This is a Python solution for the following algorithmic problem that can be found on LeetCode website (LeetCode problem 301).

Given a string s that contains parentheses and letters, remove the minimum number of invalid parentheses to make the input string valid.

Return all the possible results. You may return the answer in any order.

Return all the possible results. You may return the answer in any order.

**Example 1:**

Input: s = "()())()"

Output: ["(())()","()()()"]

Output: ["(())()","()()()"]

The core ideas behind this solution:

- Using recursion, we can generate all possible combinations of parentheses. At the each step (each new character), we either concatenate the current character to s_current accumulator or skip it.
- As we go, we check if the currently accumulated string s_current is a valid output string or not
- We store all of the valid output strings into the helper dictionary dict1
- At the end, we filter out all but longest valid strings

Code:

```
'''
Solution for algorithmic problem: "Remove Invalid Parentheses", posted on LeetCode (problem 301).
Author:
Goran Trlin
Find this and more code examples on:
https://playandlearntocode.com
'''
class Solution:
dict1 = {} # dictionary to help filtering the duplicate entries
max_length = 0 # we're only interested in valid strings of maximum possible length
def is_valid(self, unclosed):
''''
Simple check of string validity, just based on the number of unclosed parentheses
'''
if unclosed == 0:
return True
else:
return False
def r1(self, s, s_current, unclosed_count, new_pos, new_char):
'''
Main recursive function for processing the input string.
:param s: the original string
:param s_current: current string, passed by the recursive caller
:param unclosed_count: total number of unclosed parentheses
:param new_pos: position of the next character in the original string
:param new_char: next character in the original string
:return:
'''
if self.is_valid(unclosed_count) == True:
# print('Found valid:' + s_current)
if (len(s_current) > self.max_length):
self.max_length = len(s_current)
# mark this key in the dict.:
self.dict1[s_current] = 1
# terminal condition:
if new_pos > len(s) - 1:
# print ('Exiting at pos.' + str(new_pos))
return ''
# boundary condition, end of string:
next_char = ''
if new_pos + 1 > len(s) - 1:
next_char = ''
else:
next_char = s[new_pos + 1]
if new_char == '(' or new_char == ')':
# for "(" and ")", proceed with two paths - in one add the character, in the other one, omit it
diff = 0 # whether to change the number of unclosed parentheses or not
if new_char == '(':
diff = 1
if new_char == ')':
diff = -1
if unclosed_count < 0:
# makes sure that invalid parentheses get filtered out:
diff = 100
self.r1(s, s_current, unclosed_count, new_pos + 1, next_char)
self.r1(s, s_current + new_char, unclosed_count + diff, new_pos + 1, next_char)
else:
self.r1(s, s_current + new_char, unclosed_count, new_pos + 1, next_char)
# use this function signature inside of LeetCode editor (camelCase is required in the editor):
# def removeInvalidParentheses(self, s: str):
def remove_invalid_parentheses(self, s: str):
'''
Core function of this example.
:param s:
:return:
'''
if len(s) == 0:
return []
# reset before the call:
self.dict1 = {}
self.max_length = 0
# start the recursive call:
self.r1(s, '', 0, 0, s[0])
final_list = []
# filter out too short valid pairs (only leave the ones with the min. number of parentheses removed):
for key in self.dict1:
if len(key) == self.max_length:
final_list.append(key)
return final_list
```

The code for this project can be found on GitHub.