<p>Given a string <code>s</code>, partition <code>s</code> such that every <span data-keyword="substring-nonempty">substring</span> of the partition is a <span data-keyword="palindrome-string">palindrome</span>.</p>

<p>Return <em>the <strong>minimum</strong> cuts needed for a palindrome partitioning of</em> <code>s</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;aab&quot;
<strong>Output:</strong> 1
<strong>Explanation:</strong> The palindrome partitioning [&quot;aa&quot;,&quot;b&quot;] could be produced using 1 cut.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;a&quot;
<strong>Output:</strong> 0
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;ab&quot;
<strong>Output:</strong> 1
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= s.length &lt;= 2000</code></li>
	<li><code>s</code> consists of lowercase English letters only.</li>
</ul>

class Solution:
    def minCut(self, s: str) -> int:

        n = len(s)        
        @cache
        def dp(s):
            if is_pal[s][n - 1]:
                return 0
            
            ans = math.inf
            for i in range(s, n):
                if is_pal[s][i]:
                    ans = min(ans, 1 + dp(i + 1))
            
            return ans


        n = len(s)
        is_pal = [[False] * n for _ in range(n)]

        for i in range(n - 1, -1, -1):
            for j in range(i, n):
                if s[i] == s[j]:
                    is_pal[i][j] = j - i <= 2 or is_pal[i + 1][j - 1]
        
        return dp(0)

# class Solution:
#     def minCut(self, s: str) -> int:
        
#         # n = len(s)
#         # is_pal = [[False] * n for _ in range(n)]

#         # for i in range(n - 1, -1, -1):
#         #     for j in range(i, n):
#         #         if s[i] == s[j]:
#         #             is_pal[i][j] = j - i <= 2 or is_pal[i + 1][j - 1]
        
#         @cache
#         def is_palindrome(l, r):
#             if l >= r: return True
#             if s[l] != s[r]: return False
#             return is_palindrome(l + 1, r - 1)

        
#         @cache
#         def f(i):
#             if i >= len(s):
#                 return 0

#             best = math.inf

#             for j in range(i, len(s)):
#                 if is_palindrome(i, j):
#                     best = min(best, 1 + f(j + 1))

#             return best
        
#         return f(0) - 1

# class Solution:
#     def minCut(self, s: str) -> int:

#         # n = len(s)
#         # is_pal = [[False] * n for _ in range(n)]

#         # for i in range(n - 1, -1, -1):
#         #     for j in range(i, n):
#         #         if s[i] == s[j]:
#         #             is_pal[i][j] = j - i <= 2 or is_pal[i + 1][j - 1]
        
#         @cache
#         def is_palindrome(l, r):
#             if l >= r: return True
#             if s[l] != s[r]: return False
#             return is_palindrome(l + 1, r - 1)
        
#         @cache
#         def f(l, r):
#             if is_palindrome(l, r):
#                 return 0
            
#             result = math.inf
            
#             for i in range(l, r):
#                 res = 1 + f(l, i) + f(i + 1, r)
#                 result = min(result, res)
#             return result
            
#         return f(0, len(s) - 1)
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