How travelling portals finds best routes from A to B?

Question

I am really stuck on this, days are gone and I still can't even imagine, how it can be possible.

I am sure you know some travelling portals. You set source and destination and it prints possible trains/buses/flights you can use. This in itself is so easy. Absolutely no problem with any database system.

Problem comes when there is not only direct paths. How they search paths with changes? How they can find best ways from a lot of possible ways with only minimal delay? Do they use some special databases?