**π‘ Problem Formulation:** This article addresses the challenge of optimizing plane usage for skydiving events occurring over `k`

days. Given the number of skydivers and varying group sizes, the goal is to write a program in Python that computes the minimum aircraft space required to accommodate all skydivers within the allotted timeframe. Input would typically consist of a list of group sizes and the integer `k`

, representing days, with output as the minimum plane size needed.

## Method 1: Binary Search

This method employs binary search to efficiently identify the minimum plane space needed for skydivers in `k`

days. By iteratively halving the search space, it converges on the smallest possible plane size that can still accommodate all groups within the given timeframe. The function specification would generally involve the list of group sizes and the integer ‘k’ representing the days as inputs.

Here’s an example:

def can_fit(groups, k, mid): days_required = 1 total = 0 for group in groups: if total + group > mid: days_required += 1 total = group else: total += group if days_required > k or group > mid: return False return True def min_space(groups, k): left, right = max(groups), sum(groups) while left < right: mid = left + (right - left) // 2 if can_fit(groups, k, mid): right = mid else: left = mid + 1 return left # Example input groups_of_skydivers = [10, 15, 20, 17] k_days = 2 # Output print(min_space(groups_of_skydivers, k_days))

Output: 37

This script returns the minimum plane space by iteratively halving the search range for the space required until the smallest fitting space is found. It includes an inner function, `can_fit()`

, which determines whether a given plane size, `mid`

, can fit all groups within `k`

days.

## Method 2: Dynamic Programming

Dynamic Programming approach solves subproblems to find the minimum space required for the skydivers. It uses a matrix to store the minimum spaces needed for different numbers of days and skydivers, reusing these precomputed values to build up the solution for the main problem.

Here’s an example:

# Dynamic Programming code to be included here.

Output: Minimum plane space specific output.

## Method 3: Greedy Algorithm

With the Greedy Approach, the program calculates the minimum space by always opting for the best immediate solution at each step with the hope of finding the global optimum. It arranges groups in the most space-efficient way without considering future consequences.

Here’s an example:

# Greedy Algorithm code to be included here.

Output: Minimum plane space specific output.

## Method 4: Brute Force

The Brute Force Method exhaustively searches through all possible combinations of group arrangements to determine the minimum space. This is the most straightforward but often computationally expensive method.

Here’s an example:

# Brute Force code to be included here.

Output: Minimum plane space specific output.

## Bonus One-Liner Method 5: Optimized Space Estimation

This quick estimation method uses an optimized calculation to provide an approximate answer quickly. However, it is only recommended for use when an approximate solution is acceptable.

Here’s an example:

# One-Liner Optimized Space Estimation code to be included here.

Output: Estimated plane space specific output.

## Summary/Discussion

**Method 1: Binary Search.**This method is efficient for large input sizes as it significantly reduces the number of necessary comparisons. However, it requires the problem to allow for binary search (i.e., the plane space being a monotonically increasing function of the number of days).**Method 2: Dynamic Programming.**This method is powerful when there are overlapping subproblems and optimal substructure, allowing to save computation time by storing and reusing solutions. However, it may have higher space complexity depending on the implementation.**Method 3: Greedy Algorithm.**Greedy algorithms are generally easy to implement and can be very efficient, but they might not always lead to the globally optimal solution for certain problems.**Method 4: Brute Force.**The Brute Force approach is simple to understand and implement, and it guarantees an accurate solution. However, its inefficiency renders it impractical for large input sizes.**Bonus Method 5: Optimized Space Estimation.**It’s a fast, albeit approximate, method suited for making quick decisions when exact values are unnecessary. This method is not recommended when precision is paramount.