# 494. Target Sum

## Description

You are given an integer array nums and an integer target.

You want to build an expression out of nums by adding one of the symbols '+' and '-' before each integer in nums and then concatenate all the integers.

• For example, if nums = [2, 1], you can add a '+' before 2 and a '-' before 1 and concatenate them to build the expression "+2-1".

Return the number of different expressions that you can build, which evaluates to target.

Example 1:

Input: nums = [1,1,1,1,1], target = 3
Output: 5
Explanation: There are 5 ways to assign symbols to make the sum of nums be target 3.
-1 + 1 + 1 + 1 + 1 = 3
+1 - 1 + 1 + 1 + 1 = 3
+1 + 1 - 1 + 1 + 1 = 3
+1 + 1 + 1 - 1 + 1 = 3
+1 + 1 + 1 + 1 - 1 = 3


Example 2:

Input: nums = [1], target = 1
Output: 1


Constraints:

• 1 <= nums.length <= 20
• 0 <= nums[i] <= 1000
• 0 <= sum(nums[i]) <= 1000
• -1000 <= target <= 1000

## Solutions

Dynamic programming.

It is similar to the 0-1 Knapsack problem, except that the index may appear negative, which requires special handling.

• class Solution {
public int findTargetSumWays(int[] nums, int target) {
int s = 0;
for (int v : nums) {
s += v;
}
if (s < target || (s - target) % 2 != 0) {
return 0;
}
int n = (s - target) / 2;
int[] dp = new int[n + 1];
dp[0] = 1;
for (int v : nums) {
for (int j = n; j >= v; --j) {
dp[j] += dp[j - v];
}
}
return dp[n];
}
}

• class Solution {
public:
int findTargetSumWays(vector<int>& nums, int target) {
int s = accumulate(nums.begin(), nums.end(), 0);
if (s < target || (s - target) % 2 != 0) return 0;
int n = (s - target) / 2;
vector<int> dp(n + 1);
dp[0] = 1;
for (int& v : nums)
for (int j = n; j >= v; --j)
dp[j] += dp[j - v];
return dp[n];
}
};

• class Solution:
def findTargetSumWays(self, nums: List[int], target: int) -> int:
s = sum(nums)
if s < target or (s - target) % 2 != 0:
return 0
n = (s - target) // 2
dp = [0] * (n + 1)
dp[0] = 1
for v in nums:
for j in range(n, v - 1, -1):
dp[j] += dp[j - v]
return dp[-1]


• func findTargetSumWays(nums []int, target int) int {
s := 0
for _, v := range nums {
s += v
}
if s < target || (s-target)%2 != 0 {
return 0
}
n := (s - target) / 2
dp := make([]int, n+1)
dp[0] = 1
for _, v := range nums {
for j := n; j >= v; j-- {
dp[j] += dp[j-v]
}
}
return dp[n]
}

• /**
* @param {number[]} nums
* @param {number} target
* @return {number}
*/
var findTargetSumWays = function (nums, target) {
let s = 0;
for (let v of nums) {
s += v;
}
if (s < target || (s - target) % 2 != 0) {
return 0;
}
const m = nums.length;
const n = (s - target) / 2;
let dp = new Array(n + 1).fill(0);
dp[0] = 1;
for (let i = 1; i <= m; ++i) {
for (let j = n; j >= nums[i - 1]; --j) {
dp[j] += dp[j - nums[i - 1]];
}
}
return dp[n];
};


• impl Solution {
pub fn find_target_sum_ways(nums: Vec<i32>, target: i32) -> i32 {
let mut sum = 0;
for e in &nums {
sum += *e;
}

// -x + (sum - x) = target <-> -2 * x + sum = target <-> 2 * x = sum - target
if sum < target || (sum - target) % 2 != 0 {
// There is no way to get any expression in this case
return 0;
}
let n = nums.len();
let m = (sum - target) / 2;

let mut dp: Vec<Vec<i32>> = vec![vec![0; m as usize + 1]; n + 1];

// Initialize the dp vector
dp[0][0] = 1;

// Begin the actual dp phase
for i in 1..=n {
for j in 0..=m as usize {
// nums[i - 1] is not included
dp[i][j] = dp[i - 1][j];
if nums[i - 1] <= (j as i32) {
// nums[i - 1] is included
dp[i][j] += dp[i - 1][j - (nums[i - 1] as usize)];
}
}
}

dp[n][m as usize]
}
}


• function findTargetSumWays(nums: number[], target: number): number {
const s = nums.reduce((a, b) => a + b, 0);
if (s < target || (s - target) % 2) {
return 0;
}
const [m, n] = [nums.length, ((s - target) / 2) | 0];
const f: number[][] = Array.from({ length: m + 1 }, () => Array(n + 1).fill(0));
f[0][0] = 1;
for (let i = 1; i <= m; i++) {
for (let j = 0; j <= n; j++) {
f[i][j] = f[i - 1][j];
if (j >= nums[i - 1]) {
f[i][j] += f[i - 1][j - nums[i - 1]];
}
}
}
return f[m][n];
}