Wiggle Sort II
Given an unsorted array `nums`, reorder it such that $nums[0] < nums[1] > nums[2] < nums[3] \dots$. Must be solved in $O(n)$ time with $O(1)$ extra space.
Why Interviewers Ask This
Microsoft asks Wiggle Sort II to evaluate a candidate's ability to handle complex in-place array manipulations under strict constraints. This problem specifically tests mastery of partitioning algorithms, understanding of index mapping logic, and the capacity to optimize for O(n) time and O(1) space simultaneously. It reveals whether an engineer can balance theoretical correctness with practical memory efficiency.
How to Answer This Question
Structure your solution using a three-phase logical framework: Analysis, Partitioning, and Reordering. First, analyze the constraints; explicitly state that sorting takes O(n log n), so you must find a linear-time alternative. Second, implement a median-finding algorithm like Quickselect to identify the pivot element in O(n) average time without modifying the array order globally. Third, execute the critical reordering step using a virtual index mapping technique. Instead of creating a new array, map the sorted positions to specific indices in the original array (e.g., placing larger elements at odd indices and smaller ones at even indices) using a three-way partition or a specific swap strategy. Finally, validate edge cases where duplicates exist, ensuring the wiggle property holds strictly. Throughout, verbalize your thought process regarding why standard sorting fails and how your chosen approach satisfies Microsoft's strict performance requirements.
Key Points to Cover
- Explicitly rejecting standard sorting due to O(n log n) complexity
- Correctly implementing Quickselect to find the median in O(n)
- Using virtual index mapping to achieve O(1) space complexity
- Handling duplicate elements to prevent equality violations in the wiggle pattern
- Verbalizing the trade-off between algorithmic complexity and memory usage
Sample Answer
To solve Wiggle Sort II within O(n) time and O(1) space, we cannot simply sort the array, as that would violate the time complexity constraint. My approach begins by finding the median element of the unsorted array. I will use the Quickselect algorithm, which partitions the array around a pivot to find the k-th smallest element in average O(n) time. Once we have the median, we need to rearrange the array such that elements smaller than the median go to even indices and larger elements go to odd indices. Since we cannot allocate extra space, we must perform this rearrangement in-place. The challenge is handling duplicates correctly to avoid adjacent equal values. I will employ a virtual index mapping strategy. We treat the array as if it were sorted, but we map the actual indices to their target positions using a formula like (2 * i + 1) % (n | 1). This effectively interleaves the smaller and larger halves of the array. By performing a three-way partition around the median value and swapping elements based on this mapped index logic, we ensure nums[0] < nums[1] > nums[2] < nums[3]. This method guarantees the wiggle property while adhering to the strict memory and time limits required for high-scale systems at Microsoft.
Common Mistakes to Avoid
- Attempting to sort the array first, which violates the O(n) time requirement
- Creating a new auxiliary array to store the result, failing the O(1) space constraint
- Ignoring edge cases where multiple identical median values exist, causing adjacent duplicates
- Failing to explain the mathematical logic behind the virtual index mapping during the interview
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