# Minimum operations to make counts of remainders same in an array

Given an array **arr[]** of **N** integers and an integer **M** where **N % M = 0**. The task is to find the minimum number of operations that need to be performed on the array to make **c _{0} = c_{1} = ….. = c_{M – 1} = N / M** where

**c**is the

_{r}**number of elements**in the given array

**having remainder r**when divided by

**M**. In each operation, any array element can be incremented by

**1**.

**Examples:**

Input:arr[] = {1, 2, 3}, M = 3

Output:0

After performing the modulus operation on the given array, the array becomes {0, 1, 2}

And count of c_{0}= c_{1}= c_{2}= n / m = 1.

So, no any additional operations are required.

Input:arr[] = {3, 2, 0, 6, 10, 12}, M = 3

Output:3

After performing the modulus operation on the given array, the array becomes {0, 2, 0, 0, 1, 0}

And count of c_{0}= 4, c_{1}= 1 and c_{2}= 1. To make c_{0}= c_{1}= c_{2}= n / m = 2.

Add 1 to 6 and 2 to 12 then the array becomes {3, 2, 0, 7, 10, 14} and c_{0}= c_{1}= c_{2}= n / m = 2.

**Approach:** For each **i** from **0** to **m – 1**, find all the elements of the array that are congruent to **i modulo m** and store their indices in a list. Also, create a vector called extra, and let **k = n / m**.

We have to cycle from **0 to m – 1** twice. For each **i** from **0 to m – 1**, if there are more elements than **k** in the list, remove the extra elements from this list and add them to extra. If instead there are lesser elements than **k** then remove the last few elements from the vector extra. For every removed index **idx**, increase **arr[idx]** by **(i – arr[idx]) % m**.

It is obvious that after the first **m** iterations, every list will have size at most **k** and after **m** more iterations all lists will have the same sizes i.e. **k**.

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 minimum ` `// number of operations required ` `int` `minOperations(` `int` `n, ` `int` `a[], ` `int` `m) ` `{ ` ` ` `int` `k = n / m; ` ` ` ` ` `// To store modulos values ` ` ` `vector<vector<` `int` `> > val(m); ` ` ` `for` `(` `int` `i = 0; i < n; ++i) { ` ` ` `val[a[i] % m].push_back(i); ` ` ` `} ` ` ` ` ` `long` `long` `ans = 0; ` ` ` `vector<pair<` `int` `, ` `int` `> > extra; ` ` ` ` ` `for` `(` `int` `i = 0; i < 2 * m; ++i) { ` ` ` `int` `cur = i % m; ` ` ` ` ` `// If it's size greater than k ` ` ` `// it needed to be decreased ` ` ` `while` `(` `int` `(val[cur].size()) > k) { ` ` ` `int` `elem = val[cur].back(); ` ` ` `val[cur].pop_back(); ` ` ` `extra.push_back(make_pair(elem, i)); ` ` ` `} ` ` ` ` ` `// If it's size is less than k ` ` ` `// it needed to be increased ` ` ` `while` `(` `int` `(val[cur].size()) < k && !extra.empty()) { ` ` ` `int` `elem = extra.back().first; ` ` ` `int` `mmod = extra.back().second; ` ` ` `extra.pop_back(); ` ` ` `val[cur].push_back(elem); ` ` ` `ans += i - mmod; ` ` ` `} ` ` ` `} ` ` ` ` ` `return` `ans; ` `} ` ` ` `// Drive code ` `int` `main() ` `{ ` ` ` `int` `m = 3; ` ` ` ` ` `int` `a[] = { 3, 2, 0, 6, 10, 12 }; ` ` ` `int` `n = ` `sizeof` `(a) / ` `sizeof` `(a[0]); ` ` ` `cout << minOperations(n, a, m); ` ` ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Python3

# Python3 implementation of the approach

# Function to return the minimum

# number of operations required

def minOperations(n, a, m):

k = n // m

# To store modulos values

val = [[] for i in range(m)]

for i in range(0, n):

val[a[i] % m].append(i)

ans = 0

extra = []

for i in range(0, 2 * m):

cur = i % m

# If it’s size greater than k

# it needed to be decreased

while len(val[cur]) > k:

elem = val[cur].pop()

extra.append((elem, i))

# If it’s size is less than k

# it needed to be increased

while (len(val[cur]) < k and
len(extra) > 0):

elem = extra[-1][0]

mmod = extra[-1][1]

extra.pop()

val[cur].append(elem)

ans += i – mmod

return ans

# Drive code

if __name__ == “__main__”:

m = 3

a = [3, 2, 0, 6, 10, 12]

n = len(a)

print(minOperations(n, a, m))

# This code is contributed by Rituraj Jain

**Output:**

3

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