**π‘ Problem Formulation:** Our task is to determine the farthest possible distance one can travel using a collection of n bikes, where each bike can travel a specific maximum distance before it fails. This is particularly crucial when attempting a set distance with varied bike endurance levels. Input may include an array or list `[d1, d2, d3, ..., dn]`

of distances that each of the n bikes can cover, and the output will be a single number representing the maximum distance that can be traveled using these bikes.

## Method 1: Greedy Algorithm

As a straightforward approach to this problem, a greedy algorithm selects the bike with the longest available distance at each step until we’ve utilized all bikes. While not always optimal for every scenario, it works well when the problem guarantees the choice of the next best option at each step leads to a global optimum.

Here’s an example:

def max_distance_greedy(bike_distances): return sum(sorted(bike_distances, reverse=True)) # Example usage bike_distances = [10, 20, 30, 40, 50] print(max_distance_greedy(bike_distances))

Output: 150

The `max_distance_greedy`

function sorts the list of bike distances in descending order and sums them up to find the total maximum distance. This algorithm is efficient for finding the global optimum provided each choice is the best local decision.

## Method 2: Dynamic Programming

Dynamic Programming can optimize problems by breaking them into subproblems. For the bike distances, this means using previously calculated maximum distances to find the maximum distance with an additional bike. This method ensures an optimal solution by constructing solutions from the bottom up.

Here’s an example:

def max_distance_dp(bike_distances): n = len(bike_distances) dp = [0] * (n + 1) for i in range(1, n + 1): dp[i] = dp[i-1] + bike_distances[i-1] return dp[n] # Example usage bike_distances = [10, 20, 30, 40, 50] print(max_distance_dp(bike_distances))

Output: 150

The `max_distance_dp`

function employs an array `dp`

to keep track of the maximum distances up to the i-th bike. The final value `dp[n]`

will have the total maximum distance after all bikes have been considered.

## Method 3: Recursion with Memoization

Using recursion, we can explore different combinations of distances, but this might lead to the same calculations being done multiple times. To avoid this, memoization caches results, which can be reused, therefore improving efficiency.

Here’s an example:

def max_distance_recursive(bike_distances, n, memo): if n == 0: return 0 if n in memo: return memo[n] memo[n] = max_distance_recursive(bike_distances, n-1, memo) + bike_distances[n-1] return memo[n] # Example usage bike_distances = [10, 20, 30, 40, 50] memo = {} print(max_distance_recursive(bike_distances, len(bike_distances), memo))

Output: 150

The function `max_distance_recursive`

recursively calculates the maximum distance, while the `memo`

dictionary stores already computed results to prevent redundant calculations. This technique wisely balances space and time complexity.

## Method 4: Iterative Accumulation

An iterative solution iterates through the distances, accumulating the total distance covered. This method trivially solves the problem and ensures that each bike’s distance is considered exactly once.

Here’s an example:

def max_distance_iterative(bike_distances): total_distance = 0 for distance in bike_distances: total_distance += distance return total_distance # Example usage bike_distances = [10, 20, 30, 40, 50] print(max_distance_iterative(bike_distances))

Output: 150

The function `max_distance_iterative`

declares a variable `total_distance`

and adds each bike’s distance to it within a loop. This straightforward method is easy to understand and implement.

## Bonus One-Liner Method 5: Pythonic Way

For those who appreciate Python’s ability to compress algorithms into one line, the sum function directly applied to the list of distances provides a quick solution.

Here’s an example:

bike_distances = [10, 20, 30, 40, 50] print(sum(bike_distances))

Output: 150

This example leverages Python’s built-in `sum()`

function to calculate the total distance covered by all bikes with a very concise, readable, and performant line of code.

## Summary/Discussion

**Method 1:**Greedy Algorithm. Efficient and simple. Best works when each local optimal solution leads to a global optimal solution.**Method 2:**Dynamic Programming. Ensures optimal solution by constructing from simpler subproblems. More complex, but handles larger sets efficiently.**Method 3:**Recursion with Memoization. Good balance between computational efficiency and intuitive understanding through recursion. May be slower than iterative for some data sets.**Method 4:**Iterative Accumulation. Straightforward implementation. Can be slower for large data sets due to the explicit loop.**Method 5:**Pythonic Way. Clean and concise. Perfect for simple summation tasks. However, lacks scalability for more complex variations of the problem.