<p>In a project, you have a list of required skills <code>req_skills</code>, and a list of people. The <code>i<sup>th</sup></code> person <code>people[i]</code> contains a list of skills that the person has.</p>

<p>Consider a sufficient team: a set of people such that for every required skill in <code>req_skills</code>, there is at least one person in the team who has that skill. We can represent these teams by the index of each person.</p>

<ul>
	<li>For example, <code>team = [0, 1, 3]</code> represents the people with skills <code>people[0]</code>, <code>people[1]</code>, and <code>people[3]</code>.</li>
</ul>

<p>Return <em>any sufficient team of the smallest possible size, represented by the index of each person</em>. You may return the answer in <strong>any order</strong>.</p>

<p>It is <strong>guaranteed</strong> an answer exists.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre><strong>Input:</strong> req_skills = ["java","nodejs","reactjs"], people = [["java"],["nodejs"],["nodejs","reactjs"]]
<strong>Output:</strong> [0,2]
</pre><p><strong class="example">Example 2:</strong></p>
<pre><strong>Input:</strong> req_skills = ["algorithms","math","java","reactjs","csharp","aws"], people = [["algorithms","math","java"],["algorithms","math","reactjs"],["java","csharp","aws"],["reactjs","csharp"],["csharp","math"],["aws","java"]]
<strong>Output:</strong> [1,2]
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= req_skills.length &lt;= 16</code></li>
	<li><code>1 &lt;= req_skills[i].length &lt;= 16</code></li>
	<li><code>req_skills[i]</code> consists of lowercase English letters.</li>
	<li>All the strings of <code>req_skills</code> are <strong>unique</strong>.</li>
	<li><code>1 &lt;= people.length &lt;= 60</code></li>
	<li><code>0 &lt;= people[i].length &lt;= 16</code></li>
	<li><code>1 &lt;= people[i][j].length &lt;= 16</code></li>
	<li><code>people[i][j]</code> consists of lowercase English letters.</li>
	<li>All the strings of <code>people[i]</code> are <strong>unique</strong>.</li>
	<li>Every skill in <code>people[i]</code> is a skill in <code>req_skills</code>.</li>
	<li>It is guaranteed a sufficient team exists.</li>
</ul>

class Solution:
    def smallestSufficientTeam2(self, req_skills, people):  # dp, based on dict
        n, m = len(req_skills), len(people)
        key = {v: i for i, v in enumerate(req_skills)}
        dp = {0: []}
        for i, p in enumerate(people):
            his_skill = 0
            for skill in p:
                if skill in key:
                    his_skill |= 1 << key[skill]
            for skill_set, need in list(dp.items()):
                with_him = skill_set | his_skill
                if with_him == skill_set: continue
                if with_him not in dp or len(dp[with_him]) > len(need) + 1:
                    dp[with_him] = need + [i]
        return dp[(1 << n) - 1]
    
    def smallestSufficientTeam(self, req_skills: List[str], people: List[List[str]]) -> List[int]:  # DFS
        mp = {skill : i for i, skill in enumerate(req_skills)} # digitized skills
        
        cand = []
        for skills in people: 
            val = 0
            for skill in skills: 
                val |= 1 << mp[skill] # digitized skill
            cand.append(val)
        
        @cache
        def fn(i, mask): 
            """Return smallest sufficient team of people[i:] for skills in mask."""
            if mask == 0: return []
            if i == len(people): return [0]*100 # impossible
            if not (mask & cand[i]): return fn(i+1, mask)
            return min(fn(i+1, mask), [i] + fn(i+1, mask & ~cand[i]), key=len)
        
        return fn(0, (1 << len(req_skills)) - 1)
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