<p>You are given a string <code>num</code>, representing a large integer. Return <em>the <strong>largest-valued odd</strong> integer (as a string) that is a <strong>non-empty substring</strong> of </em><code>num</code><em>, or an empty string </em><code>""</code><em> if no odd integer exists</em>.</p>
<p>A <strong>substring</strong> is a contiguous sequence of characters within a string.</p>
<strong>Explanation:</strong> The only non-empty substrings are "5", "2", and "52". "5" is the only odd number.