Given a set of n integers, divide the set in two subsets of n/2 sizes each such that the absolute difference of the sum of two subsets is as minimum as possible. If n is even, then sizes of two subsets must be strictly n/2 and if n is odd, then size of one subset must be (n-1)/2 and size of other subset must be (n+1)/2. For example, let given set be {3, 4, 5, -3, 100, 1, 89, 54, 23, 20}, the size of set is 10. Output for this set should be {4, 100, 1, 23, 20} and {3, 5, -3, 89, 54}. Both output subsets are of size 5 and sum of elements in both subsets is same (148 and 148). Let us consider another example where n is odd. Let given set be {23, 45, -34, 12, 0, 98, -99, 4, 189, -1, 4}. The output subsets should be {45, -34, 12, 98, -1} and {23, 0, -99, 4, 189, 4}. The sums of elements in two subsets are 120 and 121 respectively.
- The following solution tries every possible subset of half size. If one subset of half size is formed, the remaining elements form the other subset. We initialize current set as empty and one by one build it.
- There are two possibilities for every element, either it is part of current set, or it is part of the remaining elements (other subset). We consider both possibilities for every element. When the size of current set becomes n/2, we check whether this solutions is better than the best solution available so far. If it is, then we update the best solution.
Following is the implementation for Tug of War problem. It prints the required arrays.
| Algorithm | Time Complexity | Space Complexity |
|---|---|---|
| Tug of War | O(2^n) | O(n) |
#include <bits/stdc++.h>
using namespace std;
// function that tries every possible solution by calling itself recursively
void TOWUtil(int* arr, int n, bool* curr_elements, int no_of_selected_elements,
bool* soln, int* min_diff, int sum, int curr_sum, int curr_position)
{
// checks whether the it is going out of bound
if (curr_position == n)
return;
// checks that the numbers of elements left are not less than the
// number of elements required to form the solution
if ((n/2 - no_of_selected_elements) > (n - curr_position))
return;
// consider the cases when current element is not included in the solution
TOWUtil(arr, n, curr_elements, no_of_selected_elements,
soln, min_diff, sum, curr_sum, curr_position+1);
// add the current element to the solution
no_of_selected_elements++;
curr_sum = curr_sum + arr[curr_position];
curr_elements[curr_position] = true;
// checks if a solution is formed
if (no_of_selected_elements == n/2)
{
// checks if the solution formed is better than the best solution so far
if (abs(sum/2 - curr_sum) < *min_diff)
{
*min_diff = abs(sum/2 - curr_sum);
for (int i = 0; i<n; i++)
soln[i] = curr_elements[i];
}
}
else
{
// consider the cases where current element is included in the solution
TOWUtil(arr, n, curr_elements, no_of_selected_elements, soln,
min_diff, sum, curr_sum, curr_position+1);
}
// removes current element before returning to the caller of this function
curr_elements[curr_position] = false;
}
// main function that generate an arr
void tugOfWar(int *arr, int n)
{
// the boolean array that contains the inclusion and exclusion of an element
// in current set. The number excluded automatically form the other set
bool* curr_elements = new bool[n];
// The inclusion/exclusion array for final solution
bool* soln = new bool[n];
int min_diff = INT_MAX;
int sum = 0;
for (int i=0; i<n; i++)
{
sum += arr[i];
curr_elements[i] = soln[i] = false;
}
// Find the solution using recursive function TOWUtil()
TOWUtil(arr, n, curr_elements, 0, soln, &min_diff, sum, 0, 0);
// Print the solution
cout << "The first subset is: ";
for (int i=0; i<n; i++)
{
if (soln[i] == true)
cout << arr[i] << " ";
}
cout << "\nThe second subset is: ";
for (int i=0; i<n; i++)
{
if (soln[i] == false)
cout << arr[i] << " ";
}
}
// Driver program to test above functions
int main()
{
int arr[] = {23, 45, -34, 12, 0, 98, -99, 4, 189, -1, 4};
int n = sizeof(arr)/sizeof(arr[0]);
tugOfWar(arr, n);
return 0;
}// C# program for Tug of war
using System;
class GFG
{
public int min_diff;
// function that tries every possible solution
// by calling itself recursively
void TOWUtil(int []arr, int n, Boolean []curr_elements,
int no_of_selected_elements, Boolean []soln,
int sum, int curr_sum, int curr_position)
{
// checks whether the it is going out of bound
if (curr_position == n)
return;
// checks that the numbers of elements left
// are not less than the number of elements
// required to form the solution
if ((n / 2 - no_of_selected_elements) >
(n - curr_position))
return;
// consider the cases when current element
// is not included in the solution
TOWUtil(arr, n, curr_elements,
no_of_selected_elements, soln, sum,
curr_sum, curr_position + 1);
// add the current element to the solution
no_of_selected_elements++;
curr_sum = curr_sum + arr[curr_position];
curr_elements[curr_position] = true;
// checks if a solution is formed
if (no_of_selected_elements == n / 2)
{
// checks if the solution formed is
// better than the best solution so
// far
if (Math.Abs(sum / 2 - curr_sum) <
min_diff)
{
min_diff = Math.Abs(sum / 2 -
curr_sum);
for (int i = 0; i < n; i++)
soln[i] = curr_elements[i];
}
}
else
{
// consider the cases where current
// element is included in the
// solution
TOWUtil(arr, n, curr_elements,
no_of_selected_elements,
soln, sum, curr_sum,
curr_position + 1);
}
// removes current element before
// returning to the caller of this
// function
curr_elements[curr_position] = false;
}
// main function that generate an arr
void tugOfWar(int []arr)
{
int n = arr.Length;
// the boolean array that contains the
// inclusion and exclusion of an element
// in current set. The number excluded
// automatically form the other set
Boolean[] curr_elements = new Boolean[n];
// The inclusion/exclusion array for
// final solution
Boolean[] soln = new Boolean[n];
min_diff = int.MaxValue;
int sum = 0;
for (int i = 0; i < n; i++)
{
sum += arr[i];
curr_elements[i] = soln[i] = false;
}
// Find the solution using recursive
// function TOWUtil()
TOWUtil(arr, n, curr_elements, 0,
soln, sum, 0, 0);
// Print the solution
Console.Write("The first subset is: ");
for (int i = 0; i < n; i++)
{
if (soln[i] == true)
Console.Write(arr[i] + " ");
}
Console.Write("\nThe second subset is: ");
for (int i = 0; i < n; i++)
{
if (soln[i] == false)
Console.Write(arr[i] + " ");
}
}
// Driver Code
public static void Main (String[] args)
{
int []arr = {23, 45, -34, 12, 0, 98,
-99, 4, 189, -1, 4};
GFG a = new GFG();
a.tugOfWar(arr);
}
}
// Java program for Tug of war
import java.util.*;
import java.lang.*;
import java.io.*;
class TugOfWar
{
public int min_diff;
// function that tries every possible solution
// by calling itself recursively
void TOWUtil(int arr[], int n, boolean curr_elements[],
int no_of_selected_elements, boolean soln[],
int sum, int curr_sum, int curr_position)
{
// checks whether the it is going out of bound
if (curr_position == n)
return;
// checks that the numbers of elements left
// are not less than the number of elements
// required to form the solution
if ((n / 2 - no_of_selected_elements) >
(n - curr_position))
return;
// consider the cases when current element
// is not included in the solution
TOWUtil(arr, n, curr_elements,
no_of_selected_elements, soln, sum,
curr_sum, curr_position+1);
// add the current element to the solution
no_of_selected_elements++;
curr_sum = curr_sum + arr[curr_position];
curr_elements[curr_position] = true;
// checks if a solution is formed
if (no_of_selected_elements == n / 2)
{
// checks if the solution formed is
// better than the best solution so
// far
if (Math.abs(sum / 2 - curr_sum) <
min_diff)
{
min_diff = Math.abs(sum / 2 -
curr_sum);
for (int i = 0; i < n; i++)
soln[i] = curr_elements[i];
}
}
else
{
// consider the cases where current
// element is included in the
// solution
TOWUtil(arr, n, curr_elements,
no_of_selected_elements,
soln, sum, curr_sum,
curr_position + 1);
}
// removes current element before
// returning to the caller of this
// function
curr_elements[curr_position] = false;
}
// main function that generate an arr
void tugOfWar(int arr[])
{
int n = arr.length;
// the boolean array that contains the
// inclusion and exclusion of an element
// in current set. The number excluded
// automatically form the other set
boolean[] curr_elements = new boolean[n];
// The inclusion/exclusion array for
// final solution
boolean[] soln = new boolean[n];
min_diff = Integer.MAX_VALUE;
int sum = 0;
for (int i = 0; i < n; i++)
{
sum += arr[i];
curr_elements[i] = soln[i] = false;
}
// Find the solution using recursive
// function TOWUtil()
TOWUtil(arr, n, curr_elements, 0,
soln, sum, 0, 0);
// Print the solution
System.out.print("The first subset is: ");
for (int i = 0; i < n; i++)
{
if (soln[i] == true)
System.out.print(arr[i] + " ");
}
System.out.print("\nThe second subset is: ");
for (int i = 0; i < n; i++)
{
if (soln[i] == false)
System.out.print(arr[i] + " ");
}
}
// Driver program to test above functions
public static void main (String[] args)
{
int arr[] = {23, 45, -34, 12, 0, 98,
-99, 4, 189, -1, 4};
TugOfWar a = new TugOfWar();
a.tugOfWar(arr);
}
}
# Python3 program for above approach
# function that tries every possible
# solution by calling itself recursively
def TOWUtil(arr, n, curr_elements, no_of_selected_elements,
soln, min_diff, Sum, curr_sum, curr_position):
# checks whether the it is going
# out of bound
if (curr_position == n):
return
# checks that the numbers of elements
# left are not less than the number of
# elements required to form the solution
if ((int(n / 2) - no_of_selected_elements) >
(n - curr_position)):
return
# consider the cases when current element
# is not included in the solution
TOWUtil(arr, n, curr_elements, no_of_selected_elements,
soln, min_diff, Sum, curr_sum, curr_position + 1)
# add the current element to the solution
no_of_selected_elements += 1
curr_sum = curr_sum + arr[curr_position]
curr_elements[curr_position] = True
# checks if a solution is formed
if (no_of_selected_elements == int(n / 2)):
# checks if the solution formed is better
# than the best solution so far
if (abs(int(Sum / 2) - curr_sum) < min_diff[0]):
min_diff[0] = abs(int(Sum / 2) - curr_sum)
for i in range(n):
soln[i] = curr_elements[i]
else:
# consider the cases where current
# element is included in the solution
TOWUtil(arr, n, curr_elements, no_of_selected_elements,
soln, min_diff, Sum, curr_sum, curr_position + 1)
# removes current element before returning
# to the caller of this function
curr_elements[curr_position] = False
# main function that generate an arr
def tugOfWar(arr, n):
# the boolean array that contains the
# inclusion and exclusion of an element
# in current set. The number excluded
# automatically form the other set
curr_elements = [None] * n
# The inclusion/exclusion array
# for final solution
soln = [None] * n
min_diff = [999999999999]
Sum = 0
for i in range(n):
Sum += arr[i]
curr_elements[i] = soln[i] = False
# Find the solution using recursive
# function TOWUtil()
TOWUtil(arr, n, curr_elements, 0,
soln, min_diff, Sum, 0, 0)
# Print the solution
print("The first subset is: ")
for i in range(n):
if (soln[i] == True):
print(arr[i], end = " ")
print()
print("The second subset is: ")
for i in range(n):
if (soln[i] == False):
print(arr[i], end = " ")
# Driver Code
if __name__ == '__main__':
arr = [23, 45, -34, 12, 0, 98,
-99, 4, 189, -1, 4]
n = len(arr)
tugOfWar(arr, n)
The second subset is: "); for (var i = 0; i < n; i++) { if (soln[i] == false) document.write(arr[i] + " "); } } // Driver program to test above functions var arr = [ 23, 45, -34, 12, 0, 98, -99, 4, 189, -1, 4 ]; tugOfWar(arr); </script>
The first subset is: 45 -34 12 98 -1
The second subset is: 23 0 -99 4 189 4