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Number of quadruples where the first three terms are in AP and last three terms are in GP

  • Last Updated : 23 Feb, 2022

Given an array arr[] of N integers. The task is to find the number of index quadruples (i, j, k, l) such that a[i], a[j] and a[k] are in AP and a[j], a[k] and a[l] are in GP. All the quadruples have to be distinct.
Examples: 
 

Input: arr[] = {2, 6, 4, 9, 2} 
Output:
Indexes of elements in the quadruples are (0, 2, 1, 3) and (4, 2, 1, 3) and corresponding quadruples are (2, 4, 6, 9) and (2, 4, 6, 9)
Input: arr[] = {1, 1, 1, 1} 
Output: 24 
 

 

A naive approach is to solve the above problem using four nested loops. Check for the first three elements if they are in AP or not and then check whether the last three elements are in GP or not. If both the conditions satisfy, then they increase the count by 1. 
Time Complexity: O(n4
An efficient approach is to use combinatorics to solve the above problem. Initially keep a count of the number of occurrences of every array element. Run two nested loops, and consider both elements to be the second and third number. Hence the first element will be a[j] – (a[k] – a[j]) and the fourth element will be a[k] * a[k] / a[j] if it is an integer value. Hence the number of quadruples using these two index j and k will be count of first number * count of fourth number with the second and third element being fixed. 
Below is the implementation of the above approach: 
 

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the count of quadruples
int countQuadruples(int a[], int n)
{
 
    // Hash table to count the number of occurrences
    unordered_map<int, int> mpp;
 
    // Traverse and increment the count
    for (int i = 0; i < n; i++)
        mpp[a[i]]++;
 
    int count = 0;
 
    // Run two nested loop for second and third element
    for (int j = 0; j < n; j++) {
        for (int k = 0; k < n; k++) {
 
            // If they are same
            if (j == k)
                continue;
 
            // Initially decrease the count
            mpp[a[j]]--;
            mpp[a[k]]--;
 
            // Find the first element using common difference
            int first = a[j] - (a[k] - a[j]);
 
            // Find the fourth element using GP
            // y^2 = x * z property
            int fourth = (a[k] * a[k]) / a[j];
 
            // If it is an integer
            if ((a[k] * a[k]) % a[j] == 0) {
 
                // If not equal
                if (a[j] != a[k])
                    count += mpp[first] * mpp[fourth];
 
                // Same elements
                else
                    count += mpp[first] * (mpp[fourth] - 1);
            }
 
            // Later increase the value for
            // future calculations
            mpp[a[j]]++;
            mpp[a[k]]++;
        }
    }
    return count;
}
 
// Driver code
int main()
{
    int a[] = { 2, 6, 4, 9, 2 };
    int n = sizeof(a) / sizeof(a[0]);
 
    cout << countQuadruples(a, n);
 
    return 0;
}

Java




// Java implementation of the approach
import java.util.*;
 
class GFG
{
 
    // Function to return the count of quadruples
    static int countQuadruples(int a[], int n)
    {
 
        // Hash table to count the number of occurrences
        HashMap<Integer, Integer> mp = new HashMap<Integer, Integer>();
 
        // Traverse and increment the count
        for (int i = 0; i < n; i++)
            if (mp.containsKey(a[i]))
            {
                mp.put(a[i], mp.get(a[i]) + 1);
            }
            else
            {
                mp.put(a[i], 1);
            }
 
        int count = 0;
 
        // Run two nested loop for second and third element
        for (int j = 0; j < n; j++)
        {
            for (int k = 0; k < n; k++)
            {
 
                // If they are same
                if (j == k)
                    continue;
 
                // Initially decrease the count
                mp.put(a[j], mp.get(a[j]) - 1);
                mp.put(a[k], mp.get(a[k]) - 1);
 
                // Find the first element using common difference
                int first = a[j] - (a[k] - a[j]);
 
                // Find the fourth element using GP
                // y^2 = x * z property
                int fourth = (a[k] * a[k]) / a[j];
 
                // If it is an integer
                if ((a[k] * a[k]) % a[j] == 0)
                {
 
                    // If not equal
                    if (a[j] != a[k])
                    {
                        if (mp.containsKey(first) && mp.containsKey(fourth))
                            count += mp.get(first) * mp.get(fourth);
                    }
                     
                    // Same elements
                    else if (mp.containsKey(first) && mp.containsKey(fourth))
                        count += mp.get(first) * (mp.get(fourth) - 1);
                }
 
                // Later increase the value for
                // future calculations
                if (mp.containsKey(a[j]))
                {
                    mp.put(a[j], mp.get(a[j]) + 1);
                }
                else
                {
                    mp.put(a[j], 1);
                }
                if (mp.containsKey(a[k]))
                {
                    mp.put(a[k], mp.get(a[k]) + 1);
                }
                else
                {
                    mp.put(a[k], 1);
                }
            }
        }
        return count;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int a[] = { 2, 6, 4, 9, 2 };
        int n = a.length;
 
        System.out.print(countQuadruples(a, n));
    }
}
 
// This code is contributed by 29AjayKumar

Python3




# Python3 implementation of the approach
 
# Function to return the count of quadruples
def countQuadruples(a, n) :
 
    # Hash table to count the number
    # of occurrences
    mpp = dict.fromkeys(a, 0);
 
    # Traverse and increment the count
    for i in range(n) :
        mpp[a[i]] += 1;
 
    count = 0;
 
    # Run two nested loop for second
    # and third element
    for j in range(n) :
        for k in range(n) :
 
            # If they are same
            if (j == k) :
                continue;
 
            # Initially decrease the count
            mpp[a[j]] -= 1;
            mpp[a[k]] -= 1;
 
            # Find the first element using
            # common difference
            first = a[j] - (a[k] - a[j]);
             
            if first not in mpp :
                mpp[first] = 0;
                 
            # Find the fourth element using
            # GP y^2 = x * z property
            fourth = (a[k] * a[k]) // a[j];
             
            if fourth not in mpp :
                mpp[fourth] = 0;
                 
            # If it is an integer
            if ((a[k] * a[k]) % a[j] == 0) :
 
                # If not equal
                if (a[j] != a[k]) :
                    count += mpp[first] * mpp[fourth];
 
                # Same elements
                else :
                    count += (mpp[first] *
                             (mpp[fourth] - 1));
             
            # Later increase the value for
            # future calculations
            mpp[a[j]] += 1;
            mpp[a[k]] += 1;
             
    return count;
 
# Driver code
if __name__ == "__main__" :
 
    a = [ 2, 6, 4, 9, 2 ];
    n = len(a) ;
 
    print(countQuadruples(a, n));
 
# This code is contributed by Ryuga

C#




// C# implementation of the approach
using System;
using System.Collections.Generic;
 
class GFG
{
 
    // Function to return the count of quadruples
    static int countQuadruples(int []a, int n)
    {
 
        // Hash table to count the number of occurrences
        Dictionary<int, int> mp = new Dictionary<int, int>();
 
        // Traverse and increment the count
        for (int i = 0; i < n; i++)
            if (mp.ContainsKey(a[i]))
            {
                mp[a[i]] = mp[a[i]] + 1;
            }
            else
            {
                mp.Add(a[i], 1);
            }
 
        int count = 0;
 
        // Run two nested loop for second and third element
        for (int j = 0; j < n; j++)
        {
            for (int k = 0; k < n; k++)
            {
 
                // If they are same
                if (j == k)
                    continue;
 
                // Initially decrease the count
                mp[a[j]] = mp[a[j]] - 1;
                mp[a[k]] = mp[a[k]] - 1;
 
                // Find the first element using common difference
                int first = a[j] - (a[k] - a[j]);
 
                // Find the fourth element using GP
                // y^2 = x * z property
                int fourth = (a[k] * a[k]) / a[j];
 
                // If it is an integer
                if ((a[k] * a[k]) % a[j] == 0)
                {
 
                    // If not equal
                    if (a[j] != a[k])
                    {
                        if (mp.ContainsKey(first) && mp.ContainsKey(fourth))
                            count += mp[first] * mp[fourth];
                    }
                     
                    // Same elements
                    else if (mp.ContainsKey(first) && mp.ContainsKey(fourth))
                        count += mp[first] * (mp[fourth] - 1);
                }
 
                // Later increase the value for
                // future calculations
                if (mp.ContainsKey(a[j]))
                {
                    mp[a[j]] = mp[a[j]] + 1;
                }
                else
                {
                    mp.Add(a[j], 1);
                }
                if (mp.ContainsKey(a[k]))
                {
                    mp[a[k]] = mp[a[k]] + 1;
                }
                else
                {
                    mp.Add(a[k], 1);
                }
            }
        }
        return count;
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        int []a = { 2, 6, 4, 9, 2 };
        int n = a.Length;
 
        Console.Write(countQuadruples(a, n));
    }
}
 
// This code is contributed by 29AjayKumar

Javascript




<script>
 
// JavaScript implementation of the approach
 
    // Function to return the count of quadruples
    function countQuadruples(a, n)
    {
   
        // Hash table to count the
        // number of occurrences
        let mp = new Map();
   
        // Traverse and increment the count
        for (let i = 0; i < n; i++)
            if (mp.has(a[i]))
            {
                mp.set(a[i], mp.get(a[i]) + 1);
            }
            else
            {
                mp.set(a[i], 1);
            }
   
        let count = 0;
   
        // Run two nested loop for second
        // and third element
        for (let j = 0; j < n; j++)
        {
            for (let k = 0; k < n; k++)
            {
   
                // If they are same
                if (j == k)
                    continue;
   
                // Initially decrease the count
                mp.set(a[j], mp.get(a[j]) - 1);
                mp.set(a[k], mp.get(a[k]) - 1);
   
                // Find the first element using
                // common difference
                let first = a[j] - (a[k] - a[j]);
   
                // Find the fourth element using GP
                // y^2 = x * z property
                let fourth = (a[k] * a[k]) / a[j];
   
                // If it is an integer
                if ((a[k] * a[k]) % a[j] == 0)
                {
   
                    // If not equal
                    if (a[j] != a[k])
                    {
                        if (mp.has(first) && mp.has(fourth))
                            count +=
                            mp.get(first) * mp.get(fourth);
                    }
                       
                    // Same elements
                    else if (mp.has(first) && mp.has(fourth))
                        count +=
                        mp.get(first) * (mp.get(fourth) - 1);
                }
   
                // Later increase the value for
                // future calculations
                if (mp.has(a[j]))
                {
                    mp.set(a[j], mp.get(a[j]) + 1);
                }
                else
                {
                    mp.set(a[j], 1);
                }
                if (mp.has(a[k]))
                {
                    mp.set(a[k], mp.get(a[k]) + 1);
                }
                else
                {
                    mp.set(a[k], 1);
                }
            }
        }
        return count;
    }
       
    // Driver code
     
    let a = [ 2, 6, 4, 9, 2 ];
        let n = a.length;
   
        document.write(countQuadruples(a, n));
                   
</script>
Output: 
2

 

Time Complexity: O(N2
Auxiliary Space: O(N)
 


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