<p>Given a string <code>num</code> that contains only digits and an integer <code>target</code>, return <em><strong>all possibilities</strong> to insert the binary operators </em><code>&#39;+&#39;</code><em>, </em><code>&#39;-&#39;</code><em>, and/or </em><code>&#39;*&#39;</code><em> between the digits of </em><code>num</code><em> so that the resultant expression evaluates to the </em><code>target</code><em> value</em>.</p>

<p>Note that operands in the returned expressions <strong>should not</strong> contain leading zeros.</p>

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

<pre>
<strong>Input:</strong> num = &quot;123&quot;, target = 6
<strong>Output:</strong> [&quot;1*2*3&quot;,&quot;1+2+3&quot;]
<strong>Explanation:</strong> Both &quot;1*2*3&quot; and &quot;1+2+3&quot; evaluate to 6.
</pre>

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

<pre>
<strong>Input:</strong> num = &quot;232&quot;, target = 8
<strong>Output:</strong> [&quot;2*3+2&quot;,&quot;2+3*2&quot;]
<strong>Explanation:</strong> Both &quot;2*3+2&quot; and &quot;2+3*2&quot; evaluate to 8.
</pre>

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

<pre>
<strong>Input:</strong> num = &quot;3456237490&quot;, target = 9191
<strong>Output:</strong> []
<strong>Explanation:</strong> There are no expressions that can be created from &quot;3456237490&quot; to evaluate to 9191.
</pre>

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

<ul>
	<li><code>1 &lt;= num.length &lt;= 10</code></li>
	<li><code>num</code> consists of only digits.</li>
	<li><code>-2<sup>31</sup> &lt;= target &lt;= 2<sup>31</sup> - 1</code></li>
</ul>

from collections import deque

class Solution:
    def addOperators(self, num: str, target: int) -> List[str]:
        # naive solution is to just try every combination of ops
        # when do we know a path doesn't work?
        
        #build up from base 'segments' to form possible nums 

        # multiplication is key one since that ttakes precedence
        # need to group numbers into possible mult groups
        # basically multiplication of

        # recursion only adds groups, does multiplicative possibilities inside
        # just need to check every contiguous group

        # calculate out possible multiplications for any substring starting at i and ending at j

        # val, str, stop
        prods = dict()
        prods[len(num)] = []

        # inclusive
        for i in reversed(range(len(num))):
            prods[i] = [(int(num[i]), num[i], i + 1)]
            if i + 1 < len(num):
                if num[i] == '0':
                    for prod, s, end in prods[i + 1]:
                        prods[i].append((0, '0*' + s, end))
                else:
                    for j in range(i + 1, len(num)):
                        prods[i].append((int(num[i:j+1]), num[i:j+1], j+1))
                        val = int(num[i:j])
                        for prod, s, end in prods[j]:
                            prods[i].append((val * prod, num[i:j] + '*' + s, end))
        res = []
        paths = deque(prods[0])
        while paths:
            next_path = paths.popleft()
            if next_path[2] == len(num):
                if next_path[0] == target:
                    res.append(next_path[1])
            else:
                for cont in prods[next_path[2]]:
                    paths.append((next_path[0] + cont[0], next_path[1] + '+' + cont[1], cont[2]))
                    paths.append((next_path[0] - cont[0], next_path[1] + '-' + cont[1], cont[2]))
        return res


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