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Union of Arrays with Duplicates: A Comprehensive Tutorial

The union of two or more arrays combines the elements of all the arrays into a single array. When dealing with arrays that contain duplicate elements, the standard set-based union operation (removing duplicates) doesn't always suffice. Often, you need to preserve the duplicates, reflecting their multiplicity across the input arrays. This tutorial will explore different approaches to creating a union of arrays with duplicates, providing explanations and code examples in Python, JavaScript, and Java.

*Understanding the Problem*

Let's say we have the following arrays:



The desired union with duplicates would be:



Notice that we've simply appended `array2` to `array1` without removing any duplicates. This represents a basic interpretation of union with duplicates. We'll explore more nuanced versions later.

*Methods for Creating Union with Duplicates*

Here are several approaches to achieving the union with duplicates, ranging from straightforward to more sophisticated and efficient:

*1. Simple Concatenation (Direct Approach)*

This is the most basic approach: simply append or concatenate the arrays. It is the fastest and most straightforward for many use cases.

*Python:*



*JavaScript:*



*Java:*



*Explanation:* This method simply creates a new array by combining the elements of the input arrays in the order they are provided. It's the most efficient in terms of computational complexity, as it only involves iterating through the arrays once during the concatenation process.

*Time Complexity:* O(n + m), where n is the length of `array1` and m is the length of `array2`.
*Space Complexity:* O(n + m), as it creates a new array to store the union.

*2. Counting Occurrences (Histogram-Based Union)*

This approach provides a more nuanced interpretation of "union with duplicates." It focuses on ensuring that the *maxi ...

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