<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 defaultdict

class Solution:
    def addOperators(self, num: str, target: int) -> List[str]:
        n = len(num)
        prod_values = [
            [
                defaultdict(set)
                for j in range(n)
            ]
            for i in range(n)
        ]
        sum_values = [
            [
                defaultdict(set)
                for j in range(n)
            ]
            for i in range(n)
        ]
        intervals = sorted(
            [
                (i, j)
                for i in range(n)
                for j in range(i, n)
            ],
            key=lambda p: p[1] - p[0] + 1
        )
        for i, j in intervals:
            if j - i + 1 == 1:
                prod_values[i][j][int(num[i])].add(num[i])
                continue
            elif num[i] != '0':
                prod_values[i][j][int(num[i:j + 1])].add(num[i:j + 1])
            for k in range(i, j):
                for v1, exp1_list in prod_values[i][k].items():
                    for v2, exp2_list in prod_values[k + 1][j].items():
                        for exp1 in exp1_list:
                            for exp2 in exp2_list:
                                prod_values[i][j][v1 * v2].add(exp1 + '*' + exp2)
        for i, j in intervals:
            for v, exp_list in prod_values[i][j].items():
                for exp in exp_list:
                    sum_values[i][j][v].add(exp)
                    sum_values[i][j][-v].add('-' + exp)
            for k in range(i, j):
                for v1, exp1_list in sum_values[i][k].items():
                    for v2, exp2_list in sum_values[k + 1][j].items():
                        for exp1 in exp1_list:
                            for exp2 in exp2_list:
                                sum_values[i][j][v1 + v2].add(exp1 + '+' + exp2)
        return list(set(exp.replace('+-', '-') for exp in sum_values[0][n - 1][target] if exp[0] != '-'))
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