<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>4</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>4</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:
        hm = collections.defaultdict(set)
        n = len(target)
        MOD = 1070777777
        BASE = 26
        pre_hash = [0]*(n+1)
        for i,b in enumerate(target):
            pre_hash[i+1] = (pre_hash[i]*BASE+ord(b)-ord("a"))%MOD
            
        def sub_hash(l,r):
            try:
                return (pre_hash[r] - pre_hash[l] * pow(BASE,r-l,MOD)) % MOD
            except:
                return None
        for w in words:
            h = 0
            for i in range(len(w)):
                h = (h*BASE + ord(w[i])-ord("a")) % MOD
                hm[i+1].add(h)
        def can(i,j):
            l = j-i
            return sub_hash(i,j) in hm[l]
        left = 0; right = n
        while left < right:
            mid = left + (right-left+1)//2
            if can(0,mid):
                left = mid
            else:
                right = mid-1
        if can(0,left):
            curr = farthest = left
            jumps = 1
            i = 0
        else:
            return -1
        while i <= farthest:
            #print(f"i={i},curr={curr},farthest={farthest},jumps={jumps}")
            left = farthest+1;right = n
            while left < right:
                mid = left + (right-left+1)//2
                if can(i,mid):
                    left = mid
                else:
                    right = mid-1
            if can(i,left):
                farthest = left
            if i == curr:
                curr = farthest
                jumps += 1
            i += 1
        if farthest == n:
            return jumps-1
        return -1
            
            
            
        
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