We run an algorithm to look at all the possible connecting cities and calculate the price of purchasing roundtrips to that connecting city separately (i.e. from origin to layover and from layover to destination). We don't just limit ourselves to layovers specified by airline partners, so we search the entire graph.

The savings we show are in comparison to the cheapest ticket you'll find on Kayak, Expedia, etc. On about 66% of the searches performed (by search popularity), there are no savings, but we show you the cheapest ticket you can find on other sites. On 33% of the searches, we can find savings, and we show you how to book those savings. Typical savings are $100-200, but sometimes it can be up to $500 or more for long-haul flights.

This is hard to communicate visually, so we were hoping to get some advice from HN on how to do this!

Also, to give context, this is a school project that we (a group of 6 Harvard students) hacked together in 2 months.