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Finding the Original Array from a Doubled Array: A Detailed Tutorial

This tutorial provides a comprehensive guide on how to recover the original array from a given "doubled" array. A doubled array is one where each element of the original array is duplicated, and then all elements are shuffled. We'll cover different approaches, analyze their time complexity, discuss potential edge cases, and provide clear code examples in Python.

*Understanding the Problem*

The core problem is that you're given an array `doubled`, and you need to determine if it can be formed by taking some original array `original`, doubling each element of `original`, and then shuffling the resulting array. If it can be formed this way, you need to reconstruct the `original` array.

*Example:*

`doubled = [4, 8, 2, 6, 3, 1, 2, 4]`
`original = [1, 2, 3, 4]` (Each element is doubled to form 2, 4, 6, 8, and then combined with original to form the doubled array)

*Constraints and Considerations:*

The input `doubled` array must have an even number of elements. This is because each element in `original` contributes two elements to `doubled`.
The `doubled` array may contain duplicate values.
We need to handle cases where it's impossible to form the doubled array from an original array.
The `original` array's order doesn't matter, so we can sort `doubled` to simplify the matching process.

*Approaches*

We'll explore two main approaches:

1. *Sorting and Greedy Matching:* This approach leverages sorting to efficiently match elements. It's generally preferred due to its relatively good performance.
2. *Hashmap/Counter Based Approach:* This method uses a dictionary or counter to track the frequency of each element and iteratively finds pairs.

*1. Sorting and Greedy Matching*

This approach is based on the observation that if `x` is present in `original`, then `2x` must be present in `doubled`. We sort the `doubled` array. This allows us to iterate throug ...

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