Graphs and Topological Sorting in the Functional Paradigm

What is a Graph?

From Wikipedia: 

In mathematics, a graph is a representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges.

Simply put, a graph is just a bunch of points with links between them. A road map is a simple example: roads  being edges, and intersections being vertices.  In fact, Google maps uses graphs for just this purpose! Graphs are widely used in a wide variety of places. Facebook uses graphs to model your friend connections and likes. In fact, the entire internet is just a giant graph; websites act as vertices, with hyperlinks as edges. Graphs are highly useful structures, as they can be used to model many different types of situations, and as such, they will be the focus of this blog post. I am going to discuss one way to represent a graph in the Haskell programming language, and how to functionally solve a common problem using graphs.

Graphs are often represented visually like this:

Graph representing abstract data

This graph links the first six letters of the alphabet in an arbitrary way. This data doesn't really mean anything, but it will serve as a simple foray into the world of graphs, and provides an initial graph to work towards representing in Haskell.

Let's get right to it; here's the data structure we'll be using, along with some convenience methods:


First we define the actual Graph a data type: It's simply a set vertices and edges in the form of 2-tuples (The tuple (a, b) connects vertex a to vertex b), which fits our definition. I've also defined the removeEdge method, which does just what you'd expect. The outbound and inbound methods find the outbound and inbound connections to any point in the graph, respectively. They make use of the polymorphic connections method in order to get this done in a small amount of code. Finally, the Graph module exports the relevant functions at the top of the file.

Now that we've got our framework in order, we can go ahead and build the graph we mentioned above:


We import the Graph module and define a simple Letter data type, then build our Graph from it. The set of vertices are the letters A, B, C, D, E, and F, and the edges are modeled as above.

Now that we know how to build graphs, we can start modeling more important information with them.

Modeling Actual Scenarios using Graphs

Suppose some of the characters from NBC's Parks and Recreation have just finished competing in a dance competition, and we know the following about their rankings:

Leslie beat April.
April beat Ann.
Ron beat April.
Ron beat Ann.
April beat Andy.
Leslie beat Ron.
Andy beat Jerry.
Ron beat Andy.
Ann beat Jerry.
Leslie beat Andy.
Ann beat Andy.

This is a little hard to understand, so why don't we model it as a graph to make it a little more readable? Each person can be represented as a vertex, with outgoing edges representing connections to the people they beat.

A graph of dance competition results

It would be nice to be able to be able to read scenarios like this from a text file containing the important data and parse it into a graph. Let's go ahead and set up a function to do this for us, so we don't have to hard-code each and every graph that we want to use:

Here's our data file, with a list of space-separated connections, one on each line:




And our parsing function:


The graphFromFile function takes a String and returns an IO (Graph String).  The function reads a file, parses it into two important pieces: verts (the set of all unique strings in the file, or, our vertices) and conns (the set of connections between strings in the file). It then builds a Graph from this data, wraps it in the IO monad with return, and gives it back.

Now you might have been wondering from the beginning of this section what the ranking from the dance competition was (maybe you even figured it out on your own!). How do we do this programmatically, using our graph?

Enter Topological Sort

Again, from Wikipedia:

In computer science, a topological sort of a directed graph is a linear ordering of its vertices such that, for every edge uv, u comes before v in the ordering.

In our case, this just means that each person must come before all of the people that he or she beat in the competition, in the ordering.

The basic procedure for topological sort is as follows:

L = {} --sorted list
S = Set of vertices with no incoming connections
while S is not empty:
    for each vertex v in S with no incoming connections:
        push v to L
        for each edge e from v to u:
            remove e from graph
            if u has no more incoming connections:
                push u to S

if edges still exist in the graph:
    error: there is at least one cycle in the graph

else return L

If you do not understand this, I urge you to work through topologically sorting a graph on paper first; it's not too tough to understand once you've done it on paper, but can get a little confusing in psuedocode.

The problem with this algorithm is that you see a ton of loops -- control structures that we do not have in Haskell. Therefore, we must rely on recursion, folds, and maps to achieve what we want to do. Here's how it looks:



Our tsort function first finds the elements in the graph with no incoming edges using the function noInbound.  We pass this into a sub-routine tsort' that takes a sorted list l, a list of vertices with no incoming connections (n:s), and a graph g.

We operate on the first element of the set of vertices with no incoming connections n, finding outEdges (the outgoing edges from n), and outNodes (the nodes that n points to). We build a new graph g' with the outEdges removed, and find the nodes in g' with no inbound connections, and add them to s.

We then recursively call tsort' with these new parameters (and prepend our current n to the sorted list), until there are no more nodes to check. At this point, if the edge list in the graph is empty, all is well and we return the list of sorted elements. Otherwise, an error is thrown stating that there is at least one cycle in the graph.

Now that we've got that, we're ready to find out how everyone ranked in the dance competition!



This produces the output:
["Leslie", "Ron", "April", "Ann", "Andy", "Jerry"]

(Of course Jerry lost.)

Conclusion

As you can see, Graphs are very useful data structures. They can be used to model a huge variety of things (see how many more you can come up with, they're everywhere!). Topological sort in particular is a pretty remarkable algorithm, and can be applied in many different situations from the one above. For example, finding paths through college courses with prerequisites. It's even used in UNIX systems to schedule processes according to their dependencies.

Hope you enjoyed the post! Until next time,

Ben

Functional Sorting in Haskell

Sorting is a fairly major topic of study when it comes to programming, and there are tons of ways to do it. I don't know how interesting these algorithms are to other people, but they have always been an interest of mine. I think they're neat because they illustrate that there can be several ways to tackle even the simplest of tasks (which, in my opinion, is one of the things that makes programming so fun).

Since I'm always looking for new things to try, I've decided to take a few of the common sorting algorithms and implement them on my own in Haskell. For those of you who aren't familiar with it, here's a brief overview.

Most programming languages (Java, C++, Python, Ruby, Javascript, etc) are defined as imperative, which basically means that programs are executed line-by-line. Many of the aforementioned languages are also defined as object-oriented, meaning that virtually everything is a part of an object. I won't get into the details about OOP here, since we're talking about Haskell, but you can read about it if you'd like.  Haskell differs from these languages by conforming to a paradigm in which everything is defined as a function. Objects are not present here, and iterative steps don't quite work in the same way --  nothing is being executed line-by-line -- all instructions are defined in functions. This is a rather mind-blowing concept, but it's extremely cool once you can get a mental grasp on it.

Back to the point of the blog post. I'll be covering how to implement three sorting algorithms in a functional manner: Bubble Sort, Insertion Sort, and finally, Quicksort. If you don't have a background in Haskell, this might all come off as programming jargon, but if you're looking for an interesting new look at these algorithms you've seen a billion times implemented imperatively, or are interested in picking up Haskell as a hobby or something, I'd recommend reading on.

Bubble Sort:

The bubble sort algorithm is as follows:

1. Take an array's first two elements (we'll call the first one x, and the second one y)

2. If x is less than y, leave those elements alone. If not, swap y with x, so that the array now reads

[y, x, (the rest of the array)]

3. Repeat this for each element in the array, swapping as it goes along.

4. Check to see if the list is sorted. If not, rerun the algorithm on the result of the last iteration of the function.

5. When list is sorted, return the result.

Here it is in Haskell:

Rather long, but effective. Bubble Sort isn't the most efficient sorting algorithm, especially since it has to do all of that checking if it's already sorted. I'm not a big fan of this one.

Insertion Sort:
Insertion sort is pretty simple to follow. We start with an array, as always, and then follow these steps:
1. Pop the first element of the array off of the array, and populate a new array with it.
2. For each element after the first in the array, insert it at it's proper sorted location in the array.
3. Return the final list of sorted numbers.

Let's see what this looks like in Haskell:

How elegant is that? This could take many lines in an imperative language, but we make use of Haskell's recursive foldr method to populate our sorted list, and the appropriately named function insert taken from the Data.List package helps us to populate it in the exact manner we're looking for.

Quicksort:
The Quicksort algorithm can be pretty intimidating, especially at first glance. This is because it makes use of recursion to accomplish it's sorting, which is a particularly tough concept to fully understand when it comes to programming. The instructions for Quicksort look a little something like this:
1. Choose any point in the array as a "pivot" point.
2. Create two new arrays, one populated with the numbers in the array less than the pivot, and another with the numbers greater than the pivot.
3. Recursively call Quicksort on both of the lists, until these lists (inevitably) turn up empty, at which point the function returns an empty list.
4. At this point, the recursion unwinds itself, and we concatenate the recursively sorted arrays together. (I would recommend reading up on this algorithm if the above instructions do not make sense; I am not the best at explaining things)

Sounds complicated, right? Here's a Haskell implementation:

What? It only takes five lines to implement Quicksort? This is why Haskell is so cool to me. Compare this with something like, say, a Java implementation, and you'll see what I mean. It's just so simple! We're simply defining quickSort as a quickSorted list of smaller numbers concatenated with a singleton list containing a pivot, concatenated with a quickSorted list of larger numbers. Easy to read, though the algorithm may seem very complicated.

Hope you enjoyed the read!

-Ben