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Count decrements to nearest smaller element required to make all array elements equal

  • Difficulty Level : Medium
  • Last Updated : 19 May, 2021

Given an array arr[] consisting of N non-negative integers, the task is to find the number of operations required to make all array elements equal. In each operation, any array element can be changed to its nearest smaller array element.

Examples:

Input: arr[] = {2, 5, 4, 3, 5, 4}
Output: 11
Explanation: 
Step 1: Replace all 5s with 4s. Therefore, arr[] = {2, 4, 4, 3, 4, 4}. Count of operations = 2
Step 2: Replace all 4s with 3s. Therefore, arr[] = {2, 3, 3, 3, 3, 3}. Count of operations = 4
Steps 3: Replace all 3s with 2s. Therefore, arr[] = {2, 2, 2, 2, 2, 2}. Count of operations = 5
Therefore, total number of operations = 11

Input : arr[] = {2, 2, 2}
Output : 0

Approach: The idea is to use a sorting algorithm and a Map data structure. Follow the steps below to solve the problem:



  1. Initialize a Map to store the frequency of each array element except for the minimum element where keys (K) are the set of unique elements and values (V) are their frequencies.
  2. Sort the Map in decreasing order based on the keys.
  3. Initialize two variables ans and prev_val with 0 to store the current answer and the prefix sum respectively.
  4. Now iterate the Map and add the corresponding frequencies of each element along with prev_val to ans as ans = ans + (freq) + prev_val
  5. While iterating the Map, increment prev_val every time by the frequency of the current element.
  6. After traversing the Map completely, print ans as the minimum number of operations.

Below is the implementation of the above approach:

C++




// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to print the minimum number of
// decrements to make all elements equals
int MinimumNoOfOperations(int arr[], int n)
{
 
    // Find minimum element
    int min_ele = INT_MAX;
    for (int i = 0; i < n; ++i) {
        min_ele = min(min_ele, arr[i]);
    }
 
    // Stores frequencies of array elements
    map<int, int, greater<int> > mp;
 
    // Update frequencies in the Map
    for (int i = 0; i < n; ++i) {
 
        if (arr[i] == min_ele)
            continue;
        else
            mp[arr[i]]++;
    }
 
    // Iterate the map
    map<int, int>::iterator it;
 
    // Stores the count of
    // decrements at each iteration
    int prev_val = 0;
 
    // Stores the total
    // count of derements
    int ans = 0;
 
    // Calculate the number of decrements
    for (it = mp.begin(); it != mp.end();
         ++it) {
 
        ans += (it->second + prev_val);
        prev_val += it->second;
    }
 
    // Return minimum operations
    return ans;
}
 
// Driver Code
int main()
{
    // Given array
    int arr[] = { 2, 5, 4, 3, 5, 4 };
 
    // Given size
    int size = sizeof(arr)
               / sizeof(arr[0]);
 
    // Function call
    cout << MinimumNoOfOperations(
        arr, size);
    return 0;
}

Java




// Java program for the
// above approach
import java.util.*;
 
class solution{
    
// Function to print the minimum
// number of decrements to make
// all elements equals
static int MinimumNoOfOperations(int arr[],
                                 int n)
{
  // Find minimum element
  int min_ele = Integer.MAX_VALUE;
   
  for (int i = 0; i < n; ++i)
  {
    min_ele = Math.min(min_ele,
                       arr[i]);
  }
 
  // Stores frequencies of array
  // elements
  TreeMap<Integer,
          Integer> mp =
          new TreeMap<Integer,
                      Integer>(
          Collections.reverseOrder());
 
  // Update frequencies in
  // the Map
  for (int i = 0; i < n; ++i)
  {
    if (arr[i] == min_ele)
      continue;
    else
      mp.put(arr[i],
      mp.getOrDefault(arr[i], 0) + 1);
  }
 
  // Stores the count of
  // decrements at each
  // iteration
  int prev_val = 0;
 
  // Stores the total
  // count of derements
  int ans = 0;
 
  // Calculate the number of
  // decrements
  for (Map.Entry<Integer,
                 Integer> it : mp.entrySet())
  {
    ans += (it.getValue() + prev_val);
    prev_val += it.getValue();
  }
 
  // Return minimum operations
  return ans;
}
 
// Driver Code
public static void main(String args[])
{
  // Given array
  int arr[] = {2, 5, 4, 3, 5, 4};
   
  // Given size
  int size = arr.length;
   
  // Function call
  System.out.println(
  MinimumNoOfOperations(arr, size));
}
}
 
// This code is contributed by bgangwar59

Python3




# Python3 program for the above approach
import sys
 
# Function to print the minimum number of
# decrements to make all elements equals
def MinimumNoOfOperations(arr, n):
     
    # Find minimum element
    min_ele = sys.maxsize
     
    for i in range(n):
        min_ele = min(min_ele, arr[i])
 
    # Stores frequencies of array elements
    mp = {}
 
    # Update frequencies in the Map
    for i in range(n):
        if (arr[i] == min_ele):
            continue
        else:
            mp[arr[i]] = mp.get(arr[i], 0) + 1
             
    # Stores the count of
    # decrements at each iteration
    prev_val = 0
 
    # Stores the total
    # count of derements
    ans = 0
 
    # Calculate the number of decrements
    for it in mp:
        ans += (mp[it] + prev_val)
        prev_val += mp[it]
 
    # Return minimum operations
    return ans
 
# Driver Code
if __name__ == '__main__':
     
    # Given array
    arr = [ 2, 5, 4, 3, 5, 4 ]
 
    # Given size
    size = len(arr)
 
    # Function call
    print(MinimumNoOfOperations(arr, size))
 
# This code is contributed by mohit kumar 29

C#




// C# program for the
// above approach
using System.Collections.Generic;
using System;
using System.Linq;
 
class GFG{
    
// Function to print the minimum
// number of decrements to make
// all elements equals
static int MinimumNoOfOperations(int []arr,
                                 int n)
{
   
  // Find minimum element
  int min_ele = 1000000000;
   
  for(int i = 0; i < n; ++i)
  {
      min_ele = Math.Min(min_ele, arr[i]);
  }
 
  // Stores frequencies of array
  // elements
  Dictionary<int,
             int> mp = new Dictionary<int,
                                      int>();
 
  // Update frequencies in
  // the Map
  for(int i = 0; i < n; ++i)
  {
    if (arr[i] == min_ele)
      continue;
    else
    {
      if (mp.ContainsKey(arr[i]) == true)
        mp[arr[i]] += 1;
      else
        mp[arr[i]] = 1;   
    }
  }
 
  // Stores the count of
  // decrements at each
  // iteration
  int prev_val = 0;
 
  // Stores the total
  // count of derements
  int ans = 0;
 
  // Calculate the number of
  // decrements
  var val = mp.Keys.ToList();
  foreach(var key in val)
  {
    ans += (mp[key] + prev_val);
    prev_val += mp[key];
  }
   
  // Return minimum operations
  return ans;
}
 
// Driver Code
public static void Main(String[] args)
{
   
  // Given array
  int []arr = { 2, 5, 4, 3, 5, 4 };
   
  // Given size
  int size = arr.Length;
   
  // Function call
  Console.WriteLine(MinimumNoOfOperations(
    arr, size));
}
}
 
// This code is contributed by Stream_Cipher

Javascript




<script>
// Javascript program for the above approach
 
// Function to print the minimum number of
// decrements to make all elements equals
function MinimumNoOfOperations(arr, n)
{
 
    // Find minimum element
    var min_ele = 1000000000;
    for (var i = 0; i < n; ++i) {
        min_ele = Math.min(min_ele, arr[i]);
    }
 
    // Stores frequencies of array elements
    var mp = new Map();
 
    // Update frequencies in the Map
    for (var i = 0; i < n; ++i) {
 
        if (arr[i] == min_ele)
            continue;
        else
        {
            if(mp.has(arr[i]))
            {
                mp.set(arr[i], mp.get(arr[i])+1);
            }
            else
            {
                mp.set(arr[i], 1);
            }
        }
    }
 
    // Stores the count of
    // decrements at each iteration
    var prev_val = 0;
 
    // Stores the total
    // count of derements
    var ans = 0;
    var keys = [];
 
    mp.forEach((value, key) => {
        keys.push(key);
    });
    keys.sort((a,b)=>b-a);
    // Calculate the number of decrements
    keys.forEach(value => {
        ans += (mp.get(value) + prev_val);
        prev_val += mp.get(value);
    });
 
    // Return minimum operations
    return ans;
}
 
// Driver Code
// Given array
var arr = [2, 5, 4, 3, 5, 4];
// Given size
var size = arr.length
// Function call
document.write( MinimumNoOfOperations(
    arr, size));
 
</script>
Output: 
11

 

Time Complexity: O(N) where N is the size of the array.
Auxiliary Space: O(N)

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