Partition Equal Subset Sum
Given a non-empty array `nums` containing only positive integers, find if the array can be partitioned into two subsets such that the sum of elements in both subsets is equal. This is a 0/1 Knapsack problem variant.
Why Interviewers Ask This
Stripe evaluates this question to assess a candidate's ability to transform a real-world partitioning challenge into a dynamic programming solution. They specifically look for the skill of recognizing NP-complete problems and optimizing them using memoization or space reduction, reflecting their need for engineers who write efficient, scalable code for high-volume financial transactions.
How to Answer This Question
1. Clarify constraints: Immediately check if the total sum is odd; if so, return false instantly as equal partitioning is impossible. This shows attention to edge cases.
2. Define the goal: Explain that finding two equal subsets is equivalent to finding a subset with a sum exactly equal to half the total sum, framing it as a variation of the 0/1 Knapsack problem.
3. Choose the algorithm: Propose Dynamic Programming. Briefly explain the state definition (can we reach sum 's' using first 'i' items?) and the transition logic (include current item vs. exclude it).
4. Optimize space: Discuss reducing the 2D DP table to a 1D boolean array to demonstrate awareness of memory efficiency, which is critical at Stripe.
5. Walk through an example: Trace the logic with a small array like [1, 5, 11, 5] to show how the target sum builds up iteratively, ensuring your thought process is transparent.
Key Points to Cover
- Recognizing that an odd total sum makes the answer immediately false
- Correctly mapping the partition problem to the subset sum variant of 0/1 Knapsack
- Explaining the state transition logic clearly without writing code immediately
- Demonstrating space optimization by reducing a 2D DP table to a 1D array
- Validating the solution with a concrete trace using a specific numerical example
Sample Answer
To solve the Partition Equal Subset Sum problem, my first step is to calculate the total sum of the input array. If this sum is odd, we can immediately return false because it is mathematically impossible to split an odd number into two equal integers. If the sum is even, our target becomes half of that total sum. The problem then transforms into a classic 0/1 Knapsack scenario: can we find a subset of numbers that sums exactly to this target?
I would approach this using Dynamic Programming. I'll define a boolean array where each index represents whether a specific sum is achievable. Initially, only index zero is true since a sum of zero is always possible with an empty set. We then iterate through each number in the array. For each number, we update our achievable sums by checking if adding the current number to any previously reachable sum lands us within our target range. To optimize space, instead of maintaining a full 2D matrix, I use a single 1D array updated backwards to prevent reusing the same element multiple times for the same subset.
For example, with nums = [1, 5, 11, 5], the total sum is 22, making the target 11. We start with {0}. Processing 1 gives us {0, 1}. Processing 5 adds 5 and 6, resulting in {0, 1, 5, 6}. Finally, processing 11 allows us to reach 11 directly from 0. Since our target is met, we return true. This approach runs in O(n * sum) time and O(sum) space, balancing readability with performance.
Common Mistakes to Avoid
- Failing to check for an odd total sum first, wasting time on unnecessary computation
- Confusing this with the Unbounded Knapsack problem and allowing elements to be reused multiple times
- Implementing a recursive solution without memoization, leading to exponential time complexity TLE
- Updating the 1D DP array forwards instead of backwards, causing incorrect results due to double counting
Practice This Question with AI
Answer this question orally or via text and get instant AI-powered feedback on your response quality, structure, and delivery.