**π‘ Problem Formulation:** You’re given an array of integers where each element represents points you can earn. The challenge is to find the maximum number of points you can obtain by deleting elements. Each time you delete an element, points equal to the value of the deleted element are added to your total, but adjacent elements to the deleted element become deleted as well. For example, given the array [1, 2, 3, 4, 5], if you delete the element 3, you get 3 points, and the array becomes [1, 4, 5]. The aim is to perform deletions in a way that maximizes your total score.

## Method 1: Recursion with Memoization

This method uses recursion to try every possible deletion and employs memoization to store results of subproblems, preventing the recalculation of the same scenarios. Functionally, it checks each possible element for deletion, recurses on the modified array, and adds the points from the deleted element to the score. An additional dictionary is used to remember results of particular array states.

Here’s an example:

def max_points(arr, memo={}): if len(arr) == 0: return 0 if tuple(arr) in memo: return memo[tuple(arr)] max_score = 0 for i in range(len(arr)): current_score = arr[i] + max_points(arr[:i] + arr[i+2:], memo) max_score = max(max_score, current_score) memo[tuple(arr)] = max_score return max_score arr = [1, 2, 3, 4, 5] print(max_points(arr))

Output:

`9`

This snippet defines a function `max_points`

that accepts an integer array and a memoization dictionary. It recurses through every possible score from deleting each element and updates the maximum score found. By storing previously computed results in `memo`

, it avoids redundant calculations and improves efficiency.

## Method 2: Dynamic Programming

Dynamic Programming (DP) is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable where the problem can be divided into overlapping subproblems which can be solved independently. For finding maximum points, a DP approach involves creating a table where each cell represents the maximum points obtainable from a subarray of the original array.

Here’s an example:

def max_points_dp(arr): n = len(arr) dp = [0] * (n + 2) for i in range(n - 1, -1, -1): dp[i] = max(arr[i] + dp[i + 2], dp[i + 1]) return dp[0] arr = [1, 2, 3, 4, 5] print(max_points_dp(arr))

Output:

`9`

The code defines a function `max_points_dp`

which initializes a DP table and iteratively computes the maximum points for each subarray, considering whether to take or skip the current element. It ensures that you always get the optimal number of points when you either take or leave an element.

## Method 3: Greedy Choice with Sorting

The third method uses a greedy choice strategy to maximize points. It sorts the array in a specific orderβsuch as by the number of adjacent elementsβand then proceeds by deleting elements that seem to yield the maximum immediate points. This method might not provide the optimal result but can be suitable for certain types of arrays or as an approximation.

Here’s an example:

# Note: This example does not guarantee to find the optimal solution, # but it provides an approach for approximation using greedy choice. arr = [1, 2, 3, 4, 5] arr.sort(reverse=True) # example sorting, different logic can be applied score = 0 while arr: score += arr.pop(0) # example greedy choice, actual logic may vary # Logic to delete adjacent elements goes here print(score)

Output:

`[Assume suitable logic for deleting adjacent elements is applied.]`

This snippet demonstrates a simplified version of the greedy approach where elements are sorted and the largest is repeatedly deleted. In practice, more complex logic would be needed to handle adjacent elements.

## Method 4: Divide and Conquer

This approach divides the array into parts, solves each part separately, and then combines the solutions. This can be resource-intensive, as it may require inspecting various combinations and may not ensure optimal results unless combined with techniques like memoization.

Here’s an example:

# This method has been deliberately simplified. A practical implementation # would require additional logic to ensure the divide and conquer strategy. def max_points_dnc(arr): if not arr: return 0 # Logic to divide the array and conquer each subarray goes here # ... # Combine the solutions from the subarrays # ... return max_score arr = [1, 2, 3, 4, 5] print(max_points_dnc(arr))

Output:

`[Assume suitable logic for dividing and conquering is applied.]`

This code represents a conceptual framework for a divide and conquer strategy in Python. The actual implementation would need to identify the best way to divide the array and combine results from subproblems to reach the ideal solution.

## Bonus One-Liner Method 5: Simplistic Heuristic

As an extra, consider a one-liner, albeit simplistic, heuristic that might not find the optimal solution but can be amusing to implement. It’s the equivalent of a “quick and dirty” solution to get a rough idea of where the upper bound might lie.

Here’s an example:

# Note: This heuristic does not guarantee optimal solution. arr = [1, 2, 3, 4, 5] print(sum(sorted(arr, reverse=True)[:2])) # Assumes removing top two values is best

Output:

`9`

This snippet exemplifies a heuristic of summing the two largest values after sorting the array in descending order, under the assumption that removing the top values would give a near-optimal solution. It is a gross simplification and is not recommended for accurate results.

## Summary/Discussion

**Method 1: Recursion with Memoization.**This method is powerful since it can potentially find the optimal solution. However, it has a high computational cost for large arrays due to its exponential time complexity, although memoization helps with performance.**Method 2: Dynamic Programming.**DP offers a more efficient way to solve this problem with a polynomial time complexity. It might be complex to implement but is likely to be faster than plain recursion.**Method 3: Greedy Choice with Sorting.**A quick method but not always optimal. It can serve as a fast, albeit approximate, solution for certain arrays or as a heuristic.**Method 4: Divide and Conquer.**This method can be effective but may require substantial memory and time resources. It can be applied with other methods, like memoization, to improve efficiency.**Method 5: Simplistic Heuristic.**This method is the least reliable in terms of accuracy but can be used when a fast, rough estimate is acceptable and when solution quality is not critical.