**π‘ Problem Formulation:** Given a list of integers representing the lengths of sticks, the objective is to construct the triangle with the largest possible perimeter. The requirement is to identify a set of three lengths that adhere to the triangle inequality theorem and maximize the perimeter. For example, from the input `[3, 6, 2, 7]`

, the output should be `16`

, which is the perimeter of the triangle with sides `6, 7, and 3`

.

## Method 1: Brute Force Approach

This method involves iterating through every possible combination of three sticks in the list to check if they can form a triangle that satisfies the triangle inequality theorem. The largest perimeter is updated accordingly if a valid triangle is found. The function `largestPerimeterBruteForce()`

utilizes this approach effectively, albeit not the most efficient method for large datasets.

Here’s an example:

def largestPerimeterBruteForce(sticks): max_perimeter = 0 n = len(sticks) for i in range(n): for j in range(i+1, n): for k in range(j+1, n): a, b, c = sorted([sticks[i], sticks[j], sticks[k]]) if a + b > c: max_perimeter = max(max_perimeter, a + b + c) return max_perimeter print(largestPerimeterBruteForce([3, 6, 2, 7]))

The output of this code is:

16

This method checks each triplet of sticks and updates the largest perimeter found that satisfies the triangle inequality. While simple to understand, it is not optimal for large lists due to its O(n^3) time complexity.

## Method 2: Sorting and Linear Scan

By first sorting the list of stick lengths, we can significantly improve the efficiency of our search. The `largestPerimeterSorted()`

function sorts the list descendingly and then looks for a valid triangle starting with the largest stick until the third in the list. This method is more efficient than the brute force approach as it primarily runs in O(n log n) time, where n is the number of sticks.

Here’s an example:

def largestPerimeterSorted(sticks): sticks.sort(reverse=True) for i in range(len(sticks) - 2): if sticks[i] < sticks[i+1] + sticks[i+2]: return sticks[i] + sticks[i+1] + sticks[i+2] return 0 print(largestPerimeterSorted([3, 6, 2, 7]))

The output of this code is:

16

The `largestPerimeterSorted()`

function leverages sorting to reduce the search space and improve performance. By comparing each triplet in a sorted list, we can find the largest perimeter in a more efficient manner.

## Method 3: Python’s Max Function with List Comprehension

Utilizing Python’s built-in `max()`

function combined with list comprehension, we can create a concise one-liner that achieves the same goal. This approach takes advantage of Python’s high-level syntax and is very readable for those familiar with list comprehensions.

Here’s an example:

def largestPerimeterMax(sticks): return max([a+b+c for i, a in enumerate(sticks) for j, b in enumerate(sticks) for k, c in enumerate(sticks) if i<jc and a+c>b and b+c>a], default=0) print(largestPerimeterMax([3, 6, 2, 7]))

The output of this code is:

16

The function `largestPerimeterMax()`

employs list comprehension to generate all possible triangles and then applies the `max()`

function to find the largest perimeter. It is elegant but could suffer from performance issues for large inputs due to its O(n^3) time complexity.

## Method 4: Using itertools

The Python `itertools`

module provides combinations utility, which allows for a clean and expressive iteration through all possible triangle sides. The `largestPerimeterItertools()`

function uses the combinations generator to produce triplets and then finds the maximum perimeter as per the triangle inequality.

Here’s an example:

from itertools import combinations def largestPerimeterItertools(sticks): for triplet in sorted(combinations(sticks, 3), reverse=True): a, b, c = sorted(triplet) if a + b > c: return a + b + c return 0 print(largestPerimeterItertools([3, 6, 2, 7]))

The output of this code is:

16

By using the `combinations`

function from `itertools`

, the `largestPerimeterItertools()`

method brings in more readable and maintainable code, though still with a O(n^3) time complexity.

## Bonus One-Liner Method 5: Using Lambda and Filter

As a more advanced one-liner, one can use a lambda function within Python’s `max()`

function, incorporating `filter()`

to sift through the triplets that satisfy the triangle inequality theorem. Itβs the most compact and ‘pythonic’ way to solve the problem, but readability might be compromised for less experienced programmers.

Here’s an example:

sticks = [3, 6, 2, 7] print(max((a+b+c for a, b, c in combinations(sticks, 3) if a+b>c and a+c>b and b+c>a), default=0))

The output of this code is:

16

This one-liner makes clever use of generator expressions and the `max()`

function to condense the solution into a single line. While elegant, it requires a thorough understanding of Python’s functional features.

## Summary/Discussion

**Method 1:**Brute Force Approach. Straightforward. Not efficient for large datasets.**Method 2:**Sorting and Linear Scan. More efficient with O(n log n) complexity. Requires the list to be mutable.**Method 3:**Max Function with List Comprehension. Elegant use of Python syntax. Can be slow for large input sizes.**Method 4:**Using itertools. Clean and expressive code leveraging a powerful module but maintains O(n^3) complexity.**Method 5:**Lambda and Filter One-liner. Compact and ‘pythonic.’ May impact readability for those not familiar with advanced Python constructs.