494 Target Sum - Medium

Problem:

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

Problem Analysis:

• use dynamic programming and memorization.
• from the first element, recursively increment the index and backtrack to look for suitable combination
• store intermediate solution in a dictionary

Solutions:

class Solution(object):
    def findTargetSumWays(self, nums, target):
        """
        :type nums: List[int]
        :type target: int
        :rtype: int
        """
        # memorization
        dp = {} # (index, total) : # of ways

        def backtrack(i, total):
            # base case
            if i == len(nums):
                return 1 if total == target else 0
            if (i, total) in dp:
                # take from dp
                return dp[(i, total)]
            
            dp[(i, total)] = (backtrack(i+1, total + nums[i]) +
            backtrack(i + 1, total - nums[i]))

            return dp[(i, total)]

        return backtrack(0,0)

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