Fast yuletide algorithms to visit all those chimneys in time

by Paul Curzon, Queen Mary University of London

How does Santa do it? How does he visit all those children, all those chimneys, in just one night? My theory is he combines a special Scandinavian super-power with some computational wizardry.

There are about 2 billion children in the world and Santa visits them all. Clearly he has magic (flying reindeer remember) to help him do it but what kind of magic (beyond the reindeer)? And is it all about magic? Some have suggested he stops time, or moves through other dimensions, others that he just travels at an amazingly fast speed (Speedy Gonzales or The Flash style). Perhaps though he uses computer science too (though by that I don’t mean computer technology, just the power of computation).

The problem can be thought of as a computational one. The task is to visit, let’s say a billion homes (assuming an average of 2 children per household), as fast as possible. The standard solution assumes Santa visits them one at a time in order. This is what is called a linear algorithm and linear algorithms are slow. If there are n pieces of data to process (here, chimneys to descend) then we write this as having efficiency O(n). This way of writing about efficiency is called Big-O notation. O(n) just means as n increases the amount of work increases proportionately. Double the number of children and you double the workload for Santa. Currently the population doubles every 60 or 70 years or so, so clearly Santa needs to think in this way or he will eventually fail keep up, whatever magic he uses.

Perhaps, Santa uses teams of Elves as in the film Arthur Christmas, so that at each location he can deliver say presents to 1000 homes at once (though then it is the 1000 Elf helpers doing the delivering not Santa which goes against all current wisdom that Santa does it himself). It would speed things up apparently enormously to 1000 times faster. However, in computational terms that barely makes a difference. It is still a linear order of efficiency: it is still O(n) as the work still goes up proportionately with n. Double the population and Santa is in trouble still as his one night workload doubles too. O(2n) and O(1000n) both simplify to mean exactly the same as O(n). Computationally it makes little difference, and if their algorithms are to solve big problems computer scientists have to think in terms of dealing with data doubling, doubling and doubling again, just like Santa has had to over the centuries.

Divide and Conquer problem solving

When a computer scientist has a problem like this to solve, one of the first tools to reach for is called Divide and Conquer problem solving. It is a way of inventing lightening fast algorithms, that barely increase in work needed as the size of the problem doubles. The secret is to find a way to convert the problem into one that is half the size of the original, but (and this is key) that is otherwise exactly the same problem. If it is the same problem (just smaller) then that means you can solve those resulting smaller problems in the same way. You keep splitting the problem until the problems are so small they are trivial. That turns out to be a massively fast way to get a job done. It does not have to be computers doing the divide and conquer: I’ve used the approach for sorting piles of hundreds and hundreds of exam scripts into sorted order quickly, for example.

My theory is that divide and conquer is what Santa does, though it requires a particular superhero power too to work in his context, but then he is magical, so why not. How do I think it works? I think Santa is capable of duplicating himself. There is a precedent for this in the superhero world. Norse god Loki is able to copy himself to get out of scrapes, and since Santa is from the same part of the world it seems likely he could have a similar power.

If he copied himself twice then one of him could do the Northern Hemisphere and the other the Southern hemisphere. The problem has been split into an identical problem (delivering presents to lots of children) but that is half the size for each Santa (each has only half the world so half as many children to cover). That would allow him to cover the world twice as fast. However that is really no different to getting a couple of Elves to do the work. It is still O(n) in terms of the efficiency the work is done. As the population doubles he quickly ends up back in the same situation as before: too much work for each Santa. Likewise if he made a fixed number of 1000 copies of himself it would be similar to having 1000 Elves doing the deliveries. The work still increases proportional to the number of deliveries. Double the population and you still double the time it takes.

Double Santa and double again (and keep doubling)

So Santa needs to do better than that if he is to keep up with the population explosion. But divide and conquer doesn’t say halve the problem once, it says solve the new problem in the same way. So each new Santa has to copy themselves too! As they are identical copies to the original surely they can do that as easily as the first one could. Those new Santas have to do the same, and so on. They all split again and again until each has a simple problem to solve that they can just do. That might be having a single village to cover, or perhaps a single house. At that point the copying can stop and the job of delivering presents actually done. Each drops down a chimney and leaves the presents. (Now you can see how he manages to eat all those mince pies too!)

An important thing to remember is that that is not the end of it. The world is now full of Santas. Before the night is over and the job done, each Santa has to merge back with the one they split from, recursively all the way back to the original Santa. Otherwise come Christmas Day we wouldn’t be able to move for Santas. Better leave 30 minutes for that at the end!

Does this make a big difference? Well, yes (as long as all the copying can be done quickly and there is an organised way to split up the world). It makes a massive difference. The key is in thinking about how often the Santas double in number, so how often the problem is halved in size.

We start with 1 Santa who duplicates to 2, but now both can duplicate to 4, then to 8, 16, and after only 5 splittings there are already 32 Santas, then 64, 128, 256, 512 Santas, and after only 10 splittings we have over a 1000 Santas (1024 to be precise). As we saw that isn’t enough so they keep splitting. Following the same pattern, after 20 splittings we have over a million Santas to do the job. After only 30 rounds of splittings we have a billion Santas, so each can deal with a single family: a trivial problem for each.

So if a Santa can duplicate himself (along with the sleigh and reindeer) in a minute or so (Loki does it in a fraction of a second so probably this is a massive over-estimate and Santa can too), we have enough Santas to do the job in about half an hour, leaving each plenty of time to do the delivery to their destination. The splitting can also be done on the way so each Santa travels only as far as needed. Importantly this splitting process is NOT linear. It is O(log2 n) rather than O(n) and log2 n is massively smaller than n for large n. It means if we double the population of households to visit due to population explosion, the number of rounds of splitting does not need to double, the Santas just have to do one more round of splitting to cover it. The calculation log2 n (the logarithm to base 2 of n) is just a mathematicians way of saying how many times you can halve the number n before you get to 1 (or equivalently how many times you double from 1 before you get up to n). 1024 can be halved 10 times so (log2 1024) is 10. A billion can be halved about 30 times so (log2 1 billion) is about 30. Instead of a billion pieces of work we do only 30 for the splitting. Double the chimneys to 2 billion and you need only one more for a total of 31 splittings.

In computer terms divide and conquer algorithms involve methods (ie functions / procedures) calling themselves multiple times. Each call of the method, works on eg half the problem. So a method to sort data might first divide the data in half. One half is passed to one new call (copy) of the same method to sort in the same way, the other half is passed to the other call (copy). They do the same calling more copies to work on half of their part of the data, until eventually each has only one piece of data to sort (which is trivial). Work then has to be done merging the sorted halves back into sorted wholes. A billion pieces of data are sorted in only 30 rounds of recursive splitting. Double to 2 billion pieces of data and you need just 1 more round of splitting to get the sorting done.

Living in a simulation

If this mechanism for Santa to do deliveries all still seems improbable then consider that for all we know the reality of our universe may actually be a simulation (Matrix-like) in some other-dimensional computer. If so we are each just software in that simulation, each of us a method executing to make decisions about what we do in our virtual world. If that is the nature of reality, then Santa is also just a (special yuletide) software routine, and his duplicating feat is just a method calling itself recursively (as with the sort algorithm). Then the whole Christmas delivery done this way is just a simple divide and conquer algorithm running in a computer…

Given the other ways suggested for Santa to do his Christmas miracle seem even more improbable, that suggests to me that the existence of Santa provides strong evidence that we are all just software in a simulation. Not that that would make our reality, or Christmas, any less special.

More on …

• The CS4FN Christmas advent calendar

This blog is funded through EPSRC grant EP/W033615/1.