<p>You are given a <strong>0-indexed</strong> string <code>s</code>, a string <code>a</code>, a string <code>b</code>, and an integer <code>k</code>.</p>

<p>An index <code>i</code> is <strong>beautiful</strong> if:</p>

<ul>
	<li><code>0 &lt;= i &lt;= s.length - a.length</code></li>
	<li><code>s[i..(i + a.length - 1)] == a</code></li>
	<li>There exists an index <code>j</code> such that:
	<ul>
		<li><code>0 &lt;= j &lt;= s.length - b.length</code></li>
		<li><code>s[j..(j + b.length - 1)] == b</code></li>
		<li><code>|j - i| &lt;= k</code></li>
	</ul>
	</li>
</ul>

<p>Return <em>the array that contains beautiful indices in <strong>sorted order from smallest to largest</strong></em>.</p>

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

<pre>
<strong>Input:</strong> s = &quot;isawsquirrelnearmysquirrelhouseohmy&quot;, a = &quot;my&quot;, b = &quot;squirrel&quot;, k = 15
<strong>Output:</strong> [16,33]
<strong>Explanation:</strong> There are 2 beautiful indices: [16,33].
- The index 16 is beautiful as s[16..17] == &quot;my&quot; and there exists an index 4 with s[4..11] == &quot;squirrel&quot; and |16 - 4| &lt;= 15.
- The index 33 is beautiful as s[33..34] == &quot;my&quot; and there exists an index 18 with s[18..25] == &quot;squirrel&quot; and |33 - 18| &lt;= 15.
Thus we return [16,33] as the result.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;abcd&quot;, a = &quot;a&quot;, b = &quot;a&quot;, k = 4
<strong>Output:</strong> [0]
<strong>Explanation:</strong> There is 1 beautiful index: [0].
- The index 0 is beautiful as s[0..0] == &quot;a&quot; and there exists an index 0 with s[0..0] == &quot;a&quot; and |0 - 0| &lt;= 4.
Thus we return [0] as the result.
</pre>

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

<ul>
	<li><code>1 &lt;= k &lt;= s.length &lt;= 5 * 10<sup>5</sup></code></li>
	<li><code>1 &lt;= a.length, b.length &lt;= 5 * 10<sup>5</sup></code></li>
	<li><code>s</code>, <code>a</code>, and <code>b</code> contain only lowercase English letters.</li>
</ul>

class Solution:
    def beautifulIndices(self, s: str, a: str, b: str, k: int) -> List[int]:
        # "AAACAAA"
        # [0,1,2,0,1,2,3]
        def prefix(a):
            lpsIndex = 0
            lps = [0] * (len(a))
            i = 1
            while i < len(a):
                if a[lpsIndex] == a[i]:
                    lps[i] = lpsIndex + 1
                    lpsIndex += 1
                    i += 1
                elif lpsIndex == 0:
                    lps[i] = 0
                    i += 1
                else:
                    lpsIndex = lps[lpsIndex - 1]
            return lps
        def kmp(s, a, table):
            i = 0
            j = 0
            res = []
            while i < len(s):
                if s[i] == a[j]:
                    i, j = i + 1, j + 1
                elif j == 0:
                    i += 1
                else:
                    j = table[j - 1]
                if j == len(a):
                    res.append(i - len(a))
                    j = table[j - 1]
            return res
        aIndexes = kmp(s, a, prefix(a))
        bIndexes = kmp(s, b, prefix(b))
        print(aIndexes)
        print(bIndexes)
        cur = 0
        res = []
        for i in aIndexes:
            while cur < len(bIndexes) and bIndexes[cur] < i - k:
                cur += 1
            if cur < len(bIndexes) and bIndexes[cur] <= i + k:
                res.append(i)
        return res
        
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