To solve this problem, iterate through the array using backtracking, leveraging hash tables to track unique subsequences. Focus on maintaining non-decreasing order as you explore potential subsequences. The solution explores array scanning and hash lookup for efficiency and correctness.
Problem Statement
Given an integer array nums, return all possible non-decreasing subsequences of the array with at least two elements. Each subsequence should be returned in any order, and subsequences must maintain the order in the original array.
You can assume that the length of nums will not exceed 15, making a backtracking solution feasible. The goal is to find subsequences where the elements do not decrease as you move through the sequence, and the solution must handle arrays with duplicates as well.
Examples
Example 1
Input: nums = [4,6,7,7]
Output: [[4,6],[4,6,7],[4,6,7,7],[4,7],[4,7,7],[6,7],[6,7,7],[7,7]]
Example details omitted.
Example 2
Input: nums = [4,4,3,2,1]
Output: [[4,4]]
Example details omitted.
Constraints
- 1 <= nums.length <= 15
- -100 <= nums[i] <= 100
Solution Approach
Backtracking with Hash Table
Use a backtracking approach where each potential subsequence is constructed step by step. At each step, verify if the new number can be added to the subsequence without breaking the non-decreasing rule. Use a hash table to ensure uniqueness of subsequences.
Array Scanning and Element Comparison
Iterate through the array, comparing elements to their predecessors. If the current element is greater than or equal to the last element of the subsequence, append it to the subsequence and continue exploring. This ensures all subsequences are non-decreasing.
Pruning with Early Termination
Prune unnecessary branches of the recursion tree to improve efficiency. If a subsequence has reached a length of two, store it, and skip further exploration if it cannot lead to valid subsequences.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time and space complexity depends on the method used for backtracking and the number of unique subsequences generated. With a maximum array length of 15, the solution involves exploring potential subsequences, which can be exponential in nature. The use of hash tables ensures uniqueness of subsequences, reducing unnecessary recomputations.
What Interviewers Usually Probe
- The candidate effectively applies backtracking and hash table usage to avoid redundant subsequences.
- The solution scales well to the upper constraint, showing consideration of optimization techniques like pruning.
- The candidate demonstrates a clear understanding of array scanning and how it helps ensure subsequences remain non-decreasing.
Common Pitfalls or Variants
Common pitfalls
- Failure to track unique subsequences leading to repeated results.
- Not pruning the recursion tree early enough, causing excessive computational overhead.
- Incorrect handling of edge cases such as arrays with duplicate elements.
Follow-up variants
- Modify the problem to allow subsequences of length one.
- Expand the problem to allow strictly increasing subsequences only.
- Add a constraint where subsequences must sum to a target value.
How GhostInterview Helps
- GhostInterview assists in identifying key points of backtracking and array scanning for optimized subsequence generation.
- The platform provides insights into hash table usage for ensuring uniqueness, a critical part of this problem.
- GhostInterview emphasizes optimization techniques, such as pruning and early termination, to avoid unnecessary computations.
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 solving the Non-decreasing Subsequences problem?
The main pattern for solving this problem is array scanning combined with backtracking, leveraging hash tables to ensure subsequences remain unique.
How can I optimize my solution for Non-decreasing Subsequences?
Optimize by pruning unnecessary recursive branches and using hash tables to track and avoid duplicate subsequences.
Can Non-decreasing Subsequences be solved in linear time?
No, this problem involves exploring all potential subsequences, leading to an exponential time complexity depending on the array size.
What are the constraints for the Non-decreasing Subsequences problem?
The array length is constrained to be between 1 and 15, and element values are between -100 and 100.
Why is backtracking used in the Non-decreasing Subsequences problem?
Backtracking is used to explore all possible subsequences, ensuring we generate all non-decreasing subsequences of length two or more.
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