Target Sum

Problem

You are given a list of non-negative integers, a1, a2, ..., an, and a target, S. Now you have 2 symbols + and -. For each integer, you should choose one from + and - as its new symbol.

Find out how many ways to assign symbols to make sum of integers equal to target S.

Example 1:
Input: nums is [1, 1, 1, 1, 1], S is 3. 
Output: 5
Explanation: 

-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

There are 5 ways to assign symbols to make the sum of nums be target 3.

Note:

• The length of the given array is positive and will not exceed 20.
• The sum of elements in the given array will not exceed 1000.
• Your output answer is guaranteed to be fitted in a 32-bit integer.

Solution

Convert to Subproblem: Find Subset with Sum n

public class Solution {
    public int findTargetSumWays(int[] nums, int S) {
        int sum = 0;
        for (int num : nums) sum += num;
        if (sum < S || (sum + S) % 2 == 1) return 0;
        return subset(nums, (sum + S) / 2);
    }

    private int subset(int[] nums, int target) {
        int[] dp = new int[target + 1];
        dp[0] = 1;
        for (int num : nums) {
            for (int i = target; i >= num; i--) {
                dp[i] += dp[i - num];
            }
        }
        return dp[target];
    }
}

Analysis

We try to convert this problem to finding subset with sum target
Let's say we have sum(+) with all positive number and sum(-) with all negative number
sum(+) - sum(-) = S => sum(+) - sum(-) + sum(+) + sum(-) = S + sum(+) + sum(-)
2 * sum(+) = sum + S => sum(+) = (sum + S) / 2
Hence our target is (sum + S) / 2
To get how many ways, we need to find how many subsets we can construct with target
We use the idea in Partition Equal Subset Sum