Find the Number of Maximum Product Quadruples

• Difficulty Level : Medium
• Last Updated : 28 Jun, 2021

Given an array of N positive elements, find the number of quadruples, (i, j, k, m) such that i < j < k < m such that the product aiajakam is the maximum possible.

Examples:

Input : N = 7, arr = {1, 2, 3, 3, 3, 3, 5}
Output : 4
Explanation
The maximum quadruple product possible is 135, which can be
achieved by the following quadruples {i, j, k, m} such that aiajakam = 135:
1) a3a4a5a7
2) a3a4a6a7
3) a4a5a6a7
4) a3a5a6a7

Input : N = 4, arr = {1, 5, 2, 1}
Output : 1
Explanation
The maximum quadruple product possible is 10, which can be
achieved by the following quadruple {1, 2, 3, 4} as a1a2a3a4 = 10

Brute Force: O(n4)
Generate all possible quadruple and count the quadruples, giving the maximum product.

Optimized Solution
It is easy to see that the product of the four largest numbers would be the maximum. So, the problem can now be reduced to finding the number of ways of selecting the four largest elements. To do so, we maintain a frequency array that stores the frequency of each element of the array.
Suppose the largest element is X with frequency FX, then if the frequency of this element is >= 4, it is best suited to select the four elements as X, X, X, as that, given a maximum product and the number of ways to do so is FX C 4
and if the frequency is less than 4, the number of ways to select this is 1 and now the required number of elements is 4 – FX. For the second element, say Y, the number of ways are: FX C remaining_choices. Remaining choices denotes the number of additional elements we need to select after selecting the first element. If at any time, the remaining_choices = 0, it means the quadruples are selected, so we can stop the algorithm.

C++

 // CPP program to find the number of Quadruples// having maximum product#include using namespace std; // Returns the number of ways to select r objects// out of available n choicesint NCR(int n, int r){    int numerator = 1;    int denominator = 1;     // ncr = (n * (n - 1) * (n - 2) * .....    // ... (n - r + 1)) / (r * (r - 1) * ... * 1)    while (r > 0) {        numerator *= n;        denominator *= r;        n--;        r--;    }     return (numerator / denominator);} // Returns the number of quadruples having maximum productint findWays(int arr[], int n){    // stores the frequency of each element    map count;     if (n < 4)        return 0;     for (int i = 0; i < n; i++) {        count[arr[i]]++;    }     // remaining_choices denotes the remaining    // elements to select inorder to form quadruple    int remaining_choices = 4;    int ans = 1;     // traverse the elements of the map in reverse order    for (auto iter = count.rbegin(); iter != count.rend(); ++iter) {        int number = iter->first;        int frequency = iter->second;         // If Frequency of element < remaining choices,        // select all of these elements, else select only        // the number of elements required        int toSelect = min(remaining_choices, frequency);        ans = ans * NCR(frequency, toSelect);         // Decrement remaining_choices acc to the number        // of the current elements selected        remaining_choices -= toSelect;         // if the quadruple is formed stop the algorithm        if (!remaining_choices) {            break;        }    }    return ans;} // Driver Codeint main(){    int arr[] = { 1, 2, 3, 3, 3, 5 };    int n = sizeof(arr) / sizeof(arr);     int maxQuadrupleWays = findWays(arr, n);    cout << maxQuadrupleWays;     return 0;}

Java

 // Java program to find the number of Quadruples// having maximum productimport java.util.*;class Solution{// Returns the number of ways to select r objects// out of available n choicesstatic int NCR(int n, int r){    int numerator = 1;    int denominator = 1;     // ncr = (n * (n - 1) * (n - 2) * .....    // ... (n - r + 1)) / (r * (r - 1) * ... * 1)    while (r > 0) {        numerator *= n;        denominator *= r;        n--;        r--;    }     return (numerator / denominator);} // Returns the number of quadruples having maximum productstatic int findWays(int arr[], int n){    // stores the frequency of each element    HashMap count= new HashMap();     if (n < 4)        return 0;     for (int i = 0; i < n; i++) {        count.put(arr[i],(count.get(arr[i])==null?0🙁int)count.get(arr[i])));    }     // remaining_choices denotes the remaining    // elements to select inorder to form quadruple    int remaining_choices = 4;    int ans = 1;         // Getting an iterator        Iterator hmIterator = count.entrySet().iterator();           while (hmIterator.hasNext()) {            Map.Entry mapElement = (Map.Entry)hmIterator.next();            int number =(int) mapElement.getKey();            int frequency =(int)mapElement.getValue();                           // If Frequency of element < remaining choices,        // select all of these elements, else select only        // the number of elements required        int toSelect = Math.min(remaining_choices, frequency);        ans = ans * NCR(frequency, toSelect);         // Decrement remaining_choices acc to the number        // of the current elements selected        remaining_choices -= toSelect;         // if the quadruple is formed stop the algorithm        if (remaining_choices==0) {            break;        }        }        return ans;} // Driver Codepublic static void main(String args[]){    int arr[] = { 1, 2, 3, 3, 3, 5 };    int n = arr.length;     int maxQuadrupleWays = findWays(arr, n);    System.out.print( maxQuadrupleWays);}}//contributed by Arnab Kundu

Python3

 # Python3 program to find# the number of Quadruples# having maximum productfrom collections import defaultdict # Returns the number of ways# to select r objects out of# available n choicesdef NCR(n, r):     numerator = 1    denominator = 1     # ncr = (n * (n - 1) *    # (n - 2) * .....    # ... (n - r + 1)) /    # (r * (r - 1) * ... * 1)    while (r > 0):        numerator *= n        denominator *= r        n -= 1        r -= 1             return (numerator // denominator) # Returns the number of# quadruples having# maximum productdef findWays(arr, n):     # stores the frequency    # of each element    count = defaultdict (int)     if (n < 4):        return 0     for i in range (n):        count[arr[i]] += 1     # remaining_choices denotes    # the remaining elements to    # select inorder to form quadruple    remaining_choices = 4    ans = 1     # traverse the elements of    # the map in reverse order    for it in reversed(sorted(count.keys())):        number = it        frequency = count[it]         # If Frequency of element <        # remaining choices, select        # all of these elements,        # else select only the        # number of elements required        toSelect = min(remaining_choices,                       frequency)        ans = ans * NCR(frequency,                        toSelect)         # Decrement remaining_choices        # acc to the number of the        # current elements selected        remaining_choices -= toSelect         # if the quadruple is        # formed stop the algorithm        if (not remaining_choices):            break    return ans # Driver Codeif __name__ == "__main__":       arr = [1, 2, 3, 3, 3, 5]    n = len(arr)    maxQuadrupleWays = findWays(arr, n)    print (maxQuadrupleWays) # This code is contributed by Chitranayal

Javascript



Output:

1

Time Complexity: O(NlogN), where N is the size of the array.

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