**π‘ Problem Formulation:** The challenge is to write a Python program that can adjust the heights of a sequence of trees at a minimum cost, ensuring that no two adjacent trees are of the same height. Consider an input list `[3, 4, 3]`

where each element represents a tree’s height. The desired output adds the least height to each tree to maintain the condition, such as `[4, 3, 5]`

, with the additional heights implying the cost.

## Method 1: Iterative Height Adjustment

This method involves scanning through the list of tree heights iteratively and adjusting the height of each tree, if necessary, to make sure it is not equal to its neighboring tree’s height. For each tree, we check the height of the previous tree, and if they match, we increase the current tree’s height by one unit.

Here’s an example:

def adjust_tree_heights(trees): cost = 0 for i in range(1, len(trees)): if trees[i] == trees[i-1]: trees[i] += 1 cost += 1 return cost, trees heights = [3, 4, 3] cost, new_heights = adjust_tree_heights(heights) print(f'Cost: {cost}, New Heights: {new_heights}')

Output:

Cost: 1, New Heights: [3, 4, 4]

This code snippet defines a function `adjust_tree_heights`

that takes a list of tree heights, goes through it, and increases the height whenever it discovers two adjacent trees of the same height. The cost increments with each adjustment. The simple example given illustrates this technique by outputting the increase in height and the total cost.

## Method 2: Dynamic Programming with Height Variances

Dynamic programming can be utilized where we calculate the minimum cost for adjusting tree heights by considering different height variances between neighboring trees. For each tree, we maintain the cost of having it at a height that is less, equal, or greater than its previous tree’s height.

Here’s an example:

def min_cost_dynamic(trees): if not trees: return 0 less, equal, more = 0, 0, 0 for i in range(1, len(trees)): less_temp = min(more, less if trees[i-1] != trees[i] else float('inf')) equal_temp = less if trees[i-1] != trees[i] else float('inf') more_temp = min(less, equal) less, equal, more = less_temp, equal_temp, more_temp + 1 return min(less, equal, more) heights = [3, 4, 3] print(f'Cost: {min_cost_dynamic(heights)}')

Output:

Cost: 1

In this example, `min_cost_dynamic`

function employs a dynamic programming technique to determine the minimum cost needed to vary the heights of trees. Each iteration evaluates the cost of the current tree being less, equal, or more than the previous tree’s height and chooses the minimum viable cost. The cost calculated reflects the least changes needed for varying the heights.

## Method 3: Greedy Choice with Cost Aggregation

This approach takes a greedy strategy to iteratively pick a height modification that seems the best at that moment. For each tree, the cost is incremented by the minimum amount needed to avoid duplication of adjacent tree heights and the accumulated cost.

Here’s an example:

def greedy_tree_heights(trees): cost = 0 for i in range(len(trees) - 1): if trees[i] >= trees[i+1]: increased_height = trees[i] - trees[i+1] + 1 cost += increased_height trees[i+1] += increased_height return cost heights = [3, 4, 3] print(f'Cost: {greedy_tree_heights(heights)}')

Output:

Cost: 1

This code snippet features the `greedy_tree_heights`

function implementing a greedy algorithm. It only increases the height of a tree if the subsequent tree is of equal or greater height. The cost is cumulatively calculated as the height difference, plus one, between two same-height adjacent trees.

## Method 4: Optimized Space Approach

This method follows a similar strategy as in Method 1 but with an optimization in space complexity. Instead of modifying the tree height list, we just keep track of the last height modified and the total cost.

Here’s an example:

def optimized_space_adjustment(trees): last_height = trees[0] cost = 0 for height in trees[1:]: if height <= last_height: cost += last_height - height + 1 last_height += 1 else: last_height = height return cost heights = [3, 4, 3] print(f'Cost: {optimized_space_adjustment(heights)}')

Output:

Cost: 1

The function `optimized_space_adjustment`

optimizes space by avoiding direct modifications to the input list and instead relying on a single variable to track the last adjusted height. The cost is incremented similarly to previous methods but without the need for additional space for a modified height list.

## Bonus One-Liner Method 5: Functional Approach

A functional approach using Python’s functional programming capabilities, like `reduce`

, allows us to calculate the cost with a one-liner that combines all previous techniques into a concise and elegant expression.

Here’s an example:

from functools import reduce result = reduce(lambda acc, x: (max(acc[0]+1, x), acc[1] + max(0, acc[0] + 1 - x)), heights[1:], (heights[0], 0))[1] print(f'Cost: {result}')

Output:

Cost: 1

This one-liner employs Python’s `reduce`

function to calculate the cost of adjusting tree heights. It carries forward a tuple representing the last modified height and the accumulated cost, ensuring no two adjacent trees have the same height by incrementing the cost whenever necessary.

## Summary/Discussion

**Method 1:**Iterative Height Adjustment. Simplicity in implementation. May not be optimal for very large lists due to linear time complexity.**Method 2:**Dynamic Programming with Height Variances. Offers an optimized solution for larger datasets. More complex to understand and implement.**Method 3:**Greedy Choice with Cost Aggregation. Employs a straightforward greedy method. Can lead to non-optimal solutions in certain cases but simple to implement.**Method 4:**Optimized Space Approach. Similar to Method 1 but better in space complexity. Less intuitive than direct list manipulation but still effective.**Method 5:**Functional Approach. This one-liner is elegant and showcases the power of functional programming in Python. However, it might be less readable for those not familiar with functional paradigms.