Last Tuesday, my friend Casey and I were hanging out at Aldo Coffee. We planned on enjoying some espresso, doing some work, and then heading over to the Pittsburgh Coding Dojo, where we could hang out with other geekly folks. We ended up not having enough time to go to the meeting, but we decided to hack on the challenge problem anyway, using Aldo’s ever-handy free wireless to access the Internet.
The Dojo problem was PragDave’s Kata Eleven – Sorting it Out. (It’s short; read it now.) We decided to use Haskell for our implementation language.
In this post, I’ll walk through our coding session and explain how our solution evolved. To better fit the session into a blog post, I have removed a lot of back-and-forth micro iterations, and I have edited some of the code for clarity.
The first part of the problem
The first part of the problem was “Sorting Balls.” The story: You need to implement a “rack” to hold the balls drawn at random (without replacement) from a bin containing sixty balls, numbered 0 to 59. Regardless of the order in which the balls are added to the rack, you need to present them in sorted order whenever you’re asked for them.
Upon reading this part of the challenge, a couple of thoughts sprung to mind:
- Because the range of balls is so small, the problem was begging for a solution based on a counting sort.
- Because the balls are uniquely numbered and drawn without replacement, we could even use a bit vector to represent counts.
Nevertheless, we decided to ignore these thoughts and implement a more-general solution that would work for any (orderable) values, not just small ranges of integers.
Sketching the interface
The first step, then, was to sketch out an interface. Our interface mirrored the one from the problem statement but was tweaked for Haskell:
mkRack :: Rack a add :: Ord a => a -> Rack a -> Rack a balls :: Rack a -> [a]
The function mkRack makes a new rack to hold values (“balls”) of type a. It’s equivalent to
Rack.new in Ruby.
The add function adds a ball to a rack. You give it a ball and a rack, and it returns a new rack that is the same as the original rack but also contains the ball. (If you’re accustomed to stateful programming, this may seem weird. Why return a new rack instead of modifying the original rack? Because, in Haskell, you can’t change values: you can only create new values. At first, this constraint may seem limiting, but after you get used to it, you’ll find it empowering.)
Ord a qualification on the type signature of add says that it will work for any type a whose values can be ordered. The qualification is necessary because values of some types, like IO actions, cannot be compared to see which are less than the others.
The balls function is an observer: it lets you observe the balls in a rack by returning them as an ordered list.
And that’s the interface.
With the interface sketched, we gave it meaning by defining its properties.
Giving our interface meaning: defining properties using QuickCheck
QuickCheck is a powerful, easy-to-use testing tool. Instead of checking test cases, it checks properties – statements about what your code ought to do in general.
The great thing about QuickCheck properties is that they are testable documentation. They tell the world what your code is supposed to do, and they do so in a concise, formal language that just happens to be easily readable by humans and automatically testable by computers.
To specify the desired properties of our Rack interface, we first had to import QuickCheck:
Then, we defined our first property. It said, simply, that a new rack must be empty when observed:
prop_New = balls mkRack =~ 
Our second property said that, when you add a ball x to a rack, the resulting rack must contain the same balls as the original rack plus x:
prop_AddAddsElement rack x = balls (add x rack) =~ (x : balls rack)
Both of the properties above rely upon a special, order-insensitive equality test that we defined for lists of
(=~) :: [Int] -> [Int] -> Bool xs =~ ys = sort xs == sort ys
Note that under this test,
[1,2] “equals” both
[2,1], but it does not “equal” any other values.
The reason we defined this operator was to help us specify the two essential properties of add separately: (1) it must insert a ball into a rack, and (2) the new ball’s position, when observed, must preserve the rack’s ordering invariant. The previous property definition used the
=~ operator to specify the first of these two properties. The next property we defined specified the second:
prop_AddPreservesOrdering rack x = isOrdered (balls rack) ==> isOrdered (balls (add x rack))
This definition specifies that, for all racks rack and all balls x, if the balls in rack are ordered, the balls in the rack that results from adding x to rack must also be ordered. If you are familiar with proof by induction, you’ll know why we went this route. In short, if we can prove that this property holds (and, trivially, that an empty rack is ordered), we can prove that add preserves the ordering invariant.
To round out the property definition, we needed to define the isOrdered test:
isOrdered :: [Int] -> Bool isOrdered xs = xs == sort xs
And those are the properties we needed to check the correctness of our implementation. Of course, we still needed to write our implementation, and we turned to that task next.
A simple, list-based Rack implementation
For our first implementation, we decided upon a drop-dead-simple list-based representation. We would keep the elements of the list in sorted order by inserting them into the correct positions when add was called.
Here, then, was our code:
-- Our list-based implementation of a Rack type Rack a = [a] mkRack =  add x xs = insertList x xs balls = id insertList :: Ord a => a -> [a] -> [a] insertList x  = [x] insertList x (y:ys) | x < y = x : y : ys | otherwise = y : insertList x ys
We took our new implementation for a spin in GHCi:
*Rack> balls mkRack  *Rack> balls (add 3 mkRack)  *Rack> balls (add 4 (add 3 mkRack)) [3,4] *Rack> balls (add 1 (add 4 (add 3 mkRack))) [1,3,4] *Rack> balls (foldr add mkRack [4,2,6,3,-9,0,33,9]) [-9,0,2,3,4,6,9,33]
To really test our implementation, we asked QuickCheck to check its properties:
*Rack> quickCheck prop_New OK, passed 100 tests. *Rack> quickCheck prop_AddAddsElement OK, passed 100 tests. *Rack> quickCheck prop_AddPreservesOrdering OK, passed 100 tests.
I should point out that QuickCheck did not prove that our properties held. Rather, it gathered evidence that we could use to argue that our properties held. The evidence was that each of our properties’ claims was subjected to 100 randomly generated tests, and none of the tests was able to disprove a claim.
Was this evidence sufficient for us to rest satisfied that our implementation was correct? Given how simple our implementation was, I felt that the evidence was sufficient. Casey agreed, and we moved on.
With the first implementation done, we decided to try a more-sophisticated implementation.
Generalizing the interface
Since we were about to have multiple implementations, it made sense for us to define a generalized interface that any “Rack-like” implementation could use. For that, Haskell’s type classes were perfect:
-- Our interface for "Rack-like" data types class Racklike a ra | ra -> a where mkRack :: ra add :: Ord a => a -> ra -> ra balls :: ra -> [a]
The interface was essentially the same as before, except that the data type behind the rack implementation was not given by a specific type Rack a but rather by the type variable ra, which represents some type of rack container for balls of type a.
Note that ra determines a. If, for example, you know that the container type ra equals “a list of
Int values,” you know that a must equal
Int. (To represent this relationship, we used functional dependencies, a popular extension to the Haskell 98 standard.)
With the Racklike type class in place, we moved our list-based implementation inside of the interface:
-- Our list-based implementation of a Rack type ListRack a = [a] instance Racklike a (ListRack a) where mkRack =  add = insertList balls = id
Next, we modified our QuickCheck property definitions. Where before it was fine to assume that we would be testing our single, list-based implementation, now we needed to allow for testing other implementation types. We did this by adding a rackType parameter to our property definitions. We used the type, not the value, of this parameter to determine the type of rack to test:
prop_New rackType = balls (mkRack `asTypeOf` rackType) =~  prop_AddAddsElement rackType ballList x = balls (add x rack) =~ (x : balls rack) where rack = rackFromList ballList `asTypeOf` rackType prop_AddPreservesOrdering rackType ballList x = isOrdered (balls rack) ==> isOrdered (balls (add x rack)) where rack = rackFromList ballList `asTypeOf` rackType
Because we could no longer assume the rack would be represented as a list of integers, we wrote rackFromList to convert such a list into a rack:
rackFromList xs = foldr add mkRack xs
With these modifications in place, we re-ran our tests, specifying (via type annotations) that we wanted to run them for the ListRack implementation:
*Rack> quickCheck $ prop_New (undefined :: ListRack Int) OK, passed 100 tests. *Rack> quickCheck $ prop_AddAddsElement (undefined :: ListRack Int) OK, passed 100 tests. *Rack> quickCheck $ prop_AddPreservesOrdering (undefined :: ListRack Int) OK, passed 100 tests.
A tree-based Rack implementation
Now that we were free to add additional implementation types, we created one based on binary trees. We started by defining the tree data type:
data Tree a = Empty | Root (Tree a) a (Tree a) deriving (Ord, Eq, Show)
This definition says that a tree can be either empty or a root node. A root node has a single value and left and right sub-trees.
Further, root nodes must satisfy an ordering invariant: if a root node’s value is x, all of the values in its left subtree must be less than x, and all of the values in its right subtree must be greater than or equal to x. The data type doesn’t enforce this invariant, so we would need to enforce it in our implementation.
Next, we wrote the basic functions for creating, adding elements to, and observing our trees.
We needed to be able to create empty trees:
emptyTree = Empty
Inserting an element into a tree requires us to walk the tree and append the element as a new leaf node in the correct location, being mindful of our ordering invariant. Because our data structure is inherently recursive, a recursive implementation was straightforward to code:
insertTree x Empty = Root Empty x Empty insertTree x (Root left y right) | x < y = Root (insertTree x left) y right | otherwise = Root left y (insertTree x right)
Note that we don’t try to ensure that the tree is balanced. The problem statement says that the balls are randomly selected, and thus we can expect our trees, on average, to be balanced naturally.
Next, we wrote the code to observe the elements of a tree. We used a functional-programming idiom for efficiently flattening a tree into a list:
elemsTree rx = elemsTree' rx  elemsTree' Empty = id elemsTree' (Root left x right) = elemsTree' left . (x :) . elemsTree' right
Finally, we defined a new tree-based rack type and declared it to be an instance of the Racklike type class:
type TreeRack a = Tree a instance Racklike a (TreeRack a) where mkRack = emptyTree add = insertTree balls = elemsTree
With the implementation done, we took it for a test drive:
*Rack> add 1 mkRack :: TreeRack Int Root Empty 1 Empty *Rack> add 3 (add 1 mkRack) :: TreeRack Int Root Empty 1 (Root Empty 3 Empty) *Rack> balls (add 3 (add 1 mkRack) :: TreeRack Int) [1,3]
Then, for the real test, we checked that our properties held for TreeRacks:
*Rack> quickCheck $ prop_New (undefined :: TreeRack Int) OK, passed 100 tests. *Rack> quickCheck $ prop_AddAddsElement (undefined :: TreeRack Int) OK, passed 100 tests. quickCheck $ prop_AddPreservesOrdering (undefined :: TreeRack Int) OK, passed 100 tests.
Satisfied with these results, we moved on to part two of the problem.
The second part of the problem
The second part of the problem was about sorting the letters within a block of text, ignoring white space and punctuation, and converting upper case letters into lower case: “Are there any ways to perform this sort cheaply, and without using built-in libraries?”
Again, a counting sort seemed like an obvious ideal solution, but we decided to recycle our existing code since we had to leave soon. Because our Rack implementations were generic, they would work on letters just as well as on numbers or other kinds of balls:
*Rack> balls (rackFromList "this is a test" :: TreeRack Char) " aehiisssttt"
With our existing code already doing the hard work for us, it was trivial to code up the letter-sorting function:
sortLetters xs = balls (rackFromList letters :: TreeRack Char) where letters = [toLower x | x <- xs, isAlpha x]
(Note: Because of the nature of the problem, I interpreted the question’s “without using built-in libraries” to mean “without built-in sorting libraries.”)
We took the new function for a test drive, and it worked as expected:
*Rack> sortLetters "This is a test, pal." "aaehiilpsssttt"
And that ended our coding session.
Update: Tweaked the revised definition of the AddAddsElement property for greater parallelism with the original.
Update 2007-03-03: Minor edits for clarity.