
    }
  }

  // Remove the specified integer from the sets
  remove(ele) {
    if (ele <= Math.max(...this.left, Number.NEGATIVE_INFINITY)) {
      this.left.delete(ele);
    } else if (ele <= Math.max(...this.mid, Number.NEGATIVE_INFINITY)) {
      this.sum -= ele;
      this.mid.delete(ele);
    } else {
      this.right.delete(ele);
    }

    // Ensure left set size is maintained as K
    if (this.left.size < this.k) {
      let temp = Math.min(...this.mid, Number.POSITIVE_INFINITY);
      this.left.add(temp);
      this.sum -= temp;
      this.mid.delete(temp);
    }

    // Ensure mid set size is maintained as (n - 2 * K)
    if (this.mid.size < this.n - 2 * this.k) {
      let temp = Math.min(...this.right, Number.POSITIVE_INFINITY);
      this.mid.add(temp);
      this.sum += temp;
      this.right.delete(temp);
    }
  }

  // Add an integer to the stream
  addInteger(num) {
    // If total elements exceed n, remove the oldest element
    if (this.pos >= this.n) {
      this.remove(this.v[this.pos % this.n]);
    }

    // Add the current integer to the array and sets
    this.v[this.pos % this.n] = num;
    this.add(num);
    this.pos++;
  }

  // Calculate the special average
  calculateSpecialAverage() {
    // If total elements are sufficient, return the special average
    if (this.pos >= this.n) {
      return Math.floor(this.sum / (this.n - 2 * this.k));
    }
    return -1; // Insufficient elements to calculate the special average
  }
}

// Function to process queries
function processQueries(n, k) {
  let avg = new SpecialAverage(n, k);

  // Add integers to the stream
  avg.addInteger(4);
  avg.addInteger(2);
  console.log(avg.calculateSpecialAverage()); // Calculate and print the special average

  // Add another integer to the stream
  avg.addInteger(10);
  console.log(avg.calculateSpecialAverage()); // Calculate and print the updated special average
}

// Example usage:
const N = 3;
const K = 1;
processQueries(N, K); // Call the function to process queries

Output

-1
4

Time Complexity: O(log N) for addInteger() and O(1) for calculateSpecialAverage()
Auxiliary Space: O(N)