A demonstration of sorting networks being taught is available here:
Some other videos showing different situations using Sorting Networks:
As we use computers more and more, and the amount of data we use increases, we want them to process information as quickly as possible. One way to increase the speed of a computer is to write programs that use fewer computational steps (as shown in the lessons on sorting and searching algorithms). Another way to solve problems faster is to have several computers work on different parts of the same task at the same time, which is what this unit explores. Unfortunately it's not always that simple to just split the work among separate processors!
Sorting Networks are used to sort values into ascending order by comparing pairs of values; unlike a conventional sorting algorithm, a Sorting Network can have more than one comparison happening at the same time. For example, in the six-number Sorting Network that we use a lot in this unit, a total of 12 comparisons are used to sort the numbers, but up to three comparisons can be performed simultaneously. This means that the time required will be the same as what one computer by itself would take to make only 5 comparison steps. It's a bit like the situation where you might need to type in 4 pages of writing; if you have 4 people typing at the same time on 4 computers, then you can probably get the typing done 4 times faster than if one person did all the work.
A parallel Sorting Network enables us to explore how much faster we can sort values into order if we can make simultaneous comparisons. The main six-way parallel network used in these lessons sorts a list of values more than twice as quickly as a system that can only perform one comparison at a time.
Not all tasks can be completed faster by using parallel computation however. As an analogy, imagine one person digging a ditch ten metres long. If ten people each dug one metre of the ditch at the same time the task would be completed much faster. However, the same strategy could not be applied to a ditch ten metres deep, the second metre is not accessible until the first metre has been dug.
And for typing our four page document, if you have 400 people helping, you'll probably spend so much time coordinating all the work that it might not be very fast at all! Computer Scientists are still actively trying to find the best ways to break problems up so that they can be solved by computers working in parallel, finding out how much, and which parts, of the computation can be done at the same time, and which parts have to be done one after another.
In these lessons we use a fun team activity to demonstrate an approach to parallel sorting. It can be done on paper, but we like to get students to do it on a large scale, running from node to node in the network.
As an aside, although this is called a "network", it is only one of many different types of networks that we encounter in Computer Science. A common kind of “network” is a communication network, such as the telecommunication networks that mobile phones use, and of course the Internet! There are also networks for representing things like road maps and airline routes. It’s important to recognise that the Sorting Network in this activity is not one of these types of networks, it is called a comparator network, because it’s a network where each node compares two values, rather than linking different devices (such as phones and computers) together.
To use the Sorting Network students need to follow a simple algorithm and should recognise that if they do not follow this algorithm precisely, the way a computer would, then they will probably not get to a correct result, or may not get a result at all! Students will be working collaboratively to ensure that each part of the algorithm is coordinated, because if one person moves too far ahead, without stopping at the nodes they are meant to, it causes the process to fail for everyone. There are also many algorithms that are used to construct extremely efficient Sorting Networks of different sizes, and Computer Scientists study these to try and create even better ones. These can be very complex however, so when students construct their own networks they will be doing this in a simplified way.
This activity strongly supports learning about the concept of before and after (ordering) for numbers, including determining the relationship between two numbers (greater than, less than).
Often it's cheaper, and faster, to have a number of slow processors work on a computational problem, rather than one very fast one. Companies that have massive cloud servers often find it more economical to have many slower, cheaper devices rather than fewer expensive ones. Of course, this requires you to be able to split up a computational task over several processors. For some computational problems that's very easy to do, and for others it's impossible. The task we will be looking at here is between these extremes.
Having such a small operation (comparing two values) split over multiple devices means that this kind of algorithm needs to run on special hardware. It is only used for specialist applications at present, but, for example, it is sometimes done on the graphics processor (GPU) of a computer, because these processors are good at doing parallel computation of simple tasks.
Sorting Networks were invented long before powerful GPUs came along; this is an exciting thing about Computer Science - some of our discoveries are ahead of the hardware that is available, so we're ready for the hardware if it does become commonly available! Note that the approach explored in these lessons is not a conventional sorting algorithm, as the sorting that is done on a conventional system can make only one comparison at a time; conventional sorting algorithms are explored in another lesson. The main goal of these lessons is to help students explore the tradeoffs between spreading work over several processors instead of using one processor.
One approach to parallel computation that is currently popular is called "MapReduce", which is widely used in Cloud Computing systems where large amounts of computing are spread over a large number of processors.
Throughout the lessons there are links to computational thinking. Below we've noted some general links that apply to this content.
Teaching computational thinking through CSUnplugged activities supports students to learn how to describe a problem, identify what are the important details they need to solve this problem, break it down into small logical steps so that they can then create a process which solves the problem, and then evaluate this process. These skills are transferable to any other curriculum area, but are particularly relevant to developing digital systems and solving problems using the capabilities of computers.
These Computational Thinking concepts are all connected to each other and support each other, but it’s important to note that not all aspects of Computational Thinking happen in every unit or lesson. We’ve highlighted the important connections for you to observe your students in action. For more background information on what our definition of Computational Thinking is see our notes about computational thinking.
In these lessons students will be sorting a variety of things into order, but the underlying algorithm for performing these tasks will remain the same. It is an algorithm because it is a step-by-step process that will always give the right solution, as long as it is followed exactly. In this case it is a special class of algorithm called a "parallel algorithm". Students will need to follow this algorithm precisely to get to the correct solution (this is particularly clear when students try to ‘cheat’ by dashing straight to the end of the network, and then realise this means other students are now stuck in the middle! It’s a great learning opportunity when someone does this).
The Sorting Network we use in these activities is a simple representation of something much more complex: how Sorting Networks are implemented using specific hardware and software on some computers to perform parallel processing. The lines, circles, and squares we will use in our Sorting Networks hide the complicated details of the hardware and software.
Another detail we can ignore when we are using a Sorting Network is what the data we are sorting actually is, or represents. It doesn’t matter if we are sorting numbers into order, or words, or musical notes, we will still follow the same process each time. The one thing about the data that does matter however is that we can compare each item and that they have a precise way of being ordered (e.g. alphabetical order). This is described further in the section on logic.
The overall idea of a Sorting Network is actually an abstract concept as well, this is explained under the generalisation heading.
In order to create an algorithm that can solve computational problems effectively using parallel processors, we must first be able decompose the task into very small and basic operations that, when repeated many times, can build up a solution to the problem. This operation is what will be performed by each processor in the network. For the Sorting Network in these lessons this basic operation is the comparison of two values that we perform at each node. These operations need to be so basic that nodes can perform them simultaneously and independently. Parallel algorithms work best for tasks that need to do repetitive, and independent, calculations with large amounts of data.
Decomposition is one of the most important steps in creating parallel processing algorithms!
There are many links between this section and the abstraction section above, see if you can spot them!
The Sorting Networks we will look at are each constructed to take in a specific number of inputs, and that number only. We can’t use a Sorting Network that sorts 6 numbers to sort 10 numbers instead. However the generalised idea of a Sorting Network can be applied to different problems. The generalised concept of a Sorting Network is simply a comparator network (comparator just means it makes comparisons, like how we compare numbers in each of the circles in the network) that takes in a number of inputs, and sorts them into order. This general idea of a Sorting Network can then be applied to solving many different problems, by creating a Sorting Network for the specific number of inputs needed for the problem and placing its comparison nodes in a specific pattern.
There are patterns in the layout of Sorting Networks as well; recognising these helps us design larger networks. For example, the (optimal) two-way, four-way and six-way Sorting Networks follow a similar pattern in their layout. A simple pattern for generating Sorting Networks is explored at the end of lesson 3 for ages 11-14 (but this can be used with any age group if the students are interested!).
There is also a common pattern that students can observe between all the different types of information we sort with the Sorting Network, which is that they can be compared and ordered in a precise way. This is described in the logic section.
Parallel systems need to be evaluated for correctness: do they always sort values correctly? They also need to be evaluated for efficiency: how much time does this network arrangement take to sort values, and is there a faster arrangement we could use? Could this problem be solved easier by a non-parallel system?
A very important rule for the data that Sorting Networks can process is the data must have something called a transitive relation. The transitive relation means: if a is less than b, and b is less than c, then a is less than c. For example, numbers have a transitive relation: the number 5 is less than 10, and 10 is less than 15, which means that 5 must also be less than 15. Data must have this relation for a Sorting Network to be able to sort it. If items don’t have this relation then there is no logical way for us to order it!
We will also see that Sorting Networks can't be evaluated by trying every possible input (well they could be, but it could take hours, days, or even hundreds of years for big networks!), unless a very small amount of data is being sorted. So instead, we must apply logic and reason to prove why it will always sort the data correctly. In these lessons we don't get into the advanced proofs that the whole network will work, but students can apply their logical thinking skills to prove that the smallest and largest values will always end up in the correct place.