Problem Statement
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.
Examples
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
Explanation: There is 1 way to assign symbols to make the sum of nums be target 1.
+1 = 1
Constraints
- 1 <= nums.length <= 20
- 0 <= nums[i] <= 1000
- 0 <= sum(nums[i]) <= 1000
- -1000 <= target <= 1000
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming or recursion with memoization.
- We can transform this problem into a subset sum problem by observing that we're essentially partitioning the array into two subsets: one with '+' signs and one with '-' signs.
- If we denote the sum of the '+' subset as P and the sum of the '-' subset as N, then we have P + N = sum(nums) and P - N = target.
- Solving these equations, we get P = (sum(nums) + target) / 2, which means we need to find the number of subsets with sum P.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
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.
Examples
Example 1:
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:
There is 1 way to assign symbols to make the sum of nums be target 1. +1 = 1
Problem Breakdown
Constraints:
- 1 <= nums.length <= 20
- 0 <= nums[i] <= 1000
- 0 <= sum(nums[i]) <= 1000
- -1000 <= target <= 1000
Key Insights:
This problem can be solved using dynamic programming or recursion with memoization.
We can transform this problem into a subset sum problem by observing that we're essentially partitioning the array into two subsets: one with '+' signs and one with '-' signs.
If we denote the sum of the '+' subset as P and the sum of the '-' subset as N, then we have P + N = sum(nums) and P - N = target.
Solving these equations, we get P = (sum(nums) + target) / 2, which means we need to find the number of subsets with sum P.
Real-World Application
This problem has several practical applications:
Financial Planning
Finding different ways to allocate resources to achieve a specific financial goal.
Resource Allocation
Determining the number of ways to distribute resources to meet a target requirement.
Signal Processing
Analyzing different combinations of signals to achieve a desired output.