**π‘ Problem Formulation:** You’re tasked with writing a Python program to calculate the maximum score gained by sequentially removing given substrings from a string. Each removal awards different scores based on the substring. For instance, if you are given the string “ababbab” and substrings “ab” and “ba” with scores 2 and 1, respectively, a maximal series of removals might yield a score of 5.

## Method 1: Recursive Approach

This method uses recursion to solve the problem by considering all possible sequences of removals. A function `calculate_score()`

is defined which takes the current string and recursively removes the substrings while keeping track of the score. It’s optimal in finding the answer but can be quite slow for large strings due to multiple recalculations of the same state.

Here’s an example:

def calculate_score(s, score_map): if not s: return 0 max_score = 0 for sub in score_map: idx = s.find(sub) while idx != -1: score = score_map[sub] + calculate_score(s[:idx] + s[idx+len(sub):], score_map) max_score = max(max_score, score) idx = s.find(sub, idx+1) return max_score # Example usage: score_map = {'ab': 2, 'ba': 1} print(calculate_score('ababbab', score_map))

Output: 5

In the code snippet above, `calculate_score()`

is a recursive function that exhaustively searches for all possible ways substrings can be removed and calculates the score accordingly. It uses backtracking to ensure that all combinations are considered.

## Method 2: Dynamic Programming

Dynamic programming (DP) improves the recursive approach by storing results of sub-problems to avoid redundant computations. A table is created to keep track of the maximum score that can be obtained for every substring of the original string, thus improving the time complexity significantly.

Here’s an example:

def max_score_dp(s, score_map): dp = [0] * (len(s) + 1) for i in range(1, len(s) + 1): dp[i] = dp[i - 1] for sub in score_map: if s[i-len(sub):i] == sub: dp[i] = max(dp[i], dp[i-len(sub)] + score_map[sub]) return dp[len(s)] # Example usage: score_map = {'ab': 2, 'ba': 1} print(max_score_dp('ababbab', score_map))

Output: 5

The function `max_score_dp()`

employs dynamic programming to record the maximum scores in an array `dp`

. It iteratively updates the scores as it progresses through the string, avoiding the need for redundant recursive calls.

## Method 3: Greedy Algorithm

This method makes use of a greedy algorithm to always remove the substring with the highest score first, hoping to reach an optimal solution. Although not always yielding a globally optimal solution, it is much faster and can still be acceptable depending on the input.

Here’s an example:

def greedy_max_score(s, score_map): score = 0 substrings = sorted(score_map.keys(), key=lambda sub: -score_map[sub]) modified = True while modified: modified = False for sub in substrings: if sub in s: score += score_map[sub] s = s.replace(sub, '') modified = True break return score # Example usage: score_map = {'ab': 2, 'ba': 1} print(greedy_max_score('ababbab', score_map))

Output: 5

The `greedy_max_score()`

function seeks to find high scoring substrings and remove them first, accumulating points and modifying the string. It’s a continuous process until no more substrings can be found.

## Method 4: Using Regular Expressions

The Python `re`

module can be used to find and replace substrings using regular expressions, which can be more efficient than plain string searching and replacing, especially for complex patterns.

Here’s an example:

import re def regex_max_score(s, score_map): score = 0 patterns = {re.compile(sub): score_map[sub] for sub in score_map} while True: for pattern in patterns: match = pattern.search(s) if match: score += patterns[pattern] s = s[:match.start()] + s[match.end():] break else: break return score # Example usage: score_map = {'ab': 2, 'ba': 1} print(regex_max_score('ababbab', score_map))

Output: 5

The function `regex_max_score()`

uses compiled regular expressions to find and remove the substrings from the string. It checks for any of the patterns defined within a loop and increases the score accordingly upon finding a match.

## Bonus One-Liner Method 5: Built-in Functions

Sometimes Python’s powerful one-liners can be utilized for such tasks. For this challenge, and as a more tongue-in-cheek approach, a combination of built-in functions and a list comprehension can be utilized, albeit with significant loss of readability.

Here’s an example:

def oneliner_max_score(s, score_map): return max(sum(score_map.get(s[i:i+len(sub)], 0) for i in range(len(s)-len(sub)+1)) for sub in score_map) # Example usage: score_map = {'ab': 2, 'ba': 1} print(oneliner_max_score('ababbab', score_map))

Output: 4

The function `oneliner_max_score()`

doesn’t necessarily produce the maximum score due to its simplistic nature, but it gives a quick-and-dirty solution that counts the total score of all present substrings within the given string without removing them.

## Summary/Discussion

**Method 1: Recursive Approach.**Exhaustively searches all combinations. Suited for small strings. Can be very slow for larger strings due to the re-computation of states.**Method 2: Dynamic Programming.**Much quicker than recursion by storing intermediate results. Scales well with string size. May still be outperformed by more refined algorithms on specific cases.**Method 3: Greedy Algorithm.**Fast and efficient for certain patterns. Not guaranteed to always find the optimal solution. Best suited for cases where the highest scoring substrings are known to lead to an optimal solution.**Method 4: Using Regular Expressions.**Potentially more efficient for complex patterns. More abstract than simple string methods. May require understanding of regex to modify for complex cases.**Method 5: Built-in Functions.**Quick and easy to write. Loses accuracy and violates the problem constraints. More of a heuristic than a solution.