How to implement a compare-operator for a struct/class of strings?

I have a class which represents a decomposed URL.

class URL
{
    std::string proto_;
    std::string host_;
    /* other fields */
};

(for example, proto_ can be http, https, ldap; host_ can be localhost:1234, google.com).

The real, meaningful value to be compared is of course the composed URL. But constructing it is expensive and I'd like to use this class a key-type for a std::map.

How can I implement a operator<() for this class in an efficient way? How to combine the comparison of distinct objects, which are in fact forming a whole, logically?

I tried using std::tie but the results are not as I expected.