BiruLyu
6/24/2017 - 10:47 PM

## 494. Target Sum(DFS).java

``````/*The recursive solution is very slow, because its runtime is exponential

The original problem statement is equivalent to:
Find a subset of nums that need to be positive, and the rest of them negative, such that the sum is equal to target

Let P be the positive subset and N be the negative subset
For example:
Given nums = [1, 2, 3, 4, 5] and target = 3 then one possible solution is +1-2+3-4+5 = 3
Here positive subset is P = [1, 3, 5] and negative subset is N = [2, 4]

Then let's see how this can be converted to a subset sum problem:

sum(P) - sum(N) = target
sum(P) + sum(N) + sum(P) - sum(N) = target + sum(P) + sum(N)
2 * sum(P) = target + sum(nums)
So the original problem has been converted to a subset sum problem as follows:
Find a subset P of nums such that sum(P) = (target + sum(nums)) / 2

Note that the above formula has proved that target + sum(nums) must be even
We can use that fact to quickly identify inputs that do not have a solution (Thanks to @BrunoDeNadaiSarnaglia for the suggestion)
For detailed explanation on how to solve subset sum problem, you may refer to Partition Equal Subset Sum
*/
public class Solution {
public int findTargetSumWays(int[] nums, int S) {
if (nums == null || nums.length < 1) return 0;
int sum = 0;
for (int num : nums) sum += num;
return (sum < S || (sum + S) % 2 != 0) ? 0 : helper(nums, (sum + S) / 2);
}

public int helper(int[] nums, int target) {
int[] dp = new int[target + 1];
dp[0] = 1;
for (int i = 0; i < nums.length; i++) {
for (int j = target; j >= nums[i]; j--) {
dp[j] += dp[j - nums[i]];
}
}
return dp[target];
}
}``````
``````public class Solution {
public int findTargetSumWays(int[] nums, int S) {
int[] res = new int[] {0};
helper(nums, S, 0, res);
return res[0];
}

public void helper(int[] nums, int S, int start, int[] count) {
if (start == nums.length) {
if (S == 0) {
count[0]++;
}
return;
}
helper(nums, S - nums[start], start + 1, count);
helper(nums, S + nums[start], start + 1, count);
}
}``````