When I patent my boarding buzzer, you should be sure to file for your airline container idea. Brilliant! And what’s especially nice is that containers are “intermodal.” We can board our Paris-bound container at the train station; it will be hauled by rail to the airport and then lifted by crane from a siding onto the waiting containerized aircraft.

If we’re talking about what actually goes on in the classroom, there’s no rule requiring that the pupils make only pairwise comparisons of nearest neighbors. On the contrary, I think the process works so well partly because people in this situation tend to employ a hierarchical method. First you look around and decide whether you belong toward the front or toward the back, then within your half of the group you reach a similar judgment, and so on. Thus each child makes on the order of log2 n decisions.

But I find Barry’s approximate-sorting problem interesting, even if it doesn’t describe the “Line up alphabetically according to height” process. Surely someone has studied this?

I’ve written some very hasty code to play with the idea. The results seem to depend sensitively on the details of the sorting algorithm.

I define a predicate, ‘close-enough,’ which returns true iff two numbers differ by no more than 1.0. If I plug this predicate into the standard Lisp ’sort’ function, I get results that are not well sorted, to say the least:

9 11 10 15 1 13 17 4 19 8 3 2 12 14 18 20 6 5 7 16
15 7 8 18 16 17 13 12 20 19 1 11 10 9 6 4 5 3 2 14
12 11 9 4 2 16 15 18 17 20 13 14 19 6 5 7 8 10 3 1

It appears there’s not even an upper limit on the difference between adjacent elements. I don’t know offhand what algorithm the Lisp ’sort’ procedure uses, but I’d guess it’s some variant of quicksort.

Doing an insertion sort with the same ‘close-enough’ predicate gives results that look like this:

2 1 4 3 6 5 7 8 11 10 9 13 12 14 15 18 17 16 20 19
4 3 2 1 6 5 9 8 7 10 14 13 12 11 16 15 17 20 19 18
2 1 3 5 4 7 6 8 10 9 11 14 13 12 18 17 16 15 19 20

A version of the exchange-sort Barry recommends gives similar-looking results:

2 1 3 4 6 5 9 8 7 12 11 10 14 13 17 16 15 18 20 19
1 2 3 5 4 7 6 8 11 10 9 17 16 15 14 13 12 19 18 20
2 1 4 3 6 5 8 7 10 9 11 14 13 12 16 15 18 17 20 19

Next question: What’s a good measure of out-of-sortedness? Ordinarily I’d say the sum of (a_j – j)², where the index j runs from 1 through N. But here we seem to be looking for a more local measure—maybe simply the count of out-of-order pairs, or else something like the sum of (a_j – a_j+1)².

I suspect there’s more to say about this idea of sorting with errors or approximations. Maybe a subject for an item of its own.

I fly Southwest often, and I know the A-B-C drill, but not the numerical sequence within those groups. Perhaps it’s something added in recent months, or used only at certain airports?

I wonder why carry-on luggage is always above the seats? Why not have the bins below the seats? Would require less lifting.

There’s also the possibility of an auction, where passengers bid on their boarding order. Would solve the problem of who gets the bin and the window seat.

I wonder, what is the comparison threshold for judging relative height? I can imagine a scenario in which a large class, with a shortest-to-tallest difference of several inches, is unable to sort itself because nobody rates himself taller or shorter than the people on either side of him — one could conceivably even have the students lined up in exactly the wrong order!

To make the problem somewhat more mathematical, suppose you start with a permutation of the integers from 1 to N (say N=100), and then start transposing adjacent elements into “correct” relative order if they differ by more than 1. After all possible transpositions are made, you can take some measure of how far off from 1…N you are. One might ask for the “average error”, or some such statistic.

When boarding begins, the boarding group and either 1-30 or 31-60 is asked to line up. There are signs breaking the line into sections of 5 and rotatable signs indicating the letter. Everyone lines up in their appropriate group of 5, usually not bothering to be exactly precise (they don’t check when they scan your boarding pass)

In my experience, the technique works very well. People tend to spread out, alleviating the baggage problem. The bottleneck tends to be when all the window and aisle seats are full and people have to choose a middle seat (all Southwest flights are on 737s). Everyone is reluctant to spring for a middle seat, so they tend to move slow, looking for an aisle or window seat.

Another interesting aspect of Southwest’s system is the passengers who stay on the plane when it stops to take on new passengers. Because Southwest flights don’t just fly from one place to another and back repeatedly, some of the passengers don’t have to change planes. This means that occasionally, even if you have an “A” group boarding pass, you’ll still end up in a middle seat.

Southwest’s system is definitely the fastest I’ve used. I much prefer it to the others.

About the checking in: Check in online starts 24 hours before the flight leaves, so if you actually wait until you get to the airport to check in, you’re pretty much guaranteed a late B pass and a middle seat. There are a few websites which will allow you to check in automatically exactly 24 hours before your departure time.