Distances in graphs

We discussed Project 3; see the lecture capture for details.

Let’s say we’re analyzing flights in the US, maybe to examine price differences between flying out of rural vs. urban airports. Our data are in CSV format:

BTV,NYC,200
BTV,BOS,150
PVD,BOS,150
MIA,LAX,400
NYC,ATL,300
ATL,MIA,300
SEA,LAX,350
NYC,SEA,400

How can we view these data as a graph? How about something like this:

How would we determine how many flights it takes to get from Burlington (BTV) to Los Angeles (SEA)? It looks like it should take three flights: one from BTV to NYC, one from NYC to SEA, and one from SEA to LAX. But there are other possible paths: for instance, we could go BTV->NYC->ATL->MIA->LAX. We probably don’t want to do that–that’s a lot of stops! So maybe what we really want is the minimum number of flights.

Let’s start by solving an easier problem. Which airports can we get to from BTV in zero flights? Just one: BTV. I promise you, this is more useful than it sounds.

Now: which airports can we get to in one flight? By looking at the graph, it looks like we can get to NYC and BOS. In other words: in one flight, we can get to all of the airports that share an edge with the airports we could get to in zero flights.

Which airports can we get to in two flights? We can start with the airports we can get to in one flight, then look at all of their neighbors. We can get to PVD and BTV from BOS, and we can get to SEA, ATL, and BTV from NYC. We probably don’t want to include BTV in our list of airports we can get to in two flights–after all, we can get there in zero flights!

Finally, which airports can we get to in 3 flights? We can look at all of the airports from the previous step and then fly to their neighbors–but only the ones we haven’t examined yet. This gets us to MIA and LAX. So, now we know that it takes three flights to get to LAX.

function search(start, end):
  distance to start = 0
  to-visit = [start]
  while nodes in to-visit:
    node = first node from to-visit
    for neighbor in neighbors:
      if we haven't yet set neighbor's distance:
        distance to neighbor = distance to node + 1
        if neighbor == end:
          return distance to neighbor
        add neighbor to to-visit