Start by sorting the input array to streamline scanning and use a hash table to track counts of numbers. Recursively explore subset combinations while skipping pairs that violate the absolute difference condition k. This approach ensures accurate counting of all non-empty beautiful subsets without unnecessary recomputation or missed edge cases.
Problem Statement
Given an array nums of positive integers and a positive integer k, define a subset of nums as beautiful if it does not contain any two integers whose absolute difference is exactly k. Your task is to count all non-empty beautiful subsets in nums.
Return an integer representing the total number of beautiful subsets. For example, if nums = [2,4,6] and k = 2, the beautiful subsets are [2], [4], [6], and [2,6], resulting in an output of 4.
Examples
Example 1
Input: nums = [2,4,6], k = 2
Output: 4
The beautiful subsets of the array nums are: [2], [4], [6], [2, 6]. It can be proved that there are only 4 beautiful subsets in the array [2,4,6].
Example 2
Input: nums = [1], k = 1
Output: 1
The beautiful subset of the array nums is [1]. It can be proved that there is only 1 beautiful subset in the array [1].
Constraints
- 1 <= nums.length <= 18
- 1 <= nums[i], k <= 1000
Solution Approach
Sort and Track Frequency
Sort nums to handle numbers in increasing order, then create a hash table to store the count of each number. This allows efficient checking for conflicts with previously selected numbers while scanning subsets.
Backtracking with Hash Lookup
Use a recursive backtracking function that attempts to include or exclude each number. Before including a number, check the hash table to see if adding it would create a pair with absolute difference k. Skip numbers that violate the beautiful subset condition.
Count Valid Subsets
Every time a valid subset is formed that obeys the k difference rule, increment a counter. Combine results from all recursive branches to return the total number of non-empty beautiful subsets.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(n \log n) |
| Space | O(n) |
Sorting the array takes O(n log n). Backtracking explores up to 2^n subsets, but the hash lookup prevents invalid branches early, keeping practical time complexity manageable for n <= 18. Space is O(n) for the hash table and recursion stack.
What Interviewers Usually Probe
- Look for use of sorting plus hash table to avoid O(n^2) pair checks.
- Check whether subsets are counted without missing any edge cases with difference k.
- Verify handling of small arrays and duplicates using the hash table.
Common Pitfalls or Variants
Common pitfalls
- Counting subsets without checking the absolute difference k for all pairs.
- Failing to handle duplicate numbers, leading to overcounting.
- Recursing without pruning invalid branches, causing unnecessary computation.
Follow-up variants
- Count beautiful subsets with at most one pair violating the k difference.
- Find all beautiful subsets where the difference must not be a multiple of k.
- Return the sum of elements in all beautiful subsets instead of the count.
How GhostInterview Helps
- Automatically structures the backtracking recursion with hash lookup to avoid manual bookkeeping mistakes.
- Highlights when absolute difference violations occur, preventing miscounting subsets.
- Provides step-by-step subset exploration visualization to ensure every non-empty combination is correctly considered.
Topic Pages
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- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
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FAQ
What exactly defines a beautiful subset in this problem?
A beautiful subset contains no two numbers whose absolute difference is exactly k. Any subset violating this is excluded from the count.
Why is sorting nums important for solving this problem?
Sorting allows sequential scanning and ensures that checking the hash table for conflicts is efficient, avoiding unnecessary recomputation for all pairs.
Can this approach handle duplicate numbers in nums?
Yes, the hash table tracks counts of each number, ensuring that duplicates are correctly included or skipped to maintain the beautiful subset rule.
What is the recommended pattern for this problem?
Array scanning combined with hash table lookup is the key pattern, allowing efficient subset generation while enforcing the absolute difference constraint.
How does GhostInterview prevent miscounting subsets?
It guides through each recursive branch, checks the hash table for absolute difference conflicts, and accumulates only valid non-empty beautiful subsets.
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