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# Partition an array of non-negative integers into two subsets such that average of both the subsets is equal

Given an array of size N. The task is to partition the given array into two subsets such that the average of all the elements in both subsets is equal. If no such partition exists print -1. Otherwise, print the partitions. If multiple solutions exist, print the solution where the length of the first subset is minimum. If there is still a tie then print the partitions where the first subset is lexicographically smallest.

Examples:

```Input : vec[] = {1, 7, 15, 29, 11, 9}
Output : [9, 15] [1, 7, 11, 29]
Explanation : Average of the both the subsets is 12

Input : vec[] = {1, 2, 3, 4, 5, 6}
Output : [1, 6] [2, 3, 4, 5].
Explanation : Another possible solution is [3, 4] [1, 2, 5, 6],
but print the  solution whose first subset is lexicographically
smallest.```

Observation :
If we directly compute the average of a certain subset and compare it with another subset’s average, due to precision issues with compilers, unexpected results will occur. For example, 5/3 = 1.66666.. and 166/100 = 1.66. Some compilers might treat them as same, whereas some others won’t.
Let the sum of two subsets under consideration be sub1 and sub2, and let their sizes be s1 and s2. If their averages are equal, sub1/s1 = sub2/s2 . Which means sub1*s2 = sub2*s1.
Also total sum of the above two subsets = sub1+sub2, and s2= total size – s1.
On simplifying the above, we get

(sub1/s1) = (sub1+sub2)/ (s1+s2) = (total sum) / (total size).

Now this problem reduces to the fact that if we can select a particular size
of subset whose sum is equal to the current subset’s sum, we are done.

Approach :
Let us define the function partition(ind, curr_sum, curr_size), which returns true if it is possible to construct subset using elements with index equals to ind and having size equals to curr_size and sum equals to curr_sum.
This recursive relation can be defined as:

partition(ind, curr_sum, curr_size) = partition(ind+1, curr_sum, curr_size) || partition(ind+1, curr_sum – val[ind], curr_size-1).

Two parts on the right side of the above equations represent whether we including the element at index ind or not.
This is a deviation from the classic subset sum problem, in which subproblems are being evaluated again and again. Therefore we memorize the subproblems and turn it into a Dynamic Programming solution.

## C++14

 `// C++ program to Partition an array of``// non-negative integers into two subsets``// such that average of both the subsets are equal``#include ``using` `namespace` `std;` `vector > > dp;``vector<``int``> res;``vector<``int``> original;``int` `total_size;` `// Function that returns true if it is possible to``// use elements with index = ind to construct a set of s``// ize = curr_size whose sum is curr_sum.``bool` `possible(``int` `index, ``int` `curr_sum, ``int` `curr_size)``{``    ``// base cases``    ``if` `(curr_size == 0)``        ``return` `(curr_sum == 0);``    ``if` `(index >= total_size)``        ``return` `false``;` `    ``// Which means curr_sum cant be found for curr_size``    ``if` `(dp[index][curr_sum][curr_size] == ``false``)``        ``return` `false``;` `    ``if` `(curr_sum >= original[index]) {``        ``res.push_back(original[index]);``        ``// Checks if taking this element at``        ``// index i leads to a solution``        ``if` `(possible(index + 1, curr_sum -``        ``original[index],``                     ``curr_size - 1))``            ``return` `true``;` `        ``res.pop_back();``    ``}``    ``// Checks if not taking this element at``    ``// index i leads to a solution``    ``if` `(possible(index + 1, curr_sum, curr_size))``        ``return` `true``;` `    ``// If no solution has been found``    ``return` `dp[index][curr_sum][curr_size] = ``false``;``}` `// Function to find two Partitions having equal average``vector > partition(vector<``int``>& Vec)``{``    ``// Sort the vector``    ``sort(Vec.begin(), Vec.end());``    ``original.clear();``    ``original = Vec;``    ``dp.clear();``    ``res.clear();` `    ``int` `total_sum = 0;``    ``total_size = Vec.size();` `    ``for` `(``int` `i = 0; i < total_size; ++i)``        ``total_sum += Vec[i];``    ``// building the memoization table``    ``dp.resize(original.size(), vector >``(total_sum + 1, vector<``bool``>(total_size, ``true``)));` `    ``for` `(``int` `i = 1; i < total_size; i++) {``        ``// Sum_of_Set1 has to be an integer``        ``if` `((total_sum * i) % total_size != 0)``            ``continue``;``        ``int` `Sum_of_Set1 = (total_sum * i) / total_size;` `        ``// We build our solution vector if its possible``        ``// to find subsets that match our criteria``        ``// using a recursive function``        ``if` `(possible(0, Sum_of_Set1, i)) {` `            ``// Find out the elements in Vec, not in``            ``// res and return the result.``            ``int` `ptr1 = 0, ptr2 = 0;``            ``vector<``int``> res1 = res;``            ``vector<``int``> res2;``            ``while` `(ptr1 < Vec.size() || ptr2 < res.size())``            ``{``                ``if` `(ptr2 < res.size() &&``                        ``res[ptr2] == Vec[ptr1])``                ``{``                    ``ptr1++;``                    ``ptr2++;``                    ``continue``;``                ``}``                ``res2.push_back(Vec[ptr1]);``                ``ptr1++;``            ``}` `            ``vector > ans;``            ``ans.push_back(res1);``            ``ans.push_back(res2);``            ``return` `ans;``        ``}``    ``}``    ``// If we havent found any such subset.``    ``vector > ans;``    ``return` `ans;``}` `// Function to print partitions``void` `Print_Partition(vector > sol)``{``    ``// Print two partitions``    ``for` `(``int` `i = 0; i < sol.size(); i++) {``        ``cout << ``"["``;``        ``for` `(``int` `j = 0; j < sol[i].size(); j++) {``            ``cout << sol[i][j];``            ``if` `(j != sol[i].size() - 1)``                ``cout << ``" "``;``        ``}``        ``cout << ``"] "``;``    ``}``}` `// Driver code``int` `main()``{``    ``vector<``int``> Vec = { 1, 7, 15, 29, 11, 9 };` `    ``vector > sol = partition(Vec);` `    ``// If partition possible``    ``if` `(sol.size())``        ``Print_Partition(sol);``    ``else``        ``cout << -1;` `    ``return` `0;``}`

## Java

 `// Java program to Partition an array of``// non-negative integers into two subsets``// such that average of both the subsets are equal``import` `java.io.*;``import` `java.util.*;` `class` `GFG``{` `    ``static` `boolean``[][][] dp;``    ``static` `Vector res = ``new` `Vector<>();``    ``static` `int``[] original;``    ``static` `int` `total_size;` `    ``// Function that returns true if it is possible to``    ``// use elements with index = ind to construct a set of s``    ``// ize = curr_size whose sum is curr_sum.``    ``static` `boolean` `possible(``int` `index, ``int` `curr_sum,``                                        ``int` `curr_size)``    ``{` `        ``// base cases``        ``if` `(curr_size == ``0``)``            ``return` `(curr_sum == ``0``);``        ``if` `(index >= total_size)``            ``return` `false``;` `        ``// Which means curr_sum cant be found for curr_size``        ``if` `(dp[index][curr_sum][curr_size] == ``false``)``            ``return` `false``;` `        ``if` `(curr_sum >= original[index])``        ``{``            ``res.add(original[index]);` `            ``// Checks if taking this element at``            ``// index i leads to a solution``            ``if` `(possible(index + ``1``, curr_sum - original[index],``                                                ``curr_size - ``1``))``                ``return` `true``;` `            ``res.remove(res.size() - ``1``);``        ``}` `        ``// Checks if not taking this element at``        ``// index i leads to a solution``        ``if` `(possible(index + ``1``, curr_sum, curr_size))``            ``return` `true``;` `        ``// If no solution has been found``        ``return` `dp[index][curr_sum][curr_size] = ``false``;``    ``}` `    ``// Function to find two Partitions having equal average``    ``static` `Vector> partition(``int``[] Vec)``    ``{` `        ``// Sort the vector``        ``Arrays.sort(Vec);``        ``original = Vec;``        ``res.clear();` `        ``int` `total_sum = ``0``;``        ``total_size = Vec.length;` `        ``for` `(``int` `i = ``0``; i < total_size; ++i)``            ``total_sum += Vec[i];` `        ``// building the memoization table``        ``dp = ``new` `boolean``[original.length][total_sum + ``1``][total_size];` `        ``for` `(``int` `i = ``0``; i < original.length; i++)``            ``for` `(``int` `j = ``0``; j < total_sum + ``1``; j++)``                ``for` `(``int` `k = ``0``; k < total_size; k++)``                    ``dp[i][j][k] = ``true``;` `        ``for` `(``int` `i = ``1``; i < total_size; i++)``        ``{` `            ``// Sum_of_Set1 has to be an integer``            ``if` `((total_sum * i) % total_size != ``0``)``                ``continue``;``            ``int` `Sum_of_Set1 = (total_sum * i) / total_size;` `            ``// We build our solution vector if its possible``            ``// to find subsets that match our criteria``            ``// using a recursive function``            ``if` `(possible(``0``, Sum_of_Set1, i))``            ``{` `                ``// Find out the elements in Vec, not in``                ``// res and return the result.``                ``int` `ptr1 = ``0``, ptr2 = ``0``;``                ``Vector res1 = res;``                ``Vector res2 = ``new` `Vector<>();``                ``while` `(ptr1 < Vec.length || ptr2 < res.size())``                ``{``                    ``if` `(ptr2 < res.size() &&``                        ``res.elementAt(ptr2) == Vec[ptr1])``                    ``{``                        ``ptr1++;``                        ``ptr2++;``                        ``continue``;``                    ``}``                    ``res2.add(Vec[ptr1]);``                    ``ptr1++;``                ``}` `                ``Vector> ans = ``new` `Vector<>();``                ``ans.add(res1);``                ``ans.add(res2);``                ``return` `ans;``            ``}``        ``}` `        ``// If we havent found any such subset.``        ``Vector> ans = ``new` `Vector<>();``        ``return` `ans;``    ``}` `    ``// Function to print partitions``    ``static` `void` `Print_Partition(Vector> sol)``    ``{` `        ``// Print two partitions``        ``for` `(``int` `i = ``0``; i < sol.size(); i++)``        ``{``            ``System.out.print(``"["``);``            ``for` `(``int` `j = ``0``; j < sol.elementAt(i).size(); j++)``            ``{``                ``System.out.print(sol.elementAt(i).elementAt(j));``                ``if` `(j != sol.elementAt(i).size() - ``1``)``                    ``System.out.print(``" "``);``            ``}``            ``System.out.print(``"]"``);``        ``}``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int``[] Vec = { ``1``, ``7``, ``15``, ``29``, ``11``, ``9` `};``        ``Vector> sol = partition(Vec);` `        ``// If partition possible``        ``if` `(sol.size() > ``0``)``            ``Print_Partition(sol);``        ``else``            ``System.out.println(``"-1"``);``    ``}``}` `// This code is contributed by``// sanjeev2552`

## Python3

 `# Python3 program to partition an array of``# non-negative integers into two subsets``# such that average of both the subsets are equal``dp ``=` `[]``res ``=` `[]``original ``=` `[]``total_size ``=` `int``(``0``)` `# Function that returns true if it is possible``# to use elements with index = ind to construct``# a set of s ize = curr_size whose sum is curr_sum.``def` `possible(index, curr_sum, curr_size):``    ` `    ``index ``=` `int``(index)``    ``curr_sum ``=` `int``(curr_sum)``    ``curr_size ``=` `int``(curr_size)``    ` `    ``global` `dp, res``    ` `    ``# Base cases``    ``if` `curr_size ``=``=` `0``:``        ``return` `(curr_sum ``=``=` `0``)``    ``if` `index >``=` `total_size:``        ``return` `False``    ` `    ``# Which means curr_sum cant be``    ``# found for curr_size``    ``if` `dp[index][curr_sum][curr_size] ``=``=` `False``:``        ``return` `False``    ` `    ``if` `curr_sum >``=` `original[index]:``        ``res.append(original[index])``        ` `        ``# Checks if taking this element``        ``# at index i leads to a solution``        ``if` `possible(index ``+` `1``,``                    ``curr_sum ``-` `original[index],``                    ``curr_size ``-` `1``):``            ``return` `True``            ` `        ``res.pop()``    ` `    ``# Checks if not taking this element at``    ``# index i leads to a solution``    ``if` `possible(index ``+` `1``, curr_sum, curr_size):``        ``return` `True``    ` `    ``# If no solution has been found``    ``dp[index][curr_sum][curr_size] ``=` `False``    ``return` `False` `# Function to find two partitions``# having equal average``def` `partition(Vec):``    ` `    ``global` `dp, original, res, total_size``    ` `    ``# Sort the vector``    ``Vec.sort()``    ` `    ``if` `len``(original) > ``0``:``        ``original.clear()``    ` `    ``original ``=` `Vec``    ` `    ``if` `len``(dp) > ``0``:``        ``dp.clear()``    ``if` `len``(res) > ``0``:``        ``res.clear()` `    ``total_sum ``=` `0``    ``total_size ``=` `len``(Vec)` `    ``for` `i ``in` `range``(total_size):``        ``total_sum ``+``=` `Vec[i]``    ` `    ``# Building the memoization table``    ``dp ``=` `[[[``True` `for` `_ ``in` `range``(total_size)]``                 ``for` `_ ``in` `range``(total_sum ``+` `1``)]``                 ``for` `_ ``in` `range``(``len``(original))]``    ` `    ``for` `i ``in` `range``(``1``, total_size):``        ` `        ``# Sum_of_Set1 has to be an integer``        ``if` `(total_sum ``*` `i) ``%` `total_size !``=` `0``:``            ``continue``            ` `        ``Sum_of_Set1 ``=` `(total_sum ``*` `i) ``/` `total_size` `        ``# We build our solution vector if its possible``        ``# to find subsets that match our criteria``        ``# using a recursive function``        ``if` `possible(``0``, Sum_of_Set1, i):` `            ``# Find out the elements in Vec,``            ``# not in res and return the result.``            ``ptr1 ``=` `0``            ``ptr2 ``=` `0``            ``res1 ``=` `res``            ``res2 ``=` `[]``            ` `            ``while` `ptr1 < ``len``(Vec) ``or` `ptr2 < ``len``(res):``                ``if` `(ptr2 < ``len``(res) ``and``                    ``res[ptr2] ``=``=` `Vec[ptr1]):``                    ``ptr1 ``+``=` `1``                    ``ptr2 ``+``=` `1``                    ``continue``                    ` `                ``res2.append(Vec[ptr1])``                ``ptr1 ``+``=` `1` `            ``ans ``=` `[]``            ``ans.append(res1)``            ``ans.append(res2)``            ` `            ``return` `ans``    ` `    ``# If we havent found any such subset.``    ``ans ``=` `[]``    ``return` `ans` `# Driver code``Vec ``=` `[ ``1``, ``7``, ``15``, ``29``, ``11``, ``9` `]` `sol ``=` `partition(Vec)` `if` `len``(sol) > ``0``:``    ``print``(sol)``else``:``    ``print``(``"-1"``)``    ` `# This code is contributed by saishashank1`

## C#

 `// C# program to Partition an array of ``// non-negative integers into two subsets ``// such that average of both the subsets are equal ``using` `System;``using` `System.Collections; ` `class` `GFG{``  ` `static` `bool``[,,] dp;``static` `ArrayList res = ``new` `ArrayList();``static` `int``[] original;``static` `int` `total_size;` `// Function that returns true if it is possible to``// use elements with index = ind to construct a set of s``// ize = curr_size whose sum is curr_sum.``static` `bool` `possible(``int` `index, ``int` `curr_sum, ``                                ``int` `curr_size) ``{``    ` `    ``// base cases``    ``if` `(curr_size == 0)``        ``return` `(curr_sum == 0);``    ``if` `(index >= total_size)``        ``return` `false``;` `    ``// Which means curr_sum cant be``    ``// found for curr_size``    ``if` `(dp[index, curr_sum, curr_size] == ``false``)``        ``return` `false``;` `    ``if` `(curr_sum >= original[index])``    ``{``        ``res.Add(original[index]);` `        ``// Checks if taking this element at``        ``// index i leads to a solution``        ``if` `(possible(index + 1, curr_sum -``                     ``original[index], curr_size - 1))``            ``return` `true``;` `        ``res.Remove(res[res.Count - 1]);``    ``}` `    ``// Checks if not taking this element at``    ``// index i leads to a solution``    ``if` `(possible(index + 1, curr_sum, curr_size))``        ``return` `true``;``    ` `    ``dp[index, curr_sum, curr_size] = ``false``;``    ` `    ``// If no solution has been found``    ``return` `dp[index, curr_sum, curr_size];``}` `// Function to find two Partitions``// having equal average``static` `ArrayList partition(``int``[] Vec)``{``    ` `    ``// Sort the vector``    ``Array.Sort(Vec);``    ``original = Vec;``    ``res.Clear();` `    ``int` `total_sum = 0;``    ``total_size = Vec.Length;` `    ``for``(``int` `i = 0; i < total_size; ++i)``        ``total_sum += Vec[i];` `    ``// Building the memoization table``    ``dp = ``new` `bool``[original.Length,``                  ``total_sum + 1,``                  ``total_size];` `    ``for``(``int` `i = 0; i < original.Length; i++)``        ``for``(``int` `j = 0; j < total_sum + 1; j++)``            ``for``(``int` `k = 0; k < total_size; k++)``                ``dp[i, j, k] = ``true``;` `    ``for``(``int` `i = 1; i < total_size; i++) ``    ``{``        ` `        ``// Sum_of_Set1 has to be an integer``        ``if` `((total_sum * i) % total_size != 0)``            ``continue``;``            ` `        ``int` `Sum_of_Set1 = (total_sum * i) / total_size;` `        ``// We build our solution vector if its possible``        ``// to find subsets that match our criteria``        ``// using a recursive function``        ``if` `(possible(0, Sum_of_Set1, i)) ``        ``{``            ` `            ``// Find out the elements in Vec, not in``            ``// res and return the result.``            ``int` `ptr1 = 0, ptr2 = 0;``            ` `            ``ArrayList res1 = ``new` `ArrayList(res);``            ``ArrayList res2 = ``new` `ArrayList();``            ` `            ``while` `(ptr1 < Vec.Length || ptr2 < res.Count)``            ``{``                ``if` `(ptr2 < res.Count && ``                   ``(``int``)res[ptr2] == Vec[ptr1])``                ``{``                    ``ptr1++;``                    ``ptr2++;``                    ``continue``;``                ``}``                ``res2.Add(Vec[ptr1]);``                ``ptr1++;``            ``}` `            ``ArrayList ans = ``new` `ArrayList();``            ``ans.Add(res1);``            ``ans.Add(res2);``            ``return` `ans;``        ``}``    ``}` `    ``// If we havent found any such subset.``    ``ArrayList ans2 = ``new` `ArrayList();``    ``return` `ans2;``}` `// Function to print partitions``static` `void` `Print_Partition(ArrayList sol) ``{``    ` `    ``// Print two partitions``    ``for``(``int` `i = 0; i < sol.Count; i++) ``    ``{``        ``Console.Write(``"["``);``        ``for``(``int` `j = 0; j < ((ArrayList)sol[i]).Count; j++) ``        ``{``            ``Console.Write((``int``)((ArrayList)sol[i])[j]);``            ``if` `(j != ((ArrayList)sol[i]).Count - 1)``                ``Console.Write(``" "``);``        ``}``        ``Console.Write(``"] "``);``    ``}``}` `// Driver Code``public` `static` `void` `Main(``string``[] args)``{``    ``int``[] Vec = { 1, 7, 15, 29, 11, 9 };``    ``ArrayList sol = partition(Vec);` `    ``// If partition possible``    ``if` `(sol.Count > 0)``        ``Print_Partition(sol);``    ``else``        ``Console.Write(``"-1"``);``}``}` `// This code is contributed by rutvik_56`

## Javascript

 `// JS program to Partition an array of``// non-negative integers into two subsets``// such that average of both the subsets are equal`  `let dp = [];``let res = [];``let original = [];``let total_size;` `// Function that returns true if it is possible to``// use elements with index = ind to construct a set of s``// ize = curr_size whose sum is curr_sum.``function` `possible(index, curr_sum, curr_size)``{``    ``// base cases``    ``if` `(curr_size == 0)``        ``return` `(curr_sum == 0);``    ``if` `(index >= total_size)``        ``return` `false``;` `    ``// Which means curr_sum cant be found for curr_size``    ``if` `(dp[index][curr_sum][curr_size] == ``false``)``        ``return` `false``;` `    ``if` `(curr_sum >= original[index]) {``        ``res.push(original[index]);``        ``// Checks if taking this element at``        ``// index i leads to a solution``        ``if` `(possible(index + 1, curr_sum -``        ``original[index],``                     ``curr_size - 1))``            ``return` `true``;` `        ``res.pop();``    ``}``    ``// Checks if not taking this element at``    ``// index i leads to a solution``    ``if` `(possible(index + 1, curr_sum, curr_size))``        ``return` `true``;` `    ``// If no solution has been found``    ``dp[index][curr_sum][curr_size] = ``false``    ``return` `dp[index][curr_sum][curr_size];``}` `// Function to find two Partitions having equal average``function` `partition(Vec)``{``    ``// Sort the vector``    ``Vec.sort();``    ``original = [];``    ``original = Vec;``    ``dp = [];``    ``res = [];` `    ``let total_sum = 0;``    ``total_size = Vec.length;` `    ``for` `(``var` `i = 0; i < total_size; ++i)``        ``total_sum += Vec[i];``        ` `    ``// building the memoization table``    ``dp = ``new` `Array(original.length);``    ``for` `(``var` `i = 0; i < original.length; i++)``    ``{``        ``dp[i] = ``new` `Array(total_sum + 1);``        ``for` `(``var` `j = 0; j <= total_sum; j++)``        ``{``            ``dp[i][j] = ``new` `Array(total_size).fill(``true``);``        ``}``    ``}``    `  `    ``for` `(``var` `i = 1; i < total_size; i++) {``        ``// Sum_of_Set1 has to be an integer``        ``if` `((total_sum * i) % total_size != 0)``            ``continue``;``        ``var` `Sum_of_Set1 = (total_sum * i) / total_size;` `        ``// We build our solution vector if its possible``        ``// to find subsets that match our criteria``        ``// using a recursive function``        ``if` `(possible(0, Sum_of_Set1, i)) {` `            ``// Find out the elements in Vec, not in``            ``// res and return the result.``            ``var` `ptr1 = 0, ptr2 = 0;``            ``var` `res1 = res;``            ``var` `res2 = [];``            ``while` `(ptr1 < Vec.length || ptr2 < res.length)``            ``{``                ``if` `(ptr2 < res.length &&``                        ``res[ptr2] == Vec[ptr1])``                ``{``                    ``ptr1++;``                    ``ptr2++;``                    ``continue``;``                ``}``                ``res2.push(Vec[ptr1]);``                ``ptr1++;``            ``}` `            ``let ans = [];``            ``ans.push(res1);``            ``ans.push(res2);``            ``return` `ans;``        ``}``    ``}``    ``// If we havent found any such subset.``   ``return` `-1;``}`   `// Driver code``let Vec = [1, 7, 15, 29, 11, 9 ];` `let sol = partition(Vec);` `console.log(sol)`

Output:

`[9 15] [1 7 11 29]`

Time Complexity: O(n3)

Auxiliary Space: O(n3)

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