Algorithm For Longest Palindrome I made an brute force algorithm to find the longest palindrome within the string. This palindrome is a substring of the input. Below is the code.

def generate_substrings(string):
    &quot;&quot;&quot;
    avg time: O(len(string) * len(string))
    avg space: O(1) 
    
    amortized time: O(len(string) * len(string)) 
    amortized space: O(1) 
    &quot;&quot;&quot; 
    col = 0 
    for row in range(len(string)):
        col = row + 1 
        while col &lt; len(string) + 1:
            yield string[row:col]
            col = col + 1 
            
def test_generate_substrings():
    assert list(generate_substrings('')) == []
    assert list(generate_substrings('a')) == ['a']
    assert list(generate_substrings('ab')) == ['a', 'ab', 'b']

def is_palindrome(string):
    &quot;&quot;&quot;
    avg time: O(len(string)) 
    avg space: O(len(string)) 
    
    amortized time: O(len(string)) 
    amortized space: O(len(string)) 
    &quot;&quot;&quot;
    return string[::-1] == string 

def test_is_palindrome():
    assert is_palindrome('') == True 
    assert is_palindrome('ab') == False 
    assert is_palindrome('bb') == True 
    assert is_palindrome('abc') == False 
    assert is_palindrome('ddd') == True 
    
def find_palindrome(string):
    &quot;&quot;&quot;
    avg time: O(len(string) * len(string)) 
    avg space: O(len(string)) 
    
    amortized time: O(len(string) * len(string)) 
    amortized space: O(len(string))
    &quot;&quot;&quot;
    answer = ''
    vec = [] 
    
    for string in generate_substrings(string):
        if is_palindrome(string):
            vec.append(string)
    
    for string in vec:
        if len(answer) &lt;= len(string):
            answer = string 
       
    return answer 

def test_find_palindrome():
    assert find_palindrome('ab') == 'b'
    assert find_palindrome('aa') == 'aa' 

def main():
&quot;&quot;&quot;
entry point for test code 
&quot;&quot;&quot;
    test_generate_substrings() 
    test_is_palindrome() 
    test_find_palindrome() 

if __name__ == '__main__':
    main()
 
```
It's good to see code that comes with tests - well done!
I would add some more tests in places.  Specifically, we ought to test is_palindrome() with a single-letter input, as that's something that could easily get broken, and find_palindrome() needs testing with some strings where the longest palindrome is at beginning, middle or end.
I'm concerned at

assert find_palindrome('ab') == 'b'


There's nothing in the problem statement that says we couldn't return 'a' instead, so this is actually over-testing the code.  There's something similar going in in test_generate_substrings() where we require a specific order (we'd be better testing the result converted to set rather than list, to make the test more reasonable).

In terms of the algorithm, I see an obvious candidate for improvement:

for string in generate_substrings(string):
    if is_palindrome(string):
        vec.append(string)

for string in vec:
    if len(answer) &lt;= len(string):
        answer = string


We store all of the palindromes in vec, even though we only want the longest one.  So we don't really need vec:
for string in generate_substrings(string):
    if is_palindrome(string) and len(answer) &lt;= len(string):
        answer = string

We could do even better if we arrange for generate_substrings() to return all the substrings longest first.  Then we wouldn't need to generate or use all the shorter substrings:
for string in generate_substrings_longest_first(string):
    if is_palindrome(string):
        return string