Add unique variable to walker class start_lvl

Hey everyone,

I need to add unique identifiers to every level of the wp_list_categories list tree.

In situations with child categories, they are always listed as

<ul class="children">

I would like to change the start_lvl function in the Walker Class (Walker_Category) from:

function start_lvl(&$output, $depth, $args) {
		if ( 'list' != $args['style'] )
			return;
		$indent = str_repeat("\t", $depth);
		$output .= "$indent<ul class='children'>\n";
	}

To:

function start_lvl(&$output, $depth, $args) {
		if ( 'list' != $args['style'] )
			return;
		$indent = str_repeat("\t", $depth);
		$output .= "$indent<ul id='children-$UNIQUE_ID' class='children'>\n";
	}

This can be a category ID, or something similar, but each UL needs to have a fully unique ID attached to it.

The only variable I can get in there to parse is $depth, but that created non-unique IDs if there are multiple top-level cats, etc.

Any ideas?