Maximum number of groups that can receive fresh donuts distributed in batches of size K

Given an array arr[] consisting of N positive integers such that arr[i] denotes the size of the i^th group sitting in a donut shop and a positive integer K, which denotes the maximum number of donuts that can be served in a batch, the task is to find the maximum number of groups that can receive fresh donuts if the customers of the same group are served together. 

Note: All the customers in a group do not receive leftover donuts from the previous batch.

Examples:

Input: arr[] = {1, 2, 3, 4, 5, 6}, K = 3
Output: 4
Explanation: One possible way for the ordering of the groups is {6, 2, 4, 5, 1, 3}.

1. arr[0](= 6), The shops serve two batches of 3 donuts. So everyone gets fresh donuts.
2. arr[1](= 2), The shop serves 3 fresh donuts, among which 1 gets left out. So everyone gets fresh donuts.
3. arr[2](= 4), The shop serves first 1 leftover donut and then serves 3 fresh donuts.
4. arr[1](= 5), The shop serves 6 fresh donuts, among which 1 is left out. So everyone gets fresh donuts.
5. arr[1](= 1), The shop serves 1 leftover donut.
6. arr[1](= 3), The shop serves 3 fresh donuts. So everyone gets fresh donuts.

Therefore, a total of 4 groups get fresh donuts which is the maximum possible number of groups.

Input: arr[] = {1, 3, 2, 5, 2, 2, 1, 6}, K = 4
Output: 4

Naive Approach: The given problem can be solved by using backtracking for all possible ordering based on the observation that the remainder of each group size by K, needs only to be considered. Follow the steps below to solve the problem:

• Initialize an array, say V[] of size K, where V[i] denotes the number of groups with i people remaining.
• Traverse the array, arr[] using the variable i and for each arr[i], increment the value of V[arr[i] % K] by 1.
• Define a recursive function say dfs(V, left) to where left is the number of leftover donuts from the previous batch:
  • Initialize a variable, res as 0, to store the result of the current state.
  • If the value of left is 0, then iterate in the range [1, K – 1] using the variable i and perform the following steps:
    • Decrement the value of V[i] by 1 and recursively call the function with argument left-i.
    • Update res to the maximum of dfs(V, left – i) + 1 and res.
    • Increment the value of V[i] by 1 for the backtracking step.
  • Otherwise, repeat the same steps as above, but in this case, do not add 1 to the result, since the selected group will get the leftover donut.
  • Return the value of res.
• Call the recursive function defined above as dfs(V, 0) and store the returned value in a variable, say X.
• Finally, after completing the above steps, print the sum of V[0] and X as the result.

Below is the implementation of the above approach:

C++

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

// Recursive function to find the
// maximum number of groups that
// will receive fresh donuts
int dfs(int arr[], int left, int K)
{

    // Store the result for the
    // current state
    int q = 0;

    // Check if the leftover donuts
    // from the previous batch is 0
    if (left == 0) {

        // If true, then one by one
        // give the fresh donuts
        // to each group
        for (int i = 1; i < K; ++i) {
            if (arr[i] > 0) {

                // Decrement arr[i]
                arr[i]--;

                // Update the maximum
                // number of groups
                q = max(q, 1 + dfs(arr, K - i, K));

                // Increment arr[i]
                arr[i]++;
            }
        }
    }

    // Otherwise, traverse the given
    // array, arr[]
    else {

        for (int i = 1; i < K; ++i) {
            if (arr[i] > 0) {

                // Decrement arr[i]
                arr[i]--;

                int nleft
                    = (i <= left ? left - i : K + left - i);

                // Update the maximum
                // number of groups
                q = max(q, dfs(arr, nleft, K));

                // Increment arr[i]
                arr[i]++;
            }
        }
    }

    // Return the value of q
    return q;
}

// Function to find the maximum
// number of groups that will
// receive fresh donuts
int maxGroups(int K, int arr[], int n)
{

    // Stores count of remainder
    // by K
    int V[K] = { 0 };

    // Traverse the array arr[]
    for (int x = 0; x < n; x++)
        V[arr[x] % K]++;

    // Stores maximum number of groups
    int ans = V[0] + dfs(V, 0, K);

    // Return the answer
    return ans;
}

// Driver Code
int main()
{
    int arr[] = { 1, 2, 3, 4, 5, 6 };
    int n = sizeof(arr) / sizeof(arr[0]);
    int K = 3;

    cout << maxGroups(K, arr, n);

    return 0;
}

// This code is contributed by Potta Lokesh

Java

// Java program for the above approach

import java.io.*;
import java.util.*;

class GFG {

    // Recursive function to find the
    // maximum number of groups that
    // will receive fresh donuts
    public static int dfs(int[] arr,
                          int left, int K)
    {
        // Store the result for the
        // current state
        int q = 0;

        // Check if the leftover donuts
        // from the previous batch is 0
        if (left == 0) {

            // If true, then one by one
            // give the fresh donuts
            // to each group
            for (int i = 1; i < K; ++i) {
                if (arr[i] > 0) {

                    // Decrement arr[i]
                    arr[i]--;

                    // Update the maximum
                    // number of groups
                    q = Math.max(
                        q, 1 + dfs(arr, K - i, K));

                    // Increment arr[i]
                    arr[i]++;
                }
            }
        }

        // Otherwise, traverse the given
        // array, arr[]
        else {

            for (int i = 1; i < K; ++i) {
                if (arr[i] > 0) {

                    // Decrement arr[i]
                    arr[i]--;

                    int nleft = (i <= left ? left - i
                                           : K + left - i);

                    // Update the maximum
                    // number of groups
                    q = Math.max(q, dfs(arr, nleft, K));

                    // Increment arr[i]
                    arr[i]++;
                }
            }
        }

        // Return the value of q
        return q;
    }

    // Function to find the maximum
    // number of groups that will
    // receive fresh donuts
    public static int maxGroups(int K,
                                int[] arr)
    {
        // Stores count of remainder
        // by K
        int V[] = new int[K];

        // Traverse the array arr[]
        for (int x : arr)
            V[x % K]++;

        // Stores maximum number of groups
        int ans = V[0] + dfs(V, 0, K);

        // Return the answer
        return ans;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int[] arr = { 1, 2, 3, 4, 5, 6 };
        int K = 3;
        System.out.println(
            maxGroups(K, arr));
    }
}

Python3

# Python 3 program for the above approach

# Recursive function to find the
# maximum number of groups that
# will receive fresh donuts
def dfs(arr, left, K):

    # Store the result for the
    # current state
    q = 0

    # Check if the leftover donuts
    # from the previous batch is 0
    if (left == 0):

        # If true, then one by one
        # give the fresh donuts
        # to each group
        for i in range(1,K,1):
            if (arr[i] > 0):

                # Decrement arr[i]
                arr[i] -= 1

                # Update the maximum
                # number of groups
                q = max(q, 1 + dfs(arr, K - i, K))

                # Increment arr[i]
                arr[i] += 1

    # Otherwise, traverse the given
    # array, arr[]
    else:

        for i in range(1,K,1):
            if (arr[i] > 0):

                # Decrement arr[i]
                arr[i] -= 1

                nleft = left - i if i <= left else K + left - i

                # Update the maximum
                # number of groups
                q = max(q, dfs(arr, nleft, K))

                # Increment arr[i]
                arr[i] += 1

    # Return the value of q
    return q

# Function to find the maximum
# number of groups that will
# receive fresh donuts
def maxGroups(K, arr, n):

    # Stores count of remainder
    # by K
    V = [0 for i in range(K)]

    # Traverse the array arr[]
    for x in range(n):
        V[arr[x] % K] += 1

    # Stores maximum number of groups
    ans = V[0] + dfs(V, 0, K)

    # Return the answer
    return ans

# Driver Code
if __name__ == '__main__':
    arr= [1, 2, 3, 4, 5, 6]
    n = len(arr)
    K = 3

    print(maxGroups(K, arr, n))

    # This code is contributed by SURENDRA_GANGWAR.

C#

// C# program for the above approach
using System;

class GFG{

// Recursive function to find the
// maximum number of groups that
// will receive fresh donuts
public static int dfs(int[] arr,
                      int left, int K)
{

    // Store the result for the
    // current state
    int q = 0;

    // Check if the leftover donuts
    // from the previous batch is 0
    if (left == 0)
    {

        // If true, then one by one
        // give the fresh donuts
        // to each group
        for(int i = 1; i < K; ++i)
        {
            if (arr[i] > 0)
            {

                // Decrement arr[i]
                arr[i]--;

                // Update the maximum
                // number of groups
                q = Math.Max(q, 1 + dfs(arr, K - i, K));

                // Increment arr[i]
                arr[i]++;
            }
        }
    }

    // Otherwise, traverse the given
    // array, arr[]
    else
    {
        for(int i = 1; i < K; ++i)
        {
            if (arr[i] > 0)
            {

                // Decrement arr[i]
                arr[i]--;

                int nleft = (i <= left ? left - i :
                             K + left - i);

                // Update the maximum
                // number of groups
                q = Math.Max(q, dfs(arr, nleft, K));

                // Increment arr[i]
                arr[i]++;
            }
        }
    }

    // Return the value of q
    return q;
}

// Function to find the maximum
// number of groups that will
// receive fresh donuts
public static int maxGroups(int K, int[] arr)
{

    // Stores count of remainder
    // by K
    int[] V = new int[K];

    // Traverse the array arr[]
    foreach(int x in arr)
        V[x % K]++;

    // Stores maximum number of groups
    int ans = V[0] + dfs(V, 0, K);

    // Return the answer
    return ans;
}

// Driver code
public static void Main(string[] args)
{
    int[] arr = { 1, 2, 3, 4, 5, 6 };
    int K = 3;

    Console.WriteLine(maxGroups(K, arr));
}
}

// This code is contributed by sanjoy_62

Javascript

<script>

// Javascript program for the above approach

// Recursive function to find the
// maximum number of groups that
// will receive fresh donuts
function dfs(arr, left, K)
{

    // Store the result for the
    // current state
    let q = 0;

    // Check if the leftover donuts
    // from the previous batch is 0
    if (left == 0)
    {

        // If true, then one by one
        // give the fresh donuts
        // to each group
        for(let i = 1; i < K; ++i)
        {
            if (arr[i] > 0)
            {

                // Decrement arr[i]
                arr[i]--;

                // Update the maximum
                // number of groups
                q = Math.max(q, 1 + dfs(arr, K - i, K));

                // Increment arr[i]
                arr[i]++;
            }
        }
    }

    // Otherwise, traverse the given
    // array, arr[]
    else
    {
        for(let i = 1; i < K; ++i)
        {
            if (arr[i] > 0)
            {

                // Decrement arr[i]
                arr[i]--;

                let nleft = (i <= left ? left - i :
                              K + left - i);

                // Update the maximum
                // number of groups
                q = Math.max(q, dfs(arr, nleft, K));

                // Increment arr[i]
                arr[i]++;
            }
        }
    }

    // Return the value of q
    return q;
}

// Function to find the maximum
// number of groups that will
// receive fresh donuts
function maxGroups(K, arr, n)
{

    // Stores count of remainder
    // by K
    let V = new Array(K).fill(0);

    // Traverse the array arr[]
    for(let x = 0; x < n; x++)
        V[arr[x] % K]++;

    // Stores maximum number of groups
    let ans = V[0] + dfs(V, 0, K);

    // Return the answer
    return ans;
}

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

document.write(maxGroups(K, arr, n))

// This code is contributed by _Saurabh_jaiswal

</script>

Output:

4

Time Complexity: O(N + K^K)
Auxiliary Space: O(K)

Efficient Approach: The above approach has Optimal Substructure and Overlapping Subproblems, therefore, the above approach can also be optimized by memoizing the same recursive calls in a HashMap and use this state when the same problem is called recursively.

Below is the implementation of the above approach:

C++

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

// Stores the result of the same
// recursive calls
map<string, int> memo;

// Recursive function to find the
// maximum number of groups that
// will receive fresh donuts
int dfs(int V[], int left, int K)
{

    // Store the result for the
    // current state
    int q = 0;

    // Store the key and check
    // if it is present in the
    // hashmap
    string key = "";
    for(int i = 0; i < K; i++)
    {
        key = key + to_string(V[i]);
    }
    key += to_string(left);
    // If already calculated
    if (memo.find(key) != memo.end())
        return memo[key];

    // If left is 0
    else if (left == 0) {

        // Traverse the array []arr
        for (int i = 1; i < K; ++i)
            if (V[i] > 0) {

                // Decrement arr[i]
                V[i]--;

                // Update the maximum
                // number of groups
                q = max(q, 1 + dfs(V, K - i, K));

                // Increment arr[i] by 1
                V[i]++;
            }
    }

    // Otherwise, traverse the given
    // array []arr
    else {

        for (int i = 1; i < K; ++i) {
            if (V[i] > 0) {

                // Decrement arr[i]
                V[i]--;

                int nleft = i <= left ? left - i : K + left - i;

                // Update the maximum
                // number of groups
                q = max(q, dfs(V, nleft, K));

                // Increment arr[i] by 1
                V[i]++;
            }
        }
    }

    // Memoize the result and
    // return it
    if(memo.find(key) != memo.end())
        memo[key] = q;
    else
        memo[key] = q;

    return q;
}

// Function to find the maximum
// number of groups that will
// receive fresh donuts
int maxGroups(int K, int arr[])
{

    // Stores count of remainder by K
    int V[K];
    memset(V, 0, sizeof(V));

    // Traverse the array []arr
    for (int i = 0; i < 6; i++)
        V[arr[i] % K]++;

    // Store the maximum number
    // of groups
    int ans = V[0] + dfs(V, 0, K);

    // Return the answer
    return ans;
}

int main()
{
    int arr[] = { 1, 2, 3, 4, 5, 6 };
    int K = 3;
    cout << maxGroups(K, arr);

    return 0;
}

// This code is contributed by divyeshrabadiya07.

Java

// Java program for the above approach

import java.io.*;
import java.util.*;

class GFG {

    // Stores the result of the same
    // recursive calls
    static HashMap<String, Integer> memo;

    // Recursive function to find the
    // maximum number of groups that
    // will receive fresh donuts
    public static int dfs(int[] V,
                          int left, int K)
    {
        // Store the result for the
        // current state
        int q = 0;

        // Store the key and check
        // if it is present in the
        // hashmap
        String key = Arrays.toString(V);
        key += Integer.toString(left);

        // If already calculated
        if (memo.containsKey(key))
            return memo.get(key);

        // If left is 0
        else if (left == 0) {

            // Traverse the array arr[]
            for (int i = 1; i < K; ++i)
                if (V[i] > 0) {

                    // Decrement arr[i]
                    V[i]--;

                    // Update the maximum
                    // number of groups
                    q = Math.max(
                        q, 1 + dfs(V, K - i, K));

                    // Increment arr[i] by 1
                    V[i]++;
                }
        }

        // Otherwise, traverse the given
        // array arr[]
        else {

            for (int i = 1; i < K; ++i) {
                if (V[i] > 0) {

                    // Decrement arr[i]
                    V[i]--;

                    int nleft = i <= left ? left - i
                                          : K + left - i;

                    // Update the maximum
                    // number of groups
                    q = Math.max(q, dfs(V, nleft, K));

                    // Increment arr[i] by 1
                    V[i]++;
                }
            }
        }

        // Memoize the result and
        // return it
        memo.put(key, q);

        return q;
    }

    // Function to find the maximum
    // number of groups that will
    // receive fresh donuts
    public static int maxGroups(int K, int[] arr)
    {
        // Stores count of remainder by K
        int V[] = new int[K];

        // Traverse the array arr[]
        for (int x : arr)
            V[x % K]++;

        // Hashmap to memoize the results
        memo = new HashMap<String, Integer>();

        // Store the maximum number
        // of groups
        int ans = V[0] + dfs(V, 0, K);

        // Return the answer
        return ans;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int[] arr = { 1, 2, 3, 4, 5, 6 };
        int K = 3;
        System.out.println(
            maxGroups(K, arr));
    }
}

Python3

# Python3 program for the above approach

# Stores the result of the same
# recursive calls
memo = {}

# Recursive function to find the
# maximum number of groups that
# will receive fresh donuts
def dfs(V, left, K):

    # Store the result for the
    # current state
    q = 0

    # Store the key and check
    # if it is present in the
    # hashmap
    v = [str(int) for int in V]
    key = ",".join(v)

    key += str(left)

    # If already calculated
    if key in memo:
        return memo[key]

    # If left is 0
    elif left == 0:

        # Traverse the array []arr
        for i in range(1, K):
            if V[i] > 0:

                # Decrement arr[i]
                V[i]-=1

                # Update the maximum
                # number of groups
                q = max(q, 1 + dfs(V, K - i, K))

                # Increment arr[i] by 1
                V[i]+=1

    # Otherwise, traverse the given
    # array []arr
    else:
        for i in range(1, K):
            if V[i] > 0:
                # Decrement arr[i]
                V[i]-=1

                if i <= left:
                    nleft = left - i
                else:
                    nleft = K + left - i

                # Update the maximum
                # number of groups
                q = max(q, dfs(V, nleft, K))

                # Increment arr[i] by 1
                V[i]+=1

    # Memoize the result and
    # return it
    if key in memo:
        memo[key] = q
    else:
        memo[key] = q

    return q

# Function to find the maximum
# number of groups that will
# receive fresh donuts
def maxGroups(K, arr):

    # Stores count of remainder by K
    V = [0]*(K)

    # Traverse the array []arr
    for x in range(len(arr)):
        V[arr[x] % K] += 1

    # Hashmap to memoize the results
    memo = {}

    # Store the maximum number
    # of groups
    ans = V[0] + dfs(V, 0, K)

    # Return the answer
    return ans

arr = [ 1, 2, 3, 4, 5, 6 ]
K = 3
print(maxGroups(K, arr))

# This code is contributed by divyesh072019.

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

public class GFG {

    // Stores the result of the same
    // recursive calls
    static Dictionary<String, int> memo;

    // Recursive function to find the
    // maximum number of groups that
    // will receive fresh donuts
    public static int dfs(int[] V,
                          int left, int K)
    {
        // Store the result for the
        // current state
        int q = 0;

        // Store the key and check
        // if it is present in the
        // hashmap
        String key = string.Join(",", V);
        key += left.ToString();
        // If already calculated
        if (memo.ContainsKey(key))
            return memo[key];

        // If left is 0
        else if (left == 0) {

            // Traverse the array []arr
            for (int i = 1; i < K; ++i)
                if (V[i] > 0) {

                    // Decrement arr[i]
                    V[i]--;

                    // Update the maximum
                    // number of groups
                    q = Math.Max(
                        q, 1 + dfs(V, K - i, K));

                    // Increment arr[i] by 1
                    V[i]++;
                }
        }

        // Otherwise, traverse the given
        // array []arr
        else {

            for (int i = 1; i < K; ++i) {
                if (V[i] > 0) {

                    // Decrement arr[i]
                    V[i]--;

                    int nleft = i <= left ? left - i
                                          : K + left - i;

                    // Update the maximum
                    // number of groups
                    q = Math.Max(q, dfs(V, nleft, K));

                    // Increment arr[i] by 1
                    V[i]++;
                }
            }
        }

        // Memoize the result and
        // return it
        if(memo.ContainsKey(key))
            memo[key] = q;
        else
            memo.Add(key, q);

        return q;
    }

    // Function to find the maximum
    // number of groups that will
    // receive fresh donuts
    public static int maxGroups(int K, int[] arr)
    {

        // Stores count of remainder by K
        int []V = new int[K];

        // Traverse the array []arr
        foreach (int x in arr)
            V[x % K]++;

        // Hashmap to memoize the results
        memo = new Dictionary<String, int>();

        // Store the maximum number
        // of groups
        int ans = V[0] + dfs(V, 0, K);

        // Return the answer
        return ans;
    }

    // Driver Code
    public static void Main(String[] args)
    {
        int[] arr = { 1, 2, 3, 4, 5, 6 };
        int K = 3;
        Console.WriteLine(
            maxGroups(K, arr));
    }
}

// This code is contributed by 29AjayKumar

Javascript

<script>
    // Javascript program for the above approach

    // Stores the result of the same
    // recursive calls
    let memo;

    // Recursive function to find the
    // maximum number of groups that
    // will receive fresh donuts
    function dfs(V, left, K)
    {

        // Store the result for the
        // current state
        let q = 0;

        // Store the key and check
        // if it is present in the
        // hashmap
        let key = V.join(",");
        key += left.toString();

        // If already calculated
        if (memo.has(key))
            return memo[key];

        // If left is 0
        else if (left == 0) {

            // Traverse the array []arr
            for (let i = 1; i < K; ++i)
                if (V[i] > 0) {

                    // Decrement arr[i]
                    V[i]--;

                    // Update the maximum
                    // number of groups
                    q = Math.max(q, 1 + dfs(V, K - i, K));

                    // Increment arr[i] by 1
                    V[i]++;
                }
        }

        // Otherwise, traverse the given
        // array []arr
        else {

            for (let i = 1; i < K; ++i) {
                if (V[i] > 0) {

                    // Decrement arr[i]
                    V[i]--;

                    let nleft = i <= left ? left - i : K + left - i;

                    // Update the maximum
                    // number of groups
                    q = Math.max(q, dfs(V, nleft, K));

                    // Increment arr[i] by 1
                    V[i]++;
                }
            }
        }

        // Memoize the result and
        // return it
        if(memo.has(key))
            memo[key] = q;
        else
            memo[key] = q;

        return q;
    }

    // Function to find the maximum
    // number of groups that will
    // receive fresh donuts
    function maxGroups(K, arr)
    {

        // Stores count of remainder by K
        let V = new Array(K);
        V.fill(0);

        // Traverse the array []arr
        for(let x = 0; x < arr.length; x++)
            V[arr[x] % K]++;

        // Hashmap to memoize the results
        memo = new Map();

        // Store the maximum number
        // of groups
        let ans = V[0] + dfs(V, 0, K);

        // Return the answer
        return ans;
    }

    let arr = [ 1, 2, 3, 4, 5, 6 ];
      let K = 3;
      document.write(maxGroups(K, arr));

    // This code is contributed by rameshtravel07.
</script>

Output:

4

Time Complexity: O(N + K²)
Auxiliary Space: O(K)