Target Sum

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

Key Insights

• Each number in the array can be either added or subtracted, leading to a binary decision for each number.
• The problem can be framed as finding subsets of numbers that can sum to a specific value derived from the target.
• The total sum of all numbers helps in determining how to partition the numbers into two groups (positive and negative).
• Dynamic programming or backtracking can be utilized to efficiently count the number of ways to achieve the target sum.

Space and Time Complexity

Time Complexity: O(2^N) for backtracking, O(N * target) for dynamic programming
Space Complexity: O(N) for the recursion stack in backtracking, O(target) for the DP array in dynamic programming

Solution

The problem can be solved using a dynamic programming approach. We can define a DP array where each index represents the number of ways to achieve a particular sum. The idea is to iterate through each number in nums and update the DP array based on whether we are adding or subtracting the number.

1. Calculate the total sum of the array.
2. Use the relationship between the total sum and the target to determine the subset sum that needs to be achieved.
3. Use a dynamic programming approach to count the number of ways to reach this subset sum.

Code Solutions