Given an array arr[] of size N and a positive integer X, the task is to partition the array into the maximum number of subsets such that the multiplication of the smallest element of each subset with the count of elements in the subsets is greater than or equal to K. Print the maximum count of such subsets possible.
Examples:
Input: arr[] = {1, 3, 3, 7}, X = 3
Output: 3
Explanation: Partition the array into 3 subsets { {1, 3}, {3}, {7} }. Therefore, the required output is 3.Input: arr[] = {2, 4, 2, 5, 1}, X = 2
Output: 4
Approach: The problem can be solved using the Greedy technique. Follow the steps below to solve the problem:
- Sort the array elements in decreasing order.
- Traverse the array and keep track of the size of the current subset
- As the array is sorted in decreasing order, the rightmost element of the subset will be the smallest element of the current division.
- So, if (size of current subset * current element) is greater than or equal to X, then increment the count and reset the size of the current partition to 0.
- Finally, print the count obtained.
Below is the implementation of the above approach:
C++
// C++ program for the above approach #include <bits/stdc++.h> using namespace std; // Function to count maximum subsets into // which the given array can be split such // that it satisfies the given condition void maxDivisions( int arr[], int N, int X) { // Sort the array in decreasing order sort(arr, arr + N, greater< int >()); // Stores count of subsets possible int maxSub = 0; // Stores count of elements // in current subset int size = 0; // Traverse the array arr[] for ( int i = 0; i < N; i++) { // Update size size++; // If product of the smallest element // present in the current subset and // size of current subset is >= K if (arr[i] * size >= X) { // Update maxSub maxSub++; // Update size size = 0; } } cout << maxSub << endl; } // Driver Code int main() { // Given array int arr[] = { 1, 3, 3, 7 }; // Size of the array int N = sizeof (arr) / sizeof (arr[0]); // Given value of X int X = 3; maxDivisions(arr, N, X); return 0; } |
Java
// Java program for the above approach import java.util.*; class GFG { // Function to count maximum subsets into // which the given array can be split such // that it satisfies the given condition static void maxDivisions(Integer arr[], int N, int X) { // Sort the array in decreasing order Arrays.sort(arr,Collections.reverseOrder()); // Stores count of subsets possible int maxSub = 0 ; // Stores count of elements // in current subset int size = 0 ; // Traverse the array arr[] for ( int i = 0 ; i < N; i++) { // Update size size++; // If product of the smallest element // present in the current subset and // size of current subset is >= K if (arr[i] * size >= X) { // Update maxSub maxSub++; // Update size size = 0 ; } } System.out.print(maxSub + "\n" ); } // Driver Code public static void main(String[] args) { // Given array Integer arr[] = { 1 , 3 , 3 , 7 }; // Size of the array int N = arr.length; // Given value of X int X = 3 ; maxDivisions(arr, N, X); } } // This code is contributed by shikhasingrajput |
Python3
# Python3 program for the above approach # Function to count maximum subsets into # which the given array can be split such # that it satisfies the given condition def maxDivisions(arr, N, X) : # Sort the array in decreasing order arr.sort(reverse = True ) # Stores count of subsets possible maxSub = 0 ; # Stores count of elements # in current subset size = 0 ; # Traverse the array arr[] for i in range (N) : # Update size size + = 1 ; # If product of the smallest element # present in the current subset and # size of current subset is >= K if (arr[i] * size > = X) : # Update maxSub maxSub + = 1 ; # Update size size = 0 ; print (maxSub); # Driver Code if __name__ = = "__main__" : # Given array arr = [ 1 , 3 , 3 , 7 ]; # Size of the array N = len (arr); # Given value of X X = 3 ; maxDivisions(arr, N, X); # This code is contributed by AnkThon |
C#
// C# program for the above approach using System; class GFG { // Function to count maximum subsets into // which the given array can be split such // that it satisfies the given condition static void maxDivisions( int [] arr, int N, int X) { // Sort the array in decreasing order Array.Sort(arr); Array.Reverse(arr); // Stores count of subsets possible int maxSub = 0; // Stores count of elements // in current subset int size = 0; // Traverse the array arr[] for ( int i = 0; i < N; i++) { // Update size size++; // If product of the smallest element // present in the current subset and // size of current subset is >= K if (arr[i] * size >= X) { // Update maxSub maxSub++; // Update size size = 0; } } Console.WriteLine(maxSub); } // Driver Code public static void Main() { // Given array int [] arr = { 1, 3, 3, 7 }; // Size of the array int N = arr.Length; // Given value of X int X = 3; maxDivisions(arr, N, X); } } // This code is contributed by subhammahato348. |
Javascript
<script> // javascript program of the above approach // Function to count maximum subsets into // which the given array can be split such // that it satisfies the given condition function maxDivisions(arr, N, X) { // Sort the array in decreasing order arr.sort(); // Stores count of subsets possible let maxSub = 0; // Stores count of elements // in current subset let size = 0; // Traverse the array arr[] for (let i = 0; i < N; i++) { // Update size size++; // If product of the smallest element // present in the current subset and // size of current subset is >= K if (arr[i] * size >= X) { // Update maxSub maxSub++; // Update size size = 0; } } document.write(maxSub + "<br/>" ); } // Driver Code // Given array let arr = [ 1, 3, 3, 7 ]; // Size of the array let N = arr.length; // Given value of X let X = 3; maxDivisions(arr, N, X); </script> |
3
Time Complexity: O(N * log(N))
Auxiliary Space: O(1)
Another Approach:
- Sort the given array arr[] in non-decreasing order.
- Initialize two variables, ans and prod, to 0 and 1 respectively. (a) ans: count of subsets into which the given array can be split such that the multiplication of the smallest element of each subset with the count of elements in the subsets is greater than or equal to K.
(b) prod: the product of elements in the current subset. - Traverse the array arr[] from left to right:
(a) Multiply prod with arr[i] for every element in the array. (b) If the product of elements so far is greater than or equal to X, then divide the current product by arr[i] and reset prod to 1. Increment the count of subsets ans. - If there are still some elements left, then increment the count of subsets ans.
- Return ans.
Below is the implementation of the above approach:
C++
#include<bits/stdc++.h> using namespace std; // Function to count maximum subsets into // which the given array can be split such // that it satisfies the given condition int maxDivisions( int arr[], int N, int X) { // Sort the array in non-decreasing order sort(arr, arr + N); int ans = 0, prod = 1; // Traverse the array arr[] from left to right. for ( int i = 0; i < N; i++) { prod *= arr[i]; // If the product of elements so far is >= X, // then divide the current product by arr[i] // and reset prod to 1. if (prod >= X) { ans++; prod = 1; } } // If there are still some elements left, // then increment the count of subsets. if (prod > 1) { ans++; } return ans; } // Driver code int main() { // Given array int arr[] = { 1, 3, 3, 7 }; // Size of the array int N = sizeof (arr) / sizeof (arr[0]); // Given value of X int X = 3; cout << maxDivisions(arr, N, X) << endl; return 0; } |
Java
import java.util.*; public class Main { // Function to count maximum subsets into // which the given array can be split such // that it satisfies the given condition public static int maxDivisions( int arr[], int N, int X) { // Sort the array in non-decreasing order Arrays.sort(arr); int ans = 0 , prod = 1 ; // Traverse the array arr[] from left to right. for ( int i = 0 ; i < N; i++) { prod *= arr[i]; // If the product of elements so far is >= X, // then divide the current product by arr[i] // and reset prod to 1. if (prod >= X) { ans++; prod = 1 ; } } // If there are still some elements left, // then increment the count of subsets. if (prod > 1 ) { ans++; } return ans; } // Driver code public static void main(String[] args) { // Given array int arr[] = { 1 , 3 , 3 , 7 }; // Size of the array int N = arr.length; // Given value of X int X = 3 ; System.out.println(maxDivisions(arr, N, X)); } } |
Javascript
// Function to count maximum subsets into // which the given array can be split such // that it satisfies the given condition function maxDivisions(arr, N, X) { // Sort the array in non-decreasing order arr.sort((a, b) => a - b); let ans = 0, prod = 1; // Traverse the array arr[] from left to right. for (let i = 0; i < N; i++) { prod *= arr[i]; // If the product of elements so far is >= X, // then divide the current product by arr[i] // and reset prod to 1. if (prod >= X) { ans++; prod = 1; } } // If there are still some elements left, // then increment the count of subsets. if (prod > 1) { ans++; } return ans; } // Driver code // Given array const arr = [1, 3, 3, 7]; // Size of the array const N = arr.length; // Given value of X const X = 3; console.log(maxDivisions(arr, N, X)); |
C#
using System; using System.Linq; class GFG { // Function to count maximum subsets into // which the given array can be split such // that it satisfies the given condition static int maxDivisions( int [] arr, int N, int X) { // Sort the array in non-decreasing order Array.Sort(arr); int ans = 0, prod = 1; // Traverse the array arr[] from left to right. for ( int i = 0; i < N; i++) { prod *= arr[i]; // If the product of elements so far is >= X, // then divide the current product by arr[i] // and reset prod to 1. if (prod >= X) { ans++; prod = 1; } } // If there are still some elements left, // then increment the count of subsets. if (prod > 1) { ans++; } return ans; } // Driver code static void Main() { // Given array int [] arr = { 1, 3, 3, 7 }; // Size of the array int N = arr.Length; // Given value of X int X = 3; Console.WriteLine(maxDivisions(arr, N, X)); } } // This code is contributed by user_dtewbxkn77n |
Python3
def maxDivisions(arr, N, X): # Sort the array in non-decreasing order arr.sort() ans = 0 prod = 1 # Traverse the array arr[] from left to right. for i in range (N): prod * = arr[i] # If the product of elements so far is >= X, # then divide the current product by arr[i] # and reset prod to 1. if prod > = X: ans + = 1 prod = 1 # If there are still some elements left, # then increment the count of subsets. if prod > 1 : ans + = 1 return ans # Driver code arr = [ 1 , 3 , 3 , 7 ] N = len (arr) X = 3 print (maxDivisions(arr, N, X)) |
3
Time Complexity: O(N*logN)
Auxiliary Space: O(1)
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