<p>You are given an array of strings <code>words</code> and a string <code>target</code>.</p>

<p>A string <code>x</code> is called <strong>valid</strong> if <code>x</code> is a <span data-keyword="string-prefix">prefix</span> of <strong>any</strong> string in <code>words</code>.</p>

<p>Return the <strong>minimum</strong> number of <strong>valid</strong> strings that can be <em>concatenated</em> to form <code>target</code>. If it is <strong>not</strong> possible to form <code>target</code>, return <code>-1</code>.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">words = [&quot;abc&quot;,&quot;aaaaa&quot;,&quot;bcdef&quot;], target = &quot;aabcdabc&quot;</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<p>The target string can be formed by concatenating:</p>

<ul>
	<li>Prefix of length 2 of <code>words[1]</code>, i.e. <code>&quot;aa&quot;</code>.</li>
	<li>Prefix of length 3 of <code>words[2]</code>, i.e. <code>&quot;bcd&quot;</code>.</li>
	<li>Prefix of length 3 of <code>words[0]</code>, i.e. <code>&quot;abc&quot;</code>.</li>
</ul>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">words = [&quot;abababab&quot;,&quot;ab&quot;], target = &quot;ababaababa&quot;</span></p>

<p><strong>Output:</strong> <span class="example-io">2</span></p>

<p><strong>Explanation:</strong></p>

<p>The target string can be formed by concatenating:</p>

<ul>
	<li>Prefix of length 5 of <code>words[0]</code>, i.e. <code>&quot;ababa&quot;</code>.</li>
	<li>Prefix of length 5 of <code>words[0]</code>, i.e. <code>&quot;ababa&quot;</code>.</li>
</ul>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">words = [&quot;abcdef&quot;], target = &quot;xyz&quot;</span></p>

<p><strong>Output:</strong> <span class="example-io">-1</span></p>
</div>

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

<ul>
	<li><code>1 &lt;= words.length &lt;= 100</code></li>
	<li><code>1 &lt;= words[i].length &lt;= 5 * 10<sup>3</sup></code></li>
	<li>The input is generated such that <code>sum(words[i].length) &lt;= 10<sup>5</sup></code>.</li>
	<li><code>words[i]</code> consists only of lowercase English letters.</li>
	<li><code>1 &lt;= target.length &lt;= 5 * 10<sup>3</sup></code></li>
	<li><code>target</code> consists only of lowercase English letters.</li>
</ul>

class Solution:
    def minValidStrings(self, words: List[str], target: str) -> int:
        n = len(target)
        f = [n+1] * n

        base = 31
        MOD = 10**9 + 7
        p = [1] * n
        for i in range(1, n):
            p[i] = (p[i-1] * base) % MOD
        
        words_hash = []
        for i in range(len(words)):
            words_hash.append([])
            for c in words[i]:
                h = ord(c) - ord('a') + 1
                if words_hash[i]:
                    h = (h + words_hash[i][-1] * base) % MOD
                
                words_hash[i].append(h)
        
        string_hash = []
        for c in target:
            h = ord(c) - ord('a') + 1
            if string_hash:
                h = (h + string_hash[-1] * base) % MOD
            
            string_hash.append(h)

        def get_hash(h, i, j):
            if i == 0:
                return (h[j] + MOD * MOD) % MOD

            return (h[j] - h[i - 1] * p[j - i + 1] + MOD * MOD) % MOD



        for word in words:
            for i in range(len(word)):
                if i < n and word[i] == target[i]:
                    f[i] = 1
                else:
                    break
        

        for i in range(n - 1):
            if f[i] == n + 1:
                return -1
            max_l = 0
            for t, word in enumerate(words):
                inf = 0
                suf = min(n-i-1, len(word)) - 1
                res = -1
                while inf <= suf:
                    mid = (inf + suf) // 2
                    if get_hash(words_hash[t], 0, mid) == get_hash(string_hash, i+1, i+mid+1):
                        res = mid
                        inf = mid + 1
                    else:
                        suf = mid - 1
                
                max_l = max(max_l, res + 1)
                # print(target[i+1:],word,res)
                

            for j in range(i+1, i + max_l + 1):
                f[j] = min(f[j], f[i] + 1)
            # print(f)
        
        if f[n-1] == n + 1:
            return -1
        # print(f)
        return f[n-1]
20954