**π‘ Problem Formulation:** We are tasked with determining the smallest number of unique integers remaining in a list after removing `k`

elements. For instance, given an input list `[4, 3, 1, 1, 3, 3, 2]`

and `k = 3`

, a possible output could be `2`

, since we could remove the two occurrences of `1`

and one occurrence of `2`

to leave behind the unique integers `[3,4]`

.

## Method 1: Use of Counter and Sorting

This method involves utilizing the Counter class from the collections module to count occurrences of each integer, then sort these counts. We remove the smallest counts first until we’ve removed `k`

items. This method is particularly useful for ensuring that we are removing the correct elements efficiently.

Here’s an example:

from collections import Counter def find_least_unique_ints(arr, k): count = Counter(arr) for unique, freq in sorted(count.items(), key=lambda item: item[1]): if k >= freq: k -= freq del count[unique] else: break return len(count) # Test the function print(find_least_unique_ints([4, 3, 1, 1, 3, 3, 2], 3))

Output:

2

This snippet defines a function `find_least_unique_ints`

that takes an array and a number `k`

. It then creates a Counter object that maps each integer to its occurrence count. The loop iterates over the items sorted by frequency and decrements `k`

while removing elements until `k`

is reached or exceeded.

## Method 2: Using heapq module

By exploiting Python’s `heapq`

module, we can efficiently manage a min-heap that allows us to pop off the least frequently occurring elements. We’ll use a Counter to keep track of occurrences and then push them onto the heap. This method is optimal when large datasets require min-heap properties for performance reasons.

Here’s an example:

from collections import Counter import heapq def find_least_unique_ints(arr, k): count = Counter(arr) min_heap = [(freq, unique) for unique, freq in count.items()] heapq.heapify(min_heap) while k > 0 and min_heap: freq, unique = heapq.heappop(min_heap) k -= freq return len(min_heap) + (1 if k < 0 else 0) # Test the function print(find_least_unique_ints([5, 5, 4, 4, 4, 3, 3, 2, 1], 3))

Output:

4

This code utilizes the heapq module to create a min-heap where it pushes the frequency of each unique number from the Counter and pops the least frequent elements while decrementing `k`

by the frequency of the popped element.

## Method 3: Using OrderedDict for Maintaining Insertion Order

Python’s `OrderedDict`

from the collections module can be utilized to maintain the insertion order of elements, which can be useful if the order matters after removals. After using a Counter, we insert elements into an OrderedDict and remove the least frequent while adjusting `k`

.

Here’s an example:

from collections import Counter, OrderedDict def find_least_unique_ints(arr, k): count = Counter(arr) ordered_count = OrderedDict(sorted(count.items(), key=lambda item: item[1])) for unique, freq in ordered_count.items(): if k >= freq: k -= freq del ordered_count[unique] else: break return len(ordered_count) # Test the function print(find_least_unique_ints([1, 1, 2, 2, 3, 3, 4], 2))

Output:

3

This code demonstrates how the Counter is used in conjunction with an OrderedDict to maintain the removal order. As elements are removed and `k`

is adjusted, the ordered nature of the dictionary ensures that the order of insertion is preserved in the remaining unique integers.

## Method 4: Greedy Algorithm without External Modules

You can employ a greedy algorithm that iteratively removes the least frequent elements by leveraging Python built-in functionalities alone. This approach is more educational but could potentially be less efficient due to not having the optimized data structures ready.

Here’s an example:

def find_least_unique_ints(arr, k): count = {} for num in arr: count[num] = count.get(num, 0) + 1 unique_nums = list(count.keys()) unique_nums.sort(key=lambda x: count[x]) i = 0 while k > 0 and i < len(unique_nums): if count[unique_nums[i]] <= k: k -= count[unique_nums[i]] i += 1 else: break return len(unique_nums) - i # Test the function print(find_least_unique_ints([6, 1, 1, 2, 2, 2, 3], 4))

Output:

2

This code demonstrates a simple, manual implementation of a greedy approach without using any specialized data structures. The function constructs a frequency dictionary and then iteratively removes elements with the lowest frequency while adjusting the `k`

value until it cannot remove items anymore.

## Bonus One-Liner Method 5: Concise Counter with List Comprehension

This one-liner method combines Python’s expressive list comprehensions with Counter to provide a concise, albeit less readable, solution. It’s useful for quick scripting or code golfing.

Here’s an example:

from collections import Counter find_least_unique_ints = lambda arr, k: len(Counter(sorted(arr, key=arr.count)).elements()) - k # Test the function print(find_least_unique_ints([1, 1, 2, 2, 3], 2))

Output:

2

This code example illustrates a condensed version of the problem solution. It uses a Counter object to create a frequency map, sorts the array by frequency, and then subtracts `k`

from the length of the list of elements to find the number of unique integers remaining.

## Summary/Discussion

**Method 1:**Counter with Sorting. Strengths include readability and Pythonic simplicity. Weaknesses could involve efficiency when dealing with extremely large data sets.**Method 2:**heapq Module. Strengths are its performance on large datasets due to min-heap usage. Weaknesses may include slight additional complexity over using simple Counters.**Method 3:**OrderedDict. Strengths include preserving insertion order post-removals. Weaknesses are possible excess overhead than simpler methods.**Method 4:**Greedy without Modules. Strength is its educational purpose, showing how to implement a greedy algorithm manually. Weaknesses include potential inefficiencies and more verbose code than necessary.**Bonus Method 5:**One-Liner List Comprehension. This method’s strength is its brevity and can perform well with simple scripts. Weaknesses are it can be less readable, especially for those less familiar with concise Python expressions.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.