The problem asks to count all squareful arrays from a given list of numbers. An array is squareful if the sum of every adjacent pair forms a perfect square. The problem requires a backtracking approach combined with hash lookup to efficiently count valid permutations.
Problem Statement
Given an integer array nums, return the number of distinct permutations of the array such that every adjacent pair of elements sums to a perfect square.
Two permutations are considered distinct if there exists an index where the elements are different. An array is considered squareful if for every pair of adjacent elements, their sum is a perfect square.
Examples
Example 1
Input: nums = [1,17,8]
Output: 2
[1,8,17] and [17,8,1] are the valid permutations.
Example 2
Input: nums = [2,2,2]
Output: 1
Example details omitted.
Constraints
- 1 <= nums.length <= 12
- 0 <= nums[i] <= 109
Solution Approach
Backtracking with Hash Set Lookup
This approach utilizes backtracking to generate all permutations of the array, while pruning invalid paths by checking if the sum of adjacent elements is a perfect square. A hash set ensures that duplicate numbers don't lead to redundant calculations.
Perfect Square Sum Precomputation
Before performing backtracking, precompute which sums of pairs of numbers are perfect squares. This reduces the complexity of checking the sum condition during the backtracking steps, making the algorithm more efficient.
Efficient Permutation Generation with Sorting
Sort the array initially to facilitate skipping over duplicate permutations. This ensures that the backtracking approach only explores unique permutations and avoids unnecessary recalculations.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity is heavily dependent on the backtracking steps and the number of permutations generated. In the worst case, this can approach factorial time, i.e., O(n!). Precomputing the perfect square sums reduces the overhead, but the final complexity still depends on the problem size. Space complexity depends on storing the permutations and checking valid sums, which can be O(n).
What Interviewers Usually Probe
- Backtracking expertise
- Proficiency with pruning techniques
- Efficiency in handling permutations and checking perfect square sums
Common Pitfalls or Variants
Common pitfalls
- Misunderstanding how to efficiently prune the search space
- Failing to handle duplicate numbers correctly
- Incorrectly checking perfect square conditions for sums of adjacent elements
Follow-up variants
- Using dynamic programming to optimize the backtracking process
- Reducing space complexity by optimizing hash lookup operations
- Exploring alternative approaches with bit manipulation to track visited elements
How GhostInterview Helps
- Guides the interviewee through efficient backtracking techniques and pruning methods
- Provides suggestions on handling edge cases, such as duplicate numbers
- Helps optimize hash set usage for faster lookups during permutation checks
Topic Pages
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- 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 a squareful array?
A squareful array is one where the sum of every adjacent pair of elements is a perfect square.
What is the time complexity of solving this problem?
The time complexity is typically O(n!) in the worst case due to the backtracking approach and permutation generation.
How do we check if the sum of two numbers is a perfect square?
You can check if the sum is a perfect square by calculating the square root of the sum and verifying if it is an integer.
What is the role of hash sets in this solution?
Hash sets are used to avoid duplicate permutations and to efficiently look up previously computed values during the backtracking process.
Can we use dynamic programming to solve this problem?
Yes, dynamic programming can optimize the backtracking approach by storing results of subproblems, though it might not always be necessary.
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