**π‘ Problem Formulation:** The task is to determine the count of elements in a list that are greater than each individual element. For instance, given the list `[3, 1, 4, 2]`

, the desired output is `[2, 3, 0, 1]`

, because for the first element 3, there are 2 elements greater (4 and 2), for 1 there are 3 elements greater (3, 4, and 2), for 4 there are none greater, and for 2 there is 1 element greater (4).

## Method 1: Brute Force Approach

To find the frequency of numbers greater than each element in a Python list, the brute force method iterates through the list for each element, comparing it against all others to count how many are higher. This method does not require any additional imports, but it has a high time complexity of O(n^2) due to the nested loops it uses.

Here’s an example:

def count_greater_numbers(lst): greater_count = [] for x in lst: count = sum([1 for i in lst if i > x]) greater_count.append(count) return greater_count example_list = [3, 1, 4, 2] print(count_greater_numbers(example_list))

Output:

[2, 3, 0, 1]

This code defines a function that loops through each element `x`

in the given list `lst`

and uses a list comprehension together with the `sum()`

function to count the other elements greater than `x`

. The results are aggregated in a new list `greater_count`

, which is returned.

## Method 2: Using Sort and Binary Search

In this method, we improve efficiency by sorting the list and using a binary search to find the index of the current element, which gives the number of elements greater than the current one. This approach has an improved time complexity due to the O(n log n) sorting and O(log n) for binary search per element.

Here’s an example:

import bisect def count_greater_numbers_bisect(lst): sorted_list = sorted(lst) greater_count = [len(lst) - bisect.bisect_right(sorted_list, x) for x in lst] return greater_count example_list = [3, 1, 4, 2] print(count_greater_numbers_bisect(example_list))

Output:

[2, 3, 0, 1]

First, we sort the original list, which allows us to take advantage of the `bisect_right()`

function from the `bisect`

module to find the insertion point of each element `x`

in the sorted list. For each element, the function calculates the number of elements after the insertion point, which corresponds to the number of elements greater than `x`

.

## Method 3: Using Sorted List and Indexes

An alternative to binary search, after sorting the list, we can use the index of the original elements in the sorted list to determine the number of greater elements. This method works well when the original list does not contain duplicate values.

Here’s an example:

def count_greater_numbers_index(lst): sorted_list = sorted(lst) greater_count = [len(lst) - sorted_list.index(x) - 1 for x in lst] return greater_count example_list = [3, 1, 4, 2] print(count_greater_numbers_index(example_list))

Output:

[2, 3, 0, 1]

After sorting the list, the function uses a list comprehension with `index()`

method to find the position of each element in the sorted list and thus determines how many elements are greater. However, if duplicates exist, this method would not yield the correct results as `index()`

returns the first occurrence of a duplicate value.

## Method 4: Using a HashMap for Frequency Counts

By mapping each number to its frequency in the list, we can traverse the list once and, using cumulative sums, determine the frequency of greater numbers efficiently, especially when the list contains many duplicate values.

Here’s an example:

from collections import Counter def count_greater_numbers_map(lst): counter = Counter(lst) keys_sorted = sorted(counter.keys(), reverse=True) cum_sum = 0 greater_count_map = {} for key in keys_sorted: greater_count_map[key] = cum_sum cum_sum += counter[key] return [greater_count_map[x] for x in lst] example_list = [3, 1, 4, 2] print(count_greater_numbers_map(example_list))

Output:

[2, 3, 0, 1]

The function uses `Counter`

from the collections module to create a frequency map of the list items. It sorts these keys in descending order and then iterates through them, keeping a cumulative sum of frequencies to determine how many elements each key is greater than. The list comprehension then converts this mapping back to our desired format using the original list.

## Bonus One-Liner Method 5: Nested List Comprehension with `sum()`

For those who prefer concise solutions, a one-liner nested list comprehension can replace the brute force method. This approach is elegant but suffers the same performance drawbacks as the initial brute force solution due to O(n^2) complexity.

Here’s an example:

example_list = [3, 1, 4, 2] greater_count = [sum(i > x for i in example_list) for x in example_list] print(greater_count)

Output:

[2, 3, 0, 1]

This line combines the iteration and counting into a single, albeit nested, list comprehension expression. It is very readable and great for small lists, but not optimized for longer lists due to its quadratic time complexity.

## Summary/Discussion

**Method 1: Brute Force Approach.**This method is easy to understand but inefficient for larger lists due to O(n^2) complexity.**Method 2: Using Sort and Binary Search.**More efficient than brute force, suitable for larger lists. Trades off extra memory complexity for sorting.**Method 3: Using Sorted List and Indexes.**Works well for lists without duplicates but not recommended when duplicates exist.**Method 4: Using a HashMap for Frequency Counts.**An efficient solution especially when the list contains many duplicates. Linear complexity in aggregate, but requires additional space complexity for the map.**Bonus Method 5: Nested List Comprehension.**Elegant and concise but shares the brute force approachβs inefficiency for large data sets.