Rearrange array to maximize count of triplets (i, j, k) such that arr[i] > arr[j] < arr[k] and i < j < k

Given an array, arr[] of size N, the task is to rearrange the array elements to maximize the count of triplets (i, j, k) that satisfy the condition arr[i] > arr[j] < arr[k] and i < j < k.

Examples:

Input: arr[] = {1, 4, 3, 3, 2, 2, 5} 
Output:
Count of triplets: 3 
Array: {3, 1, 3, 2, 4, 2, 5} 
Explanation:
Rearrange the array to {3, 1, 3, 2, 4, 2, 5} which gives the maximum count of triplets {(3, 1, 3), (3, 2, 4), (4, 2, 5)} that satisfies the given condition.
Therefore, the required count of triplets is 3.

Input: arr[] = {1, 2, 3, 4, 5, 6, 7, 7} 
Output:
Count of triplets = 3 
Array = {5, 1, 6, 2, 7, 3, 7, 4}

Naive Approach: The simplest approach to solve the problem is to generate all possible permutations of the array and count the number of triplets that satisfy the given condition. Finally, print the count of triplets and the permutations of the given array which contain the maximum count of triplets with the given condition.

Time Complexity: O(N!)
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach, the idea is to first sort the given array and place the first half of the given array at odd indexes and the last half of the array at even indexes in a temporary array. Finally, print the temporary array and count of triplets with the given condition. Follow the steps below to solve the problem:

1. Initialize a temporary array, say temp[], to store the permutations of the given array.
2. Sort the given array, arr[].
3. Place the first half of the given array at odd indexes of the temp array.
4. Place the last half of the given array at even indexes of the temp array.
5. Finally, print the count of triplets with the given conditions in temp[] and temp array.

Below are the implementations of the above approach:

C++

// C++ program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to rearrange the array
// to maximize the count of triplets
void ctTriplets(int arr[], int N)
{
    // Sort the given array
    sort(arr, arr + N);

    // Stores the permutations
    // of the given array
    vector<int> temp(N);

    // Stores the index
    // of the given array
    int index = 0;

    // Place the first half
    // of the given array
    for (int i = 1; i < N;
         i += 2) {
        temp[i]
            = arr[index++];
    }

    // Place the last half
    // of the given array
    for (int i = 0; i < N;
         i += 2) {
        temp[i]
            = arr[index++];
    }

    // Stores the count
    // of triplets
    int ct = 0;
    for (int i = 0; i < N; i++) {

        // Check the given conditions
        if (i > 0 && i + 1 < N) {
            if (temp[i] < temp[i + 1]
                && temp[i] < temp[i - 1]) {
                ct++;
            }
        }
    }
    cout << "Count of triplets:"
         << ct << endl;
    cout << "Array:";
    for (int i = 0; i < N;
         i++) {
        cout << temp[i] << " ";
    }
}

// Driver Code
int main()
{

    int arr[] = { 1, 2, 3, 4, 5, 6 };
    int N = sizeof(arr) / sizeof(arr[0]);
    ctTriplets(arr, N);
}

Java

// Java program to implement
// the above approach
import java.io.*;
import java.util.Arrays;

class GFG{

// Function to rearrange the array
// to maximize the count of triplets
static void ctTriplets(int arr[], int N)
{

    // Sort the given array
    Arrays.sort(arr);

    // Stores the permutations
    // of the given array
    int[] temp = new int[N];

    // Stores the index
    // of the given array
    int index = 0;

    // Place the first half
    // of the given array
    for(int i = 1; i < N; i += 2)
    {
        temp[i] = arr[index++];
    }

    // Place the last half
    // of the given array
    for(int i = 0; i < N; i += 2)
    {
        temp[i] = arr[index++];
    }

    // Stores the count
    // of triplets
    int ct = 0;

    for(int i = 0; i < N; i++)
    {

        // Check the given conditions
        if (i > 0 && i + 1 < N)
        {
            if (temp[i] < temp[i + 1] &&
                temp[i] < temp[i - 1])
            {
                ct++;
            }
        }
    }

    System.out.println("Count of triplets:" + ct );
    System.out.print("Array:");

    for(int i = 0; i < N; i++)
    {
        System.out.print(temp[i] + " ");
    }
}

// Driver Code
public static void main (String[] args)
{
    int arr[] = { 1, 2, 3, 4, 5, 6 };
    int N = arr.length;

    ctTriplets(arr, N);
}
}

// This code is contributed by sanjoy_62

Python3

# Python3 program to implement
# the above approach

# Function to rearrange the array
# to maximize the count of triplets
def ctTriplets(arr, N):

    # Sort the given array
    arr.sort()

    # Stores the permutations
    # of the given array
    temp = [0] * N

    # Stores the index
    # of the given array
    index = 0

    # Place the first half
    # of the given array
    for i in range(1, N, 2):
        temp[i] = arr[index]
        index += 1

    # Place the last half
    # of the given array
    for i in range(0, N, 2):
        temp[i] = arr[index]
        index += 1

    # Stores the count
    # of triplets
    ct = 0

    for i in range(N):

        # Check the given conditions
        if (i > 0 and i + 1 < N):
            if (temp[i] < temp[i + 1] and
                temp[i] < temp[i - 1]):
                ct += 1

    print("Count of triplets:", ct)
    print("Array:", end = "")

    for i in range(N):
        print(temp[i], end = " ")

# Driver Code
arr = [ 1, 2, 3, 4, 5, 6 ]
N = len(arr)

ctTriplets(arr, N)

# This code is contributed by sanjoy_62

C#

// C# program to implement
// the above approach
using System;

class GFG{

// Function to rearrange the array
// to maximize the count of triplets
static void ctTriplets(int[] arr, int N)
{

    // Sort the given array
    Array.Sort(arr);

    // Stores the permutations
    // of the given array
    int[] temp = new int[N];

    // Stores the index
    // of the given array
    int index = 0;

    // Place the first half
    // of the given array
    for(int i = 1; i < N; i += 2)
    {
        temp[i] = arr[index++];
    }

    // Place the last half
    // of the given array
    for(int i = 0; i < N; i += 2)
    {
        temp[i] = arr[index++];
    }

    // Stores the count
    // of triplets
    int ct = 0;

    for(int i = 0; i < N; i++)
    {

        // Check the given conditions
        if (i > 0 && i + 1 < N)
        {
            if (temp[i] < temp[i + 1] &&
                temp[i] < temp[i - 1])
            {
                ct++;
            }
        }
    }

    Console.WriteLine("Count of triplets:" + ct );
    Console.Write("Array:");

    for(int i = 0; i < N; i++)
    {
        Console.Write(temp[i] + " ");
    }
}

// Driver Code
public static void Main ()
{
    int[] arr = { 1, 2, 3, 4, 5, 6 };
    int N = arr.Length;

    ctTriplets(arr, N);
}
}

// This code is contributed by sanjoy_62

Javascript

<script>

// Javascript program to implement
// the above approach

// Function to rearrange the array
// to maximize the count of triplets
function ctTriplets(arr, N)
{

    // Sort the given array
    arr.sort();

    // Stores the permutations
    // of the given array
    let temp = [];

    // Stores the index
    // of the given array
    let index = 0;

    // Place the first half
    // of the given array
    for(let i = 1; i < N; i += 2)
    {
        temp[i] = arr[index++];
    }

    // Place the last half
    // of the given array
    for(let i = 0; i < N; i += 2)
    {
        temp[i] = arr[index++];
    }

    // Stores the count
    // of triplets
    let ct = 0;

    for(let i = 0; i < N; i++)
    {

        // Check the given conditions
        if (i > 0 && i + 1 < N)
        {
            if (temp[i] < temp[i + 1] &&
                temp[i] < temp[i - 1])
            {
                ct++;
            }
        }
    }

    document.write("Count of triplets:" + ct + "<br/>");
    document.write("Array:");

    for(let i = 0; i < N; i++)
    {
        document.write(temp[i] + " ");
    }
}

// Driver Code
    let arr = [ 1, 2, 3, 4, 5, 6 ];
    let N = arr.length;

    ctTriplets(arr, N);

// This code is contributed by target_2.
</script>

Output:

Count of triplets:2
Array:4 1 5 2 6 3 

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