algoadvance

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.

Clarifying Questions:

1. Are the elements of nums always integers?
   • Yes.
2. Can nums contain negative numbers?
   • No, based on the problem constraints, nums contains only non-negative numbers.
3. Is there any constraint on the size of nums?
   • Yes, the size can be up to 20.
4. What is the maximum absolute value of target?
   • The maximum absolute value of target can be up to 1000.

Strategy:

We need to determine how many ways we can use + and - operators in front of the numbers in the nums array such that the resultant expression evaluates to target. The strategy I’ll use to solve this problem is Depth-First Search (DFS) in combination with memoization to avoid recomputation of results.

1. Define a DFS function:
   • This will recursively try both adding and subtracting the current number.
2. Memoization:
   • Use a dictionary to store and reuse intermediate results (tuples representing current index and current cumulative sum).

Code:

def findTargetSumWays(nums, target):
    memo = {}
    
    def dfs(i, total):
        if i == len(nums):
            return 1 if total == target else 0
        if (i, total) in memo:
            return memo[(i, total)]
        
        # Calculate result by adding and subtracting the current number
        add = dfs(i + 1, total + nums[i])
        subtract = dfs(i + 1, total - nums[i])
        
        memo[(i, total)] = add + subtract
        return memo[(i, total)]
    
    return dfs(0, 0)

Time Complexity:

• Time Complexity:
  • O(n * s) where n is the length of nums and s is the sum of elements in nums.
  • This is because each subproblem is defined by an index and a sum total, leading to at most n * 2s subproblems.
• Space Complexity:
  • O(n * s) for the memo dictionary and the call stack in worst case scenario.

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