To solve Partition to K Equal Sum Subsets, first calculate the target sum for each subset. Use recursive backtracking with memoization or bitmasking to explore valid subset assignments. State transition dynamic programming ensures efficient pruning, quickly identifying infeasible paths and confirming possible equal-sum partitions.
Problem Statement
Given an integer array nums and an integer k, determine whether it is possible to divide the array into k non-empty subsets such that the sum of elements in each subset is equal. Each element must belong to exactly one subset, and subsets can be arranged in any order.
For example, nums = [4,3,2,3,5,2,1] and k = 4 can be partitioned into subsets (5), (1,4), (2,3), (2,3) each summing to 5. If no such division exists, return false. The array length is limited to 16, and elements are positive integers, often repeated.
Examples
Example 1
Input: nums = [4,3,2,3,5,2,1], k = 4
Output: true
It is possible to divide it into 4 subsets (5), (1, 4), (2,3), (2,3) with equal sums.
Example 2
Input: nums = [1,2,3,4], k = 3
Output: false
Example details omitted.
Constraints
- 1 <= k <= nums.length <= 16
- 1 <= nums[i] <= 104
- The frequency of each element is in the range [1, 4].
Solution Approach
Calculate target subset sum
Sum all elements in nums and divide by k to get the required subset sum. If total sum is not divisible by k, immediately return false as no equal-sum partition is possible.
Backtracking with memoization or bitmask
Use a recursive function to assign elements to subsets, marking used elements. Track current subset sums and backtrack when adding a number exceeds the target. Memoize states using bitmasking to avoid recomputation and reduce exponential complexity.
Prune invalid paths with DP state transitions
Implement state transition dynamic programming to prune search space: skip numbers already used or subsets that exceed the target. This optimization exploits overlapping subproblems and ensures efficient identification of feasible subset assignments.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity is O(k * 2^n) in the worst case using bitmask DP, since each subset assignment can produce 2^n states. Space complexity is O(2^n) for memoization. Backtracking alone may approach O(k^n) without pruning, but DP reduces repeated computation.
What Interviewers Usually Probe
- Candidate recognizes need to calculate target sum and checks divisibility early.
- Candidate uses bitmask or memoization to avoid recomputation during subset assignments.
- Candidate applies pruning to avoid exploring invalid paths exceeding the target sum.
Common Pitfalls or Variants
Common pitfalls
- Not checking total sum divisibility by k before recursion, leading to wasted computation.
- Failing to track used elements correctly, causing duplicate element usage across subsets.
- Skipping memoization or DP optimization, resulting in exponential runtime that times out for n > 12.
Follow-up variants
- Partition into 2 equal sum subsets only (simpler case with k=2).
- Allow negative numbers, requiring sum balancing and careful DP states.
- Count the number of distinct k-partitions with equal sum instead of returning boolean.
How GhostInterview Helps
- GhostInterview guides you to quickly set up the subset sum target and validate divisibility before expensive recursion.
- It provides efficient recursive backtracking templates integrated with memoization or bitmask DP for this exact problem.
- GhostInterview highlights pruning strategies tied to state transition dynamic programming, reducing exploration of impossible paths automatically.
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 pattern used in Partition to K Equal Sum Subsets?
State transition dynamic programming is the main pattern, combined with recursive backtracking to explore subset assignments efficiently.
Can this problem be solved without DP?
Yes, simple backtracking works for small arrays, but it becomes exponential and impractical for arrays close to length 16.
How does bitmasking help in this problem?
Bitmasking efficiently encodes used elements, allowing memoization of subset states and avoiding recomputation during recursion.
What is the significance of checking total sum divisibility by k?
If the total sum is not divisible by k, no equal-sum partition is possible, so early termination saves unnecessary computation.
Are repeated numbers handled differently?
No special handling is required, but tracking used elements carefully ensures each number is counted once per subset assignment.
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