# Target Sum (DP – 21)

Problem Description:

We are given an array ‘ARR’ of size ‘N’ and a number ‘Target’. Our task is to build an expression from the given array where we can place a ‘+’ or ‘-’ sign in front of an integer. We want to place a sign in front of every integer of the array and get our required target. We need to count the number of ways in which we can achieve our required target.

Examples
```Example:

```
Practice:

Disclaimer: Don’t jump directly to the solution, try it out yourself first.

Memorization approach Tabulation approach Space Optimization
Memorization Approach
Algorithm / Intuition

The first approach that comes to our mind is to generate all subsequences and try both options of placing ‘-’ and ‘+’ signs and count the expression if it evaluates the answer.
This surely will give us the answer but can we try something familiar to the previous problems we have solved?

The answer is yes! We can use the concept we studied in the following article Count Partitions with Given Difference (DP – 18).

The following insights will help us to understand intuition better:

• If we think deeper, we can say that the given ‘target’ can be expressed as addition of two integers (say S1 and S2).

S1 + S2 = target   – (i)

• Now, from where will this S1 and S2 come?  If we are given the array as [a,b,c,d,e], we want to place ‘+’ or ‘-’ signs in front of every array element and then add it. One example is :

+a-b-c+d+e which can be written as (+a+d+e) + (-b-c).

Therefore, we can say that S1=(+a+d+e) and S2=(-b-c) for this example.

•  If we calculate the total sum of elements of the array (say totSum), we can can say that,

S1 = totSum – S2      – (ii)

• Now solving for equations (i) and (iii), we can say that

S2 = (totSum – target)/2    – (iv)

Therefore this question is modified to “Count Number of subsets with sum (totSum – target)/2 ”. This is exactly what we had discussed in the article  Count Subsets with Sum K.

Edge Cases:

The following edge cases need to be handled:

• As the array elements are positive integers including zero, we don’t want to find the case when S2 is negative or we can say that totSum is lesser than D, therefore if totSum<target, we simply return 0.
• S2 can’t be a fraction, as all elements are integers, therefore if totSum – target is odd, we can return 0.

From here on we will discuss the approach to “Count Subsets with Sum K” with the required modifications. Moreover, as the array elements can also contain 0, we will handle it as discussed in part-1 of this article.

Steps to form the recursive solution:

We will first form the recursive solution by the three points mentioned in Dynamic Programming Introduction

Step 1: Express the problem in terms of indexes.

The array will have an index but there is one more parameter “target”. We are given the initial problem to find whether there exists in the whole array a subsequence whose sum is equal to the target.

So, we can say that initially, we need to find(n-1, target) which means that we are counting the number of subsequences in the array from index 0 to n-1, whose sum is equal to the target. Similarly, we can generalize it for any index ind as follows:

Base Cases:

• If target == 0, it means that we have already found the subsequence from the previous steps, so we can return 1.
• If ind==0, it means we are at the first element, so we need to return arr[ind]==target. If the element is equal to the target we return 1 else we return 0.

Step 2: Try out all possible choices at a given index.

We need to generate all the subsequences. We will use the pick/non-pick technique as discussed in this video “Recursion on Subsequences”.

We have two choices:

• Exclude the current element in the subsequence: We first try to find a subsequence without considering the current index element. For this, we will make a recursive call to f(ind-1,target).
• Include the current element in the subsequence: We will try to find a subsequence by considering the current index as element as part of subsequence. As we have included arr[ind], the updated target which we need to find in the rest if the array will be target – arr[ind]. Therefore, we will call f(ind-1,target-arr[ind]).

Note: We will consider the current element in the subsequence only when the current element is less than or equal to the target.

Step 3:  Return sum of taken and notTaken

As we have to return the total count of subsets with the target sum, we will return the sum of taken and notTaken from our recursive call.

The final pseudocode after steps 1, 2, and 3:

Steps to memoize a recursive solution:

If we draw the recursion tree, we will see that there are overlapping subproblems. In order to convert a recursive solution the following steps will be taken:

1. Create a dp array of size [n][k+1]. The size of the input array is ‘n’, so the index will always lie between ‘0’ and ‘n-1’. The target can take any value between ‘0’ and ‘k’. Therefore we take the dp array as dp[n][k+1]
2. We initialize the dp array to -1.
3. Whenever we want to find the answer of particular parameters (say f(ind,target)), we first check whether the answer is already calculated using the dp array(i.e dp[ind][target]!= -1 ). If yes, simply return the value from the dp array.
4. If not, then we are finding the answer for the given value for the first time, we will use the recursive relation as usual but before returning from the function, we will set dp[ind][target] to the solution we get.
Code
``````
#include <bits/stdc++.h>
using namespace std;

// Function to count partitions of the array into subsets with a given target sum
int countPartitionsUtil(int ind, int target, vector<int>& arr, vector<vector<int>>& dp) {
// Base cases
if (ind == 0) {
if (target == 0 && arr[0] == 0)
return 2; // Two ways to partition: include or exclude the element
if (target == 0 || target == arr[0])
return 1; // One way to partition: include or exclude the element
return 0; // No way to partition
}

// If the result for this index and target sum is already calculated, return it
if (dp[ind][target] != -1)
return dp[ind][target];

// Calculate the number of ways without taking the current element
int notTaken = countPartitionsUtil(ind - 1, target, arr, dp);

// Calculate the number of ways by taking the current element
int taken = 0;
if (arr[ind] <= target)
taken = countPartitionsUtil(ind - 1, target - arr[ind], arr, dp);

// Store the sum of ways in the DP array and return it
return dp[ind][target] = (notTaken + taken);
}

// Function to count the number of ways to achieve the target sum
int targetSum(int n, int target, vector<int>& arr) {
int totSum = 0;
for (int i = 0; i < arr.size(); i++) {
totSum += arr[i];
}

// Checking for edge cases
if (totSum - target < 0)
return 0; // Not possible to achieve the target sum
if ((totSum - target) % 2 == 1)
return 0; // The difference between the total sum and target sum must be even

int s2 = (totSum - target) / 2; // Calculate the required sum for each subset

vector<vector<int>> dp(n, vector<int>(s2 + 1, -1)); // Initialize DP table
return countPartitionsUtil(n - 1, s2, arr, dp); // Call the helper function
}

int main() {
vector<int> arr = {1, 2, 3, 1};
int target = 3;

int n = arr.size();
cout << "The number of ways found is " << targetSum(n, target, arr) << endl;

return 0; // Return 0 to indicate successful program execution
}
```
```
``````
import java.util.*;

class TUF {
// Function to count partitions using dynamic programming
static int countPartitionsUtil(int ind, int target, int[] arr, int[][] dp) {
// Base case: If we have reached the first element
if (ind == 0) {
// Check if the target is 0 and the first element is also 0
if (target == 0 && arr[0] == 0)
return 2;
// Check if the target is equal to the first element or 0
if (target == 0 || target == arr[0])
return 1;
return 0;
}

// If the result for this subproblem has already been calculated, return it
if (dp[ind][target] != -1)
return dp[ind][target];

// Calculate the number of ways without taking the current element
int notTaken = countPartitionsUtil(ind - 1, target, arr, dp);

// Initialize the number of ways taking the current element as 0
int taken = 0;

// If the current element is less than or equal to the target, calculate 'taken'
if (arr[ind] <= target)
taken = countPartitionsUtil(ind - 1, target - arr[ind], arr, dp);

// Store the result in the dp array and return it
return dp[ind][target] = (notTaken + taken);
}

// Function to find the number of ways to achieve the target sum
static int targetSum(int n, int target, int[] arr) {
int totSum = 0;

// Calculate the total sum of elements in the array
for (int i = 0; i < arr.length; i++) {
totSum += arr[i];
}

// Checking for edge cases
if (totSum - target < 0)
return 0;
if ((totSum - target) % 2 == 1)
return 0;

// Calculate the second sum based on the total sum and the target
int s2 = (totSum - target) / 2;

// Create a 2D array to store results of subproblems
int dp[][] = new int[n][s2 + 1];

// Initialize the dp array with -1 to indicate that subproblems are not solved yet
for (int row[] : dp)
Arrays.fill(row, -1);

// Call the countPartitionsUtil function to calculate the number of ways
return countPartitionsUtil(n - 1, s2, arr, dp);
}

public static void main(String args[]) {
int arr[] = { 1, 2, 3, 1 };
int target = 3;

int n = arr.length;

// Call the targetSum function and print the result
System.out.println("The number of ways found is " + targetSum(n, target, arr));
}
}
```
```
``````
def countPartitionsUtil(ind, target, arr, dp):
# Base case: If we have reached the first element in the array.
if ind == 0:
# Check if the target is zero and the first element is also zero, in which case there are two possibilities.
if target == 0 and arr[0] == 0:
return 2
# If the target is equal to the first element, there is one possibility.
if target == 0 or target == arr[0]:
return 1
# Otherwise, there is no valid partition.
return 0

# If the result for this state is already calculated, return it.
if dp[ind][target] != -1:
return dp[ind][target]

# Calculate the number of possibilities when the current element is not taken.
notTaken = countPartitionsUtil(ind - 1, target, arr, dp)

# Initialize a variable for the number of possibilities when the current element is taken.
taken = 0
if arr[ind] <= target:
taken = countPartitionsUtil(ind - 1, target - arr[ind], arr, dp)

# Store the total number of possibilities in the DP table.
dp[ind][target] = notTaken + taken
return dp[ind][target]

def targetSum(n, target, arr):
totSum = 0
for i in range(len(arr)):
totSum += arr[i]

# Checking for edge cases
if totSum - target < 0:
return 0
if (totSum - target) % 2 == 1:
return 0

s2 = (totSum - target) // 2

dp = [[-1 for j in range(s2 + 1)] for i in range(n)]
return countPartitionsUtil(n - 1, s2, arr, dp)

def main():
arr = [1, 2, 3, 1]
target = 3
n = len(arr)
print("The number of ways found is", targetSum(n, target, arr))

if __name__ == '__main__':
main()
```
```
``````
function countPartitionsUtil(ind, target, arr, dp) {
// Base cases
if (ind === 0) {
if (target === 0 && arr[0] === 0)
return 2;
if (target === 0 || target === arr[0])
return 1;
return 0;
}

// If the result for this combination of 'ind' and 'target' has already been calculated, return it
if (dp[ind][target] !== -1)
return dp[ind][target];

// Recursive cases
const notTaken = countPartitionsUtil(ind - 1, target, arr, dp);

let taken = 0;
if (arr[ind] <= target)
taken = countPartitionsUtil(ind - 1, target - arr[ind], arr, dp);

// Store and return the result
return dp[ind][target] = notTaken + taken;
}

// Define a function to find the number of ways to reach the 'target' sum
function targetSum(n, target, arr) {
// Calculate the total sum of the array elements
let totSum = 0;
for (let i = 0; i < arr.length; i++) {
totSum += arr[i];
}

// Checking for edge cases
if (totSum - target < 0) return 0;
if ((totSum - target) % 2 === 1) return 0;

// Calculate the second subset sum
let s2 = (totSum - target) / 2;

// Create a 2D array to store dynamic programming results, initialized with -1
const dp = Array.from({ length: n }, () => Array(s2 + 1).fill(-1));

// Call the countPartitionsUtil function to calculate the result
return countPartitionsUtil(n - 1, s2, arr, dp);
}

// Main function
function main() {
const arr = [1, 2, 3, 1];
const target = 3;
const n = arr.length;

// Call the targetSum function and print the result
console.log("The number of ways found is " + targetSum(n, target, arr));
}

// Call the main function to start the program
main();
```
```

Output: The number of ways found is 2

Complexity Analysis

Time Complexity: O(N*K)

Reason: There are N*K states therefore at max ‘N*K’ new problems will be solved.

Space Complexity: O(N*K) + O(N)

Reason: We are using a recursion stack space(O(N)) and a 2D array ( O(N*K)).

Tabulation Approach
Algorithm / Intuition

To convert the memoization approach to a tabulation one, create a dp array with the same size as done in memoization. We can initialize it as 0.

First, we need to initialize the base conditions of the recursive solution.

• If target == 0, ind can take any value from 0 to n-1, therefore we need to set the value of the first column as 1.

• The first row dp[0][] indicates that only the first element of the array is considered, therefore for the target value equal to arr[0], only cell with that target will be true, so explicitly set dp[0][arr[0]] =1, (dp[0][arr[0]] means that we are considering the first element of the array with the target equal to the first element itself). Please note that it can happen that arr[0]>target, so we first check it: if(arr[0]<=target) then set dp[0][arr[0]] = 1.

• After that, we will set our nested for loops to traverse the dp array, and following the logic discussed in the recursive approach, we will set the value of each cell. Instead of recursive calls, we will use the dp array itself.
• At last, we will return dp[n-1][k] as our answer.
Code
``````
#include <bits/stdc++.h>
using namespace std;

const int mod = (int)1e9 + 7;

// Function to count the number of ways to achieve the target sum
int findWays(vector<int> &num, int tar) {
int n = num.size();

vector<vector<int>> dp(n, vector<int>(tar + 1, 0));

if (num[0] == 0)
dp[0][0] = 2;  // 2 cases - pick and not pick
else
dp[0][0] = 1;  // 1 case - not pick

if (num[0] != 0 && num[0] <= tar)
dp[0][num[0]] = 1;  // 1 case - pick

for (int ind = 1; ind < n; ind++) {
for (int target = 0; target <= tar; target++) {

int notTaken = dp[ind - 1][target];

int taken = 0;
if (num[ind] <= target)
taken = dp[ind - 1][target - num[ind]];

dp[ind][target] = (notTaken + taken) % mod;
}
}
return dp[n - 1][tar];
}

// Function to calculate the number of ways to achieve the target sum
int targetSum(int n, int target, vector<int>& arr) {
int totSum = 0;
for (int i = 0; i < n; i++) {
totSum += arr[i];
}

// Checking for edge cases
if (totSum - target < 0 || (totSum - target) % 2 != 0)
return 0;  // Not possible to achieve the target sum

return findWays(arr, (totSum - target) / 2);
}

int main() {
vector<int> arr = {1, 2, 3, 1};
int target = 3;

int n = arr.size();
cout << "The number of ways found is " << targetSum(n, target, arr) << endl;

return 0;  // Return 0 to indicate successful program execution
}
```
```
``````
import java.util.*;

class TUF {
static int mod = (int) (Math.pow(10, 9) + 7);

// Function to find the number of ways to achieve the target sum
static int findWays(int[] num, int tar) {
int n = num.length;

// Create a 2D array to store results of subproblems
int[][] dp = new int[n][tar + 1];

// Initialize the dp array for the first element of the array
if (num[0] == 0)
dp[0][0] = 2; // 2 cases - pick and not pick
else
dp[0][0] = 1; // 1 case - not pick

if (num[0] != 0 && num[0] <= tar)
dp[0][num[0]] = 1; // 1 case - pick

// Fill the dp array using dynamic programming
for (int ind = 1; ind < n; ind++) {
for (int target = 0; target <= tar; target++) {
int notTaken = dp[ind - 1][target];

int taken = 0;
if (num[ind] <= target)
taken = dp[ind - 1][target - num[ind]];

dp[ind][target] = (notTaken + taken) % mod;
}
}

return dp[n - 1][tar];
}

// Function to calculate the number of ways to achieve the target sum
static int targetSum(int n, int target, int[] arr) {
int totSum = 0;

// Calculate the total sum of elements in the array
for (int i = 0; i < n; i++) {
totSum += arr[i];
}

// Checking for edge cases
if (totSum - target < 0 || (totSum - target) % 2 == 1)
return 0;

return findWays(arr, (totSum - target) / 2);
}

public static void main(String args[]) {
int arr[] = { 1, 2, 3, 1 };
int target = 3;

int n = arr.length;

// Call the targetSum function and print the result
System.out.println("The number of ways found is " + targetSum(n, target, arr));
}
}
```
```
``````
mod = int(1e9 + 7)

def findWays(num, tar):
n = len(num)
dp = [[0 for i in range(tar + 1)] for j in range(n)]

if num[0] == 0:
dp[0][0] = 2  # 2 cases - pick and not pick
else:
dp[0][0] = 1  # 1 case - not pick

if num[0] != 0 and num[0] <= tar:
dp[0][num[0]] = 1  # 1 case - pick

for ind in range(1, n):
for target in range(tar + 1):
notTaken = dp[ind - 1][target]

taken = 0
if num[ind] <= target:
taken = dp[ind - 1][target - num[ind]]

dp[ind][target] = (notTaken + taken) % mod

return dp[n - 1][tar]

def targetSum(n, target, arr):
totSum = 0
for i in range(n):
totSum += arr[i]

# Checking for edge cases
if (totSum - target) < 0 or ((totSum - target) % 2):
return 0

return findWays(arr, (totSum - target) // 2)

def main():
arr = [1, 2, 3, 1]
target = 3
n = len(arr)

print("The number of ways found is", targetSum(n, target, arr))

if __name__ == "__main__":
main()
```
```
``````
const mod = 1e9 + 7;

// Define a function to find the number of ways to achieve the 'tar' target sum
function findWays(num, tar) {
const n = num.length;

// Create a 2D array 'dp' to store dynamic programming results, initialized with 0
const dp = Array.from({ length: n }, () => Array(tar + 1).fill(0));

if (num[0] === 0) dp[0][0] = 2;  // 2 cases - pick and not pick
else dp[0][0] = 1;  // 1 case - not pick

if (num[0] !== 0 && num[0] <= tar) dp[0][num[0]] = 1;  // 1 case - pick

for (let ind = 1; ind < n; ind++) {
for (let target = 0; target <= tar; target++) {

const notTaken = dp[ind - 1][target];

let taken = 0;
if (num[ind] <= target)
taken = dp[ind - 1][target - num[ind]];

dp[ind][target] = (notTaken + taken) % mod;
}
}

return dp[n - 1][tar];
}

// Define a function to find the number of ways to achieve the 'target' sum using elements from 'arr'
function targetSum(n, target, arr) {
let totSum = 0;
for (let i = 0; i < n; i++) {
totSum += arr[i];
}

// Checking for edge cases
if (totSum - target < 0 || (totSum - target) % 2 !== 0) return 0;

// Calculate the second subset sum
const s2 = (totSum - target) / 2;

// Call the findWays function to calculate the result
return findWays(arr, s2);
}

// Main function
function main() {
const arr = [1, 2, 3, 1];
const target = 3;
const n = arr.length;

// Call the targetSum function and print the result
console.log("The number of ways found is " + targetSum(n, target, arr));
}

// Call the main function to start the program
main();
```
```

Output: The number of ways found is 2

Complexity Analysis

Time Complexity: O(N*K)

Reason: There are two nested loops

Space Complexity: O(N*K)

Reason: We are using an external array of size ‘N*K’. Stack Space is eliminated.

Space Optimization Approach
Algorithm / Intuition

If we closely look the relation,

dp[ind][target] =  dp[ind-1][target] + dp[ind-1][target-arr[ind]]

We see that to calculate a value of a cell of the dp array, we need only the previous row values (say prev). So, we don’t need to store an entire array. Hence we can space optimize it.

Note: Whenever we create a new row ( say cur), we need to explicitly set its first element as true according to our base condition.

Code
``````
#include <bits/stdc++.h>
using namespace std;

const int mod = (int)1e9 + 7;

// Function to count the number of ways to achieve the target sum
int findWays(vector<int> &num, int tar) {
int n = num.size();

vector<int> prev(tar + 1, 0);

if (num[0] == 0)
prev[0] = 2;  // 2 cases - pick and not pick
else
prev[0] = 1;  // 1 case - not pick

if (num[0] != 0 && num[0] <= tar)
prev[num[0]] = 1;  // 1 case - pick

for (int ind = 1; ind < n; ind++) {
vector<int> cur(tar + 1, 0);
for (int target = 0; target <= tar; target++) {
int notTaken = prev[target];

int taken = 0;
if (num[ind] <= target)
taken = prev[target - num[ind]];

cur[target] = (notTaken + taken) % mod;
}
prev = cur;
}
return prev[tar];
}

// Function to calculate the number of ways to achieve the target sum
int targetSum(int n, int target, vector<int>& arr) {
int totSum = 0;
for (int i = 0; i < n; i++) {
totSum += arr[i];
}

// Checking for edge cases
if (totSum - target < 0 || (totSum - target) % 2 != 0)
return 0;  // Not possible to achieve the target sum

return findWays(arr, (totSum - target) / 2);
}

int main() {
vector<int> arr = {1, 2, 3, 1};
int n = arr.size();
int target = 3;

cout << "The number of subsets found is " << targetSum(n, target, arr) << endl;

return 0;  // Return 0 to indicate successful program execution
}
```
```
``````
import java.util.*;

class TUF {
static int mod = (int) (Math.pow(10, 9) + 7);

// Function to find the number of ways to achieve the target sum
static int findWays(int[] num, int tar) {
int n = num.length;

// Create an array to store results of subproblems for the current element
int prev[] = new int[tar + 1];

// Initialize the prev array for the first element of the array
if (num[0] == 0)
prev[0] = 2; // 2 cases - pick and not pick
else
prev[0] = 1; // 1 case - not pick

if (num[0] != 0 && num[0] <= tar)
prev[num[0]] = 1; // 1 case - pick

// Fill the prev array using dynamic programming
for (int ind = 1; ind < n; ind++) {
int cur[] = new int[tar + 1];
for (int target = 0; target <= tar; target++) {
int notTaken = prev[target];

int taken = 0;
if (num[ind] <= target)
taken = prev[target - num[ind]];

cur[target] = (notTaken + taken) % mod;
}
prev = cur;
}

return prev[tar];
}

// Function to calculate the number of ways to achieve the target sum
static int targetSum(int n, int target, int[] arr) {
int totSum = 0;

// Calculate the total sum of elements in the array
for (int i = 0; i < n; i++) {
totSum += arr[i];
}

// Checking for edge cases
if (totSum - target < 0 || (totSum - target) % 2 == 1)
return 0;

return findWays(arr, (totSum - target) / 2);
}

public static void main(String args[]) {
int arr[] = { 1, 2, 3, 1 };
int n = arr.length;
int target = 3;

// Call the targetSum function and print the result
System.out.println("The number of subsets found is " + targetSum(n, target, arr));
}
}
```
```
``````
mod = int(1e9 + 7)

# Function to find the number of ways to partition an array into two subsets
# with a given target difference using dynamic programming
def findWays(num, tar):
n = len(num)

# Initialize a list 'prev' to store results for the previous element
prev = [0 for i in range(tar + 1)]

# Initialize 'prev' based on the first element of 'num'
if num[0] == 0:
prev[0] = 2  # Two cases - pick and not pick
else:
prev[0] = 1  # One case - not pick

if num[0] != 0 and num[0] <= tar:
prev[num[0]] = 1  # One case - pick

for ind in range(1, n):
# Initialize a list 'cur' to store results for the current element
cur = [0 for i in range(tar + 1)]
for target in range(tar + 1):
notTaken = prev[target]

taken = 0
if num[ind] <= target:
taken = prev[target - num[ind]]

# Store the result in 'cur' with modulo operation
cur[target] = (notTaken + taken) % mod
prev = cur

# Return the result for the target sum
return prev[tar]

# Function to calculate the number of ways to achieve a target sum
def targetSum(n, target, arr):
totSum = 0
for i in range(n):
totSum += arr[i]

# Checking for edge cases
if (totSum - target) < 0 or ((totSum - target) % 2):
return 0

# Calculate and return the number of ways using 'findWays' function
return findWays(arr, (totSum - target) // 2)

def main():
arr = [1, 2, 3, 1]
target = 3
n = len(arr)

# Print the number of ways found
print("The number of ways found is", targetSum(n, target, arr))

if __name__ == "__main__":
main()
```
```
``````
const mod = 1e9 + 7;

// Define a function to find the number of ways to achieve the 'tar' target sum
function findWays(num, tar) {
const n = num.length;

// Initialize an array 'prev' to store dynamic programming results, initialized with 0
let prev = new Array(tar + 1).fill(0);

if (num[0] === 0) prev[0] = 2;  // 2 cases - pick and not pick
else prev[0] = 1;  // 1 case - not pick

if (num[0] !== 0 && num[0] <= tar) prev[num[0]] = 1;  // 1 case - pick

for (let ind = 1; ind < n; ind++) {
// Initialize an array 'cur' for the current iteration
let cur = new Array(tar + 1).fill(0);
for (let target = 0; target <= tar; target++) {
const notTaken = prev[target];

let taken = 0;
if (num[ind] <= target)
taken = prev[target - num[ind]];

cur[target] = (notTaken + taken) % mod;
}
// Update 'prev' to be the same as 'cur' for the next iteration
prev = [...cur];
}

return prev[tar];
}

// Define a function to find the number of ways to achieve the 'target' sum using elements from 'arr'
function targetSum(n, target, arr) {
let totSum = 0;
for (let i = 0; i < n; i++) {
totSum += arr[i];
}

// Checking for edge cases
if (totSum - target < 0 || (totSum - target) % 2 !== 0) return 0;

// Calculate the second subset sum
const s2 = (totSum - target) / 2;

// Call the findWays function to calculate the result
return findWays(arr, s2);
}

// Main function
function main() {
const arr = [1, 2, 3, 1];
const n = arr.length;
const target = 3;

// Call the targetSum function and print the result
console.log("The number of subsets found is " + targetSum(n, target, arr));
}

// Call the main function to start the program
main();
```
```

Output:The number of ways found is 2

Complexity Analysis

Time Complexity: O(N*K)

Reason: There are three nested loops

Space Complexity: O(K)

Reason: We are using an external array of size ‘K+1’ to store only one row.

Video Explanation