Target Sum

Problem

https://leetcode.com/problems/target-sum/

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

Pattern

Array, Dynamic Programming, Backtracking

Approaches

Memoization

Code

def findTargetSumWays(nums: list[int], target: int) -> int:
    memo = {}

    def target_sum_ways(i, t):
        if (i, t) in memo:
            return memo[(i, t)]

        if i == len(nums):
            memo[(i, t)] = 1 if t == 0 else 0
            return 1 if t == 0 else 0

        neg_ways = target_sum_ways(i + 1, t - nums[i])
        pos_ways = target_sum_ways(i + 1, t + nums[i])
        memo[(i + 1, t - nums[i])] = neg_ways
        memo[(i + 1, t + nums[i])] = pos_ways
        return neg_ways + pos_ways

    return target_sum_ways(0, target)

Test

>>> from target_sum__memoization import findTargetSumWays
>>> findTargetSumWays([1, 1, 1, 1, 1], 3)
5
>>> findTargetSumWays([1], 1)
1

target_sum__memoization.findTargetSumWays(nums: list[int], target: int) → int