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

            best = math.inf

            for j in range(i, len(s)):
                if is_pal[i][j]:
                    best = min(best, 1 + f(j + 1))

            return best
        
        return f(0) - 1
81125