This problem asks you to select a subsequence from an integer array such that the sum is as close as possible to a given goal. The naive approach examines all $2^n$ subsequences, which is infeasible for larger arrays. GhostInterview guides you to apply state transition dynamic programming and a meet-in-the-middle strategy, efficiently computing sums while minimizing the absolute difference from the goal.
Problem Statement
You are given an integer array nums and an integer goal. Your task is to select any subsequence of nums so that the sum of the chosen elements is as close as possible to goal.
Return the smallest possible absolute difference between the sum of a subsequence and the target goal. For example, if the subsequence sum is sum, you must minimize abs(sum - goal). Arrays can contain negative numbers, and subsequences may be empty.
Examples
Example 1
Input: nums = [5,-7,3,5], goal = 6
Output: 0
Choose the whole array as a subsequence, with a sum of 6. This is equal to the goal, so the absolute difference is 0.
Example 2
Input: nums = [7,-9,15,-2], goal = -5
Output: 1
Choose the subsequence [7,-9,-2], with a sum of -4. The absolute difference is abs(-4 - (-5)) = abs(1) = 1, which is the minimum.
Example 3
Input: nums = [1,2,3], goal = -7
Output: 7
Example details omitted.
Constraints
- 1 <= nums.length <= 40
- -107 <= nums[i] <= 107
- -109 <= goal <= 109
Solution Approach
Meet-in-the-Middle Subsequence Sum
Split nums into two halves, generate all possible subsequence sums for each half using bitmasking, and store them in sorted arrays. This reduces the problem from 2^n to 2^(n/2) and prepares for efficient merging.
Binary Search to Minimize Difference
For each sum in the first half, use binary search on the second half's sorted sums to find the closest total sum to the goal. Track the minimum absolute difference encountered during this search.
State Transition Dynamic Programming Alternative
Use DP to record all reachable sums up to the length of the array. Iterate through nums, updating a set of achievable sums by adding each number to previous sums. Finally, compute the minimum absolute difference to the goal from the set.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The meet-in-the-middle approach reduces the time complexity to O(2^(n/2) log 2^(n/2)) = O(2^(n/2) * n) due to sorting and binary search. Space complexity is O(2^(n/2)) for storing subsequence sums. DP alternatives can have O(n * sum_range) time and space, which may exceed constraints for large numbers.
What Interviewers Usually Probe
- Checks if you understand subset sum state transitions and meet-in-the-middle optimization.
- Tests your ability to reduce exponential time via splitting and sorting sums.
- Looks for correct handling of negative numbers and empty subsequences.
Common Pitfalls or Variants
Common pitfalls
- Attempting a full 2^n enumeration without splitting causes timeouts for n > 20.
- Failing to sort one half's sums prevents efficient binary search and correct minimum difference calculation.
- Overlooking empty subsequences or negative numbers can result in incorrect absolute difference computation.
Follow-up variants
- Find the closest subsequence sum constrained to at most k elements.
- Compute closest subsequence sum where elements must be consecutive in nums.
- Determine closest subsequence product instead of sum, emphasizing DP state tracking.
How GhostInterview Helps
- Provides a step-by-step meet-in-the-middle implementation for subsequence sums, minimizing trial and error.
- Illustrates how to combine DP and binary search to track achievable sums efficiently.
- Highlights the exact failure modes and corner cases, such as empty subsequences and negative numbers.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the main challenge in Closest Subsequence Sum?
The challenge is efficiently finding a subsequence sum closest to goal without enumerating all 2^n possibilities.
Why use meet-in-the-middle for this problem?
Splitting the array halves reduces time complexity from O(2^n) to O(2^(n/2) log 2^(n/2)), making large inputs feasible.
Can dynamic programming alone solve Closest Subsequence Sum?
Yes, DP can track all reachable sums, but naive DP may exceed memory limits for large absolute values.
How do negative numbers affect the solution?
Negative numbers expand possible sums in both directions, so proper handling ensures the minimum absolute difference is correct.
Does GhostInterview explain state transition dynamic programming for this problem?
Yes, it guides you to implement DP and combine it with meet-in-the-middle and binary search to compute the minimal difference efficiently.
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