<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 isPalindrome(self, s):
        for i in range(len(s)//2):
            if s[i] != s[len(s)-i-1]:
                return False
        return True

    def minCutHelper(self, s, start, alreadySeen, palindrome_matrix):
        #print(f"qrhuang debug: minCutHelper() called with s={s}")
        if len(s) == 0:
            return 0
        if s in alreadySeen:
            return alreadySeen[s]
        min_cuts = len(s)-1
        for i in range(1, len(s)+1):
            if palindrome_matrix[start][i]:
                min_cuts_rest = self.minCutHelper(s[i:], start+i, alreadySeen, palindrome_matrix)
                min_cuts = min(min_cuts, min_cuts_rest)
        alreadySeen[s] = min_cuts + 1
        return min_cuts + 1

    def minCut(self, s: str) -> int:
        palindrome_matrix = [[False for j in range(len(s)+1)] for i in range(len(s))]
        for i in range(len(s)):
            palindrome_matrix[i][0] = True
            palindrome_matrix[i][1] = True
        for length in range(2, len(s)+1):
            for i in range(len(s)-length+1):
                palindrome_matrix[i][length] = s[i] == s[i+length-1] and palindrome_matrix[i+1][length-2]
        return self.minCutHelper(s, 0, {}, palindrome_matrix) - 1
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