414/514 – The triangle inequality

February 3, 2012

On Google+, Willie Wong posted a link to this interesting example, by Brian Gawalt: BART fares and the triangle inequality.

There is a natural way of measuring distance in a subway or train system, the “price between stations” metric. It turns out that when applied to BART, the  Bay Area Rapid Transit system, this fails to be a metric, with the consequence that sometimes it is cheaper to take a detour, exiting and reentering an intermediate station, than going directly to one’s destination. As Gawalt points out:

It’s probably important to recognize the 15 cents you save by jumping out costs about 15 to 20 minutes of your life waiting for the next train to come pick you up.

Willie adds an interesting comment, that I reproduce here:

Heh, while BART fails to be a metric space (with the price between stations metric), it is interesting to note that the single-fare systems form ultrametric spaces.

The British Rail / PostOffice metrics, of course, reflect systems with concentric zones in rings for which to get from one place to another almost certainly require passing through the centre. Like London Underground for example.

The public transport in Lausanne does not form a metric space using the price-between-stations metric for another (somewhat strange) reason: the price-between-stations function is set valued: the same two stations can have different prices depending on which route the bus/train takes, even without you getting off. (This is the problem with a zone based system. For certain places there are two more or less identical routes but one goes through two or three more zones than the other: some of the zones looks like they are slightly gerrymandered.) Of course, in this case most sensible people would just buy the cheapest available fare and take the cheapest available route, showing that a zone-based system is much more like a Riemannian manifold (and commuters try to travel in geodesics)…