Steven Vachon

If you were to use this on a website like http;//example.com/test/testing/, you would get results like these:
<ol>
<li>
<ul>
<li>Before: http;//example.com/test/another-test/#anchor</li>
<li>After: ../another-test/#anchor</li>
</ul>
</li>
<li>
<ul>
<li>Before: http;//example.com/wp-content/themes/twentyten/style.css</li>
<li>After: /wp-content/themes/twentyten/style.css</li>
</ul>
</li>
<li>
<ul>
<li>Before: https;//example.com/wp-content/themes/twentyten/style.css</li>
<li>After: https;//example.com/wp-content/themes/twentyten/style.css</li>
</ul>
</li>
<li>
<ul>
<li>Before: http;//google.com/test/</li>
<li>After: //google.com/test/</li>
</ul>
</li>
<li>
<ul>
<li>Before: ../../../../../../../../#anchor</li>
<li>After: /#anchor</li>
<li>After (<code>$output_type=1</code>): ../../#anchor</li>
</ul>
</li>
</ol>
All string parsing. No directory browsing.
If you’re looking for a plugin that will automatically convert all URLs on your WordPress site, instead check out my other plugin, <a href="https://wordpress.org/extend/plugins/wp-html-compression/" rel="ugc">WP-HTML-Compression</a>. 
Before you copy this code and add it into your own, keep in mind that there will probably be future updates. Keeping the code within an installed plugin will make sure that you’re notified.

Absolute-to-Relative URLs

A function() for use in shortening URL links. This plugin is meant for dev work and does not automatically shorten URLs.

Combining HTML “minification” with cache and HTTP compression (<a href="https://wordpress.org/extend/plugins/wp-super-cache/" rel="ugc">WP Super Cache</a>, or similar) will cut down your bandwidth and ensure near-immediate content delivery while increasing your Google rankings.
This plugin will compress your HTML by shortening URLs and removing standard comments and whitespace; including new lines, carriage returns, tabs and excess spaces. Most importantly, by ignoring <code><pre></code>, <code><textarea></code>, <code><script></code> and Explorer® <code>conditional comment</code> tags, presentation will not be affected.

WP-HTML-Compression

Reduce file size by shortening URLs and safely removing all standard comments and unnecessary whitespace from an HTML document.

Absolute-to-Relative URLs vs WP-HTML-Compression