This problem is best approached using state transition dynamic programming, sorting the input array first to simplify divisibility checks. You build up subsets incrementally, tracking the largest valid divisible subset at each step. GhostInterview guides through the subset construction while explaining failure points like missing intermediate divisible pairs.
Problem Statement
Given a set of distinct positive integers nums, identify the largest subset such that for every pair (a, b) in this subset, either a divides b or b divides a. If multiple valid subsets exist, any can be returned.
You are given examples like nums = [1,2,3] where [1,2] is a valid output and nums = [1,2,4,8] where [1,2,4,8] is a valid output. Your task is to implement an efficient algorithm that constructs this largest divisible subset while handling arrays up to length 1000 with unique positive integers.
Examples
Example 1
Input: nums = [1,2,3]
Output: [1,2]
[1,3] is also accepted.
Example 2
Input: nums = [1,2,4,8]
Output: [1,2,4,8]
Example details omitted.
Constraints
- 1 <= nums.length <= 1000
- 1 <= nums[i] <= 2 * 109
- All the integers in nums are unique.
Solution Approach
Sort and Initialize DP Arrays
Start by sorting the array to ensure that larger numbers are always divisible by smaller ones where possible. Create two arrays: dp[i] to store the size of the largest divisible subset ending at i, and prev[i] to track the previous index in that subset.
Build Subsets Using State Transitions
For each element nums[i], iterate through all previous elements nums[j] where j < i and nums[i] % nums[j] == 0. Update dp[i] = max(dp[i], dp[j] + 1) and set prev[i] = j if a larger subset is found. This captures the state transition dynamic programming pattern for divisible subsets.
Reconstruct the Largest Subset
After populating dp and prev arrays, identify the index of the maximum dp value. Trace back through prev to reconstruct the largest divisible subset. Return the reconstructed subset, which satisfies all divisibility conditions.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity is O(n^2) due to the nested iteration for each element to check divisibility against prior elements. Space complexity is O(n) for the dp and prev arrays storing subset lengths and backtracking information.
What Interviewers Usually Probe
- Asks about efficient subset building beyond brute force.
- Wants you to recognize the dynamic programming state transition pattern.
- May probe alternative reconstruction methods and edge cases.
Common Pitfalls or Variants
Common pitfalls
- Forgetting to sort nums can break the divisibility chain.
- Incorrectly updating dp without checking all previous divisible elements.
- Failing to reconstruct the subset using prev pointers after computing dp.
Follow-up variants
- Return all largest divisible subsets if multiple exist.
- Handle negative integers or zero with adjusted divisibility rules.
- Optimize for space using only a single array with backtracking reconstruction.
How GhostInterview Helps
- Highlights exactly where state transitions occur and which dp values to update.
- Provides step-by-step guidance for reconstructing the subset correctly.
- Flags common failure modes like skipping divisible pairs or misordering the array.
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 Largest Divisible Subset?
The problem uses state transition dynamic programming to build up subsets based on previous valid divisible elements.
Do I need to sort the array first?
Yes, sorting ensures that every element considered for extension can only divide or be divided by smaller previous elements, simplifying DP transitions.
Can multiple correct outputs exist?
Yes, if there are multiple largest subsets satisfying divisibility, any one of them can be returned.
What is the time complexity of this approach?
The time complexity is O(n^2) because for each element, you may need to check all prior elements for divisibility.
How does GhostInterview handle subset reconstruction?
GhostInterview guides through tracing the prev array from the maximum dp index to reconstruct the largest divisible subset accurately.
Need direct help with Largest Divisible Subset instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Largest Divisible Subset from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Find the largest number divisible by three by selecting and ordering digits optimally using state transition dynamic programming techniques.
Open problem page#354 Russian Doll EnvelopesRussian Doll Envelopes is a dynamic programming problem that involves finding the longest chain of envelopes that can be nested inside one another.
Open problem page#435 Non-overlapping IntervalsDetermine the minimum number of intervals to remove from a list to ensure no intervals overlap using dynamic programming and greedy insights.
Open problem page