<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 validate_pali(self, s: str, start: int, end: int) -> bool:
        if start>=end:
            return True
        elif (start, end) in self.is_valid_cache:
            return self.is_valid_cache[(start,end)]
        else:
            if s[start]!=s[end]:
                self.is_valid_cache[(start, end)] = False
                return False
            else:
                res = self.validate_pali(s, start+1, end-1)
                self.is_valid_cache[(start, end)] = res
                return res
            
    def find_cut(self, s: str, start: int, end: int) -> int:
        if start>=end:
            return 0
        elif (start, end) in self.min_cache:
            return self.min_cache[(start, end)]
        else:
            if self.is_valid_cache[(start, end)]:
                self.min_cache[(start, end)] = 0
                return 0
            else:
                res = self.INF
                for mid in range(start, end+1):
                    if self.is_valid_cache[(start, mid)]:
                        tmp = self.find_cut(s, mid+1, end)
                        res = min(res, 1+tmp)
                self.min_cache[(start, end)] = res
                return res
    
    def minCut(self, s: str) -> int:
        self.is_valid_cache = {}
        for start in range(len(s)):
            for end in range(start, len(s)):
                if (start, end) in self.is_valid_cache:
                    continue
                self.is_valid_cache[(start, end)] = self.validate_pali(s, start, end)
        
        self.min_cache = {}
        self.INF = 10**9
        res = self.find_cut(s, 0, len(s)-1)
        return res
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