<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)

        # Precompute palindromic substrings
        dp = [[False] * N for _ in range(N)]
        for i in range(N - 1, -1, -1):
            dp[i][i] = True
            for j in range(i + 1, N):
                if s[i] == s[j] and (j - i == 1 or dp[i + 1][j - 1]):
                    dp[i][j] = True

        cache = {}

        def checker(startIndex: int) -> int:
            if startIndex == N:
                return -1  # No cuts needed beyond the end of the string

            if startIndex in cache:
                return cache[startIndex]

            min_cuts = float('inf')

            for end in range(startIndex, N):
                if dp[startIndex][end]:
                    # Make a cut after the palindrome s[startIndex:end+1]
                    cuts = 1 + checker(end + 1)
                    if cuts < min_cuts:
                        min_cuts = cuts

            cache[startIndex] = min_cuts
            return min_cuts

        return checker(0)

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