I recently learned of an entirely different algorithm for achieving the same result. (In fact, I learned of it when I gave this problem on an exam and a student came up with an unexpected solution!) This solution does not use a divide-and-conquer approach at all, but hinges on a clever data structure.
Suppose we have a bag of values (i.e. a collection where duplicates are allowed) on which we can perform the following two operations:
We’ll call the second operation a rank query because it really amounts to finding the rank or index of a given value in the bag—how many values are greater than it (and thus how many are less than or equal to it)?
If we can do these two operations in logarithmic time (i.e. logarithmic in the number of values in the bag), then we can count inversions in time. Can you see how before reading on? You might also like to think about how we could actually implement a data structure that supports these operations.
So, let’s see how to use a bag with logarithmic insertion and rank queries to count inversions. Start with an empty bag. For each element in the sequence, see how many things in the bag are strictly greater than it, and add this count to a running total; then insert the element into the bag, and repeat with the next element. That is, for each element we compute the number of inversions of which it is the right end, by counting how many elements that came before it (and are hence in the bag already) are strictly greater than it. It’s easy to see that this will count every inversion exactly once. It’s also easy to see that it will take time: for each of the elements, we do two operations (one rank query and one insertion).
In fact, we can do a lot more with this data structure than just count inversions; it sometimes comes in handy for competitive programming problems. More in a future post, perhaps!
So how do we implement this magical data structure? First of all, we can use a balanced binary search tree to store the values in the bag; clearly this will allow us to insert in logarithmic time. However, a plain binary search tree wouldn’t allow us to quickly count the number of values strictly greater than a given query value. The trick is to augment the tree so that each node also caches the size of the subtree rooted at that node, being careful to maintain these counts while inserting and balancing.
Let’s see some code! In Haskell, probably the easiest type of balanced BST to implement is a red-black tree. (If I were implementing this in an imperative language I might use splay trees instead, but they are super annoying to implement in Haskell. (At least as far as I know. I will definitely take you out for a social beverage of your choice if you can show me an elegant Haskell implementation of splay trees! This is cool but somehow feels too complex.)) However, this isn’t going to be some fancy, type-indexed, correct-by-construction implementation of red-black trees, although that is certainly fun. I am actually going to implement left-leaning red-black trees, mostly following Sedgewick; see those slides for more explanation and proof. This is one of the simplest ways I know to implement red-black trees (though it’s not necessarily the most efficient).
First, a red-black tree is either empty, or a node with a color (which we imagine as the color of the incoming edge), a cached size, a value, and two subtrees.
> {-# LANGUAGE PatternSynonyms #-}
>
> data Color = R | B
> deriving Show
>
> otherColor :: Color -> Color
> otherColor R = B
> otherColor B = R
>
> data RBTree a
> = Empty
> | Node Color Int (RBTree a) a (RBTree a)
> deriving Show
To make some of the tree manipulation code easier to read, we make some convenient patterns for matching on the structure of a tree when we don’t care about the values or cached sizes: ANY
matches any tree and its subtrees, while RED
and BLACK
only match on nodes of the appropriate color. We also make a function to extract the cached size
of a subtree.
> pattern ANY l r <- Node _ _ l _ r
> pattern RED l r <- Node R _ l _ r
> pattern BLACK l r <- Node B _ l _ r
>
> size :: RBTree a -> Int
> size Empty = 0
> size (Node _ n _ _ _) = n
The next thing to implement is the workhorse of most balanced binary tree implementations: rotations. The fiddliest bit here is managing the cached sizes appropriately. When rotating, the size of the root node remains unchanged, but the new child node, as compared to the original, has lost one subtree and gained another. Note also that we will only ever rotate around red edges, so we pattern-match on the color as a sanity check, although this is not strictly necessary. The error
cases below should never happen.
> rotateL :: RBTree a -> RBTree a
> rotateL (Node c n t1 x (Node R m t2 y t3))
> = Node c n (Node R (m + size t1 - size t3) t1 x t2) y t3
> rotateL _ = error "rotateL on non-rotatable tree!"
>
> rotateR :: RBTree a -> RBTree a
> rotateR (Node c n (Node R m t1 x t2) y t3)
> = Node c n t1 x (Node R (m - size t1 + size t3) t2 y t3)
> rotateR _ = error "rotateR on non-rotatable tree!"
To recolor
a node, we just flip its color. We can then split
a tree with two red subtrees by recoloring all three nodes. (The “split” terminology comes from the isomorphism between red-black trees and 2-3-4 trees; red edges can be thought of as “gluing” nodes together into a larger node, and this recoloring operation corresponds to splitting a 4-node into three 2-nodes.)
> recolor :: RBTree a -> RBTree a
> recolor Empty = Empty
> recolor (Node c n l x r) = Node (otherColor c) n l x r
>
> split :: RBTree a -> RBTree a
> split (Node c n l@(RED _ _) x r@(RED _ _))
> = (Node (otherColor c) n (recolor l) x (recolor r))
> split _ = error "split on non-splittable tree!"
Finally, we implement a function to “fix up” the invariants by doing rotations as necessary: if we have two red subtrees we don’t touch them; if we have only one right red subtree we rotate it to the left (this is where the name “left-leaning” comes from), and if we have a left red child which itself has a left red child, we rotate right. (This function probably seems quite mysterious on its own; see Sedgewick for some nice pictures which explain it very well!)
> fixup :: RBTree a -> RBTree a
> fixup t@(ANY (RED _ _) (RED _ _)) = t
> fixup t@(ANY _ (RED _ _)) = rotateL t
> fixup t@(ANY (RED (RED _ _) _) _) = rotateR t
> fixup t = t
We can finally implement insertion. First, to insert into an empty tree, we create a red node with size 1.
> insert :: Ord a => a -> RBTree a -> RBTree a
> insert a Empty = Node R 1 Empty a Empty
If we encounter a node with two red children, we perform a split before continuing. This may violate the red-black invariants above us, but we will fix it up later on our way back up the tree.
> insert a t@(ANY (RED _ _) (RED _ _)) = insert a (split t)
Otherwise, we compare the element to be inserted with the root, insert on the left or right as appropriate, increment the cached size, and fixup
the result. Notice that we don’t stop recursing upon encountering a value that is equal to the value to be inserted, because our goal is to implement a bag rather than a set. Here I have chosen to put values equal to the root in the left subtree, but it really doesn’t matter.
> insert a (Node c n l x r)
> | a <= x = fixup (Node c (n+1) (insert a l) x r)
> | otherwise = fixup (Node c (n+1) l x (insert a r))
Now, thanks to the cached sizes, we can count the values greater than a query value.
> numGT :: Ord a => RBTree a -> a -> Int
The empty tree contains 0 values strictly greater than anything.
> numGT Empty _ = 0
For a non-empty tree, we distinguish two cases:
> numGT (Node _ n l x r) q
If the query value q
is less than the root, then we know that the root along with everything in the right subtree is strictly greater than q
, so we can just add 1 + size r
without recursing into the right subtree. We also recurse into the left subtree to count any values greater than q
it contains.
> | q < x = numGT l q + 1 + size r
Otherwise, if q
is greater than or equal to the root, any values strictly greater than q
must be in the right subtree, so we recurse to count them.
> | otherwise = numGT r q
By inspection we can see that numGT
calls itself at most once, moving one level down the tree with each recursive call, so it makes a logarithmic number of calls, with only a constant amount of work at each call—thanks to the fact that size
takes only constant time to look up a cached value.
Finally, we can put together the pieces to count inversions. The code is quite simple: recurse through the list with an accumulating red-black tree, doing a rank query on each value, and sum the results.
> inversions :: Ord a => [a] -> Int
> inversions = go Empty
> where
> go _ [] = 0
> go t (a:as) = numGT t a + go (insert a t) as
Let’s try it out!
λ> inversions [3,5,1,4,2]
6
λ> inversions [2,2,2,2,2,1]
5
λ> :set +s
λ> inversions [3000, 2999 .. 1]
4498500
(0.19 secs, 96,898,384 bytes)
It seems to work, and is reasonably fast!
Further augment each node with a counter representing the number of copies of the given value which are contained in the bag, and maintain the invariant that each distinct value occurs in only a single node.
Rewrite inversions
without a recursive helper function, using a scan, a zip, and a fold.
It should be possible to implement bags with rank queries using fingertrees instead of building our own custom balanced tree type (though it seems kind of overkill).
My intuition tells me that it is not possible to count inversions faster than . Prove it.
Euler’s proof is clever, incisive, not hard to understand, and a great introduction to the kind of abstract reasoning we can do about graphs. There’s little wonder that it is often used as one of the first nontrivial graph theory results students are introduced to, e.g. in a discrete mathematics course. (Indeed, I will be teaching discrete mathematics in the spring and certainly plan to talk about Eulerian paths!)
Euler’s 1735 solution was not constructive, and in fact he really only established one direction of the “if and only if”:
If a graph has an Eulerian path, then it has exactly zero or two vertices with odd degree.
This can be used to rule out the existence of Eulerian paths in graphs without the right vertex degrees, which was Euler’s specific motivation. However, one suspects that Euler knew it was an if and only if, and didn’t write about the other direction (if a graph has exactly zero or two vertices with odd degree, then it has an Eulerian path) because he thought it was trivial.^{1}
The first person to publish a full proof of both directions, including an actual algorithm for finding an Eulerian path, seems to be Carl Hierholzer, whose friend published a posthumous paper in Hierholzer’s name after his untimely death in 1871, a few weeks before his 31st birthday.^{2} (Notice that this was almost 150 years after Euler’s original paper!) If the vertex degrees cooperate, finding an Eulerian path is almost embarrassingly easy according to Hierholzer’s algorithm: starting at one of the odd-degree vertices (or anywhere you like if there are none), just start walking through the graph—any which way you please, it doesn’t matter!—visiting each edge at most once, until you get stuck. Then pick another part of the graph you haven’t visited, walk through it randomly, and splice that path into your original path. Repeat until you’ve explored the whole graph. And generalizing all of this to directed graphs isn’t much more complicated.
So, in summary, this is a well-studied problem, solved hundreds of years ago, that we present to students as a first example of a nontrivial yet still simple-to-understand graph proof and algorithm. So it should be pretty easy to code, right?
Recently I came across the eulerianpath problem on Open Kattis, and I realized that although I have understood this algorithm on a theoretical level for almost two decades (I almost certainly learned it as a young undergraduate), I have never actually implemented it! So I set out to solve it.
Right away the difficulty rating of 5.7 tells us that something strange is going on. “Easy” problems—the kind of problems you can give to an undergraduate at the point in their education when they might first be presented with the problem of finding Eulerian paths—typically have a difficulty rating below 3. As I dove into trying to implement it, I quickly realized two things. First of all, given an arbitrary graph, there’s a lot of somewhat finicky work that has to be done to check whether the graph even has an Eulerian path, before running the algorithm proper:
And if the graph is directed—as it is in the eulerianpath problem on Kattis—then the above steps get even more finicky. In step 1, we have to count the in- and outdegree of each vertex separately; in step 2, we have to check that the in- and outdegrees of all vertices are equal, except for possibly two vertices where one of them has exactly one more outgoing than incoming edge (which must be the start vertex), and vice versa for the other vertex; in step 4, we have to make sure to start the DFS from the chosen start vertex, because the graph need not be strongly connected, it’s enough for the entire graph to be reachable from the start vertex.
The second thing I realized is that Hierholzer’s algorithm proper—walk around until getting stuck, then repeatedly explore unexplored parts of the graph and splice them into the path being built—is still rather vague, and it’s nontrivial to figure out how to do it, and what data structures to use, so that everything runs in time linear in the number of edges. For example, we don’t want to iterate over the whole graph—or even just the whole path built so far—to find the next unexplored part of the graph every time we get stuck. We also need to be able to do the path splicing in constant time; so, for example, we can’t just store the path in a list or array, since then splicing in a new path segment would require copying the entire path after that point to make space. I finally found a clever solution that pushes the nodes being explored on a stack; when we get stuck, we start popping nodes, placing them into an array which will hold the final path (starting from the end), and keep popping until we find a node with an unexplored outgoing edge, then switch back into exploration mode, pushing things on the stack until we get stuck again, and so on. But this is also nontrivial to code correctly since there are many lurking off-by-one errors and so on. And I haven’t even talked about how we keep track of which edges have been explored and quickly find the next unexplored edge from a vertex.
I think it’s worth writing another blog post or two with more details of how the implementation works, both in an imperative language and in a pure functional language, and I may very well do just that. But in any case, what is it about this problem that results in such a large gap between the ease of understanding its solution theoretically, and the difficulty of actually implementing it?
Actually, the way I have stated the other direction of the if and only if is technically false!—can you spot the reason why?
Though apparently someone named Listing published the basic idea of the proof, with some details omitted, some decades earlier. I’ve gotten all this from Herbert Fleischner, Eulerian Graphs and Related Topics, Annals of Discrete Mathematics 45, Elsevier 1990. Fleischner reproduces Euler’s original paper as well as Hierholzer’s, together with English translations.
Scanner
combinator library for lightweight input parsing. It uses String
everywhere, and usually this is fine, but occasionally it’s not.
A good example is the Kattis problem Army Strength (Hard). There are a number of separate test cases; each test case consists of two lines of positive integers which record the strengths of monsters in two different armies. Supposedly the armies will have a sequence of battles, where the weakest monster dies each time, with some complex-sounding rules about how to break ties. It sounds way more complicated than it really is, though: a bit of thought reveals that to find out who wins we really just need to see which army’s maximum-strength monster is strongest.
So our strategy for each test case is to read in the two lists of integers, find the maximum of each list, and compare. Seems pretty straightforward, right? Something like this:
import Control.Arrow
import Data.List.Split
main = interact $
lines >>> drop 1 >>> chunksOf 4 >>>
map (drop 2 >>> map (words >>> map read) >>> solve) >>>
unlines
solve :: [[Int]] -> String
solve [gz, mgz] = case compare (maximum gz) (maximum mgz) of
LT -> "MechaGodzilla"
_ -> "Godzilla"
Note I didn’t actually use the Scanner
abstraction here, though I could have; it’s actually easier to just ignore the numbers telling us how many test cases there are and the length of each line, and just split up the input by lines and go from there.
This seems straightforward enough, but sadly, it results in a Time Limit Exceeded (TLE) error on the third of three test cases. Apparently this program takes longer than the allowed 1 second. What’s going on?
If we look carefully at the limits for the problem, we see that there could be up to 50 test cases, each test case could have two lists of length , and the numbers in the lists can be up to . If all those are maxed out (as they probably are in the third, secret test case), we are looking at an input file many megabytes in size. At this point the time to simply read the input is a big factor. Reading the input as a String
has a lot of overhead: each character gets its own cons cell; breaking the input into lines and words requires traversing over these cons cells one by one. We need a representation with less overhead.
Now, if this were a real application, we would reach for Text
, which is made for representing textual information and can correctly handle unicode encodings and all that good stuff. However, this isn’t a real application: competitive programming problems always limit the input and output strictly to ASCII, so characters are synonymous with bytes. Therefore we will commit a “double no-no”: not only are we going to use ByteString
to represent text, we’re going to use Data.ByteString.Lazy.Char8
which simply assumes that each 8 bits is one character. As explained in a previous post, however, I think this is one of those things that is usually a no-no but is completely justified in this context.
Let’s start by just replacing some of our string manipulation with corresponding ByteString
versions:
import Control.Arrow
import qualified Data.ByteString.Lazy.Char8 as C
import Data.List.Split
main = C.interact $
C.lines >>> drop 1 >>> chunksOf 4 >>>
map (drop 2 >>> map (C.words >>> map (C.unpack >>> read)) >>> solve) >>>
C.unlines
solve :: [[Int]] -> C.ByteString
solve [gz, mgz] = case compare (maximum gz) (maximum mgz) of
LT -> C.pack "MechaGodzilla"
_ -> C.pack "Godzilla"
This already helps a lot: this version is actually accepted, taking 0.66 seconds. (Note there’s no way to find out how long our first solution would take if allowed to run to completion: once it goes over the time limit Kattis just kills the process. So we really don’t know how much of an improvement this is, but hey, it’s accepted!)
But we can do even better: it turns out that read
also has a lot of overhead, and if we are specifically reading Int
values we can do something much better. The ByteString
module comes with a function
readInt :: C.ByteString -> Maybe (Int, C.ByteString)
Since, in this context, we know we will always get an integer with nothing left over, we can replace C.unpack >>> read
with C.readInt >>> fromJust >>> fst
. Let’s try it:
import Control.Arrow
import qualified Data.ByteString.Lazy.Char8 as C
import Data.List.Split
import Data.Maybe (fromJust)
main = C.interact $
C.lines >>> drop 1 >>> chunksOf 4 >>>
map (drop 2 >>> map (C.words >>> map readInt) >>> solve) >>>
C.unlines
where
readInt = C.readInt >>> fromJust >>> fst
solve :: [[Int]] -> C.ByteString
solve [gz, mgz] = case compare (maximum gz) (maximum mgz) of
LT -> C.pack "MechaGodzilla"
_ -> C.pack "Godzilla"
Now we’re talking — this version completes in a blazing 0.04 seconds!
We can take these principles and use them to make a variant of the Scanner
module from last time which uses (lazy, ASCII) ByteString
instead of String
, including the use of the readInt
functions to read Int
values quickly. You can find it here.
Data.Enumeration
module defines a type Enumeration a
, represented simply by a function Integer -> a
which picks out the value of type a
at a given index. This representation has a number of advantages, including the ability to quickly index into very large enumerations, and the convenience that comes from having Functor
, Applicative
, and Alternative
instances for Enumeration
.
I’ve just uploaded version 0.2 of the package, which adds a new Data.Enumeration.Invertible
module with a new type, IEnumeration a
, representing invertible enumerations. Whereas a normal enumeration is just a function from index to value, an invertible enumeration is a bijection between indices and values. In particular, alongside the Integer -> a
function for picking out the value at an index, an invertible enumeration also stores an inverse function a -> Integer
(called locate
) for finding the index of a given value.
On the one hand, this comes at a cost: because the type parameter a
now occurs both co- and contravariantly, IEnumeration
i s no longer an instance of Functor
, Applicative
, or Alternative
. There is a mapE
combinator provided for mapping IEnumeration a
to IEnumeration b
, but in order to work it needs both an a -> b
function and an inverse b -> a
.
On the other hand, we also gain something: of course the ability to look up the index of a value is nifty, and beyond that we also get a combinator
functionOf :: IEnumeration a -> IEnumeration b -> IEnumeration (a -> b)
which works as long as the IEnumeration a
is finite. This is not possible to implement with normal, non-invertible enumerations: we have to take an index and turn it into a function a -> b
, but that function has to take an a
as input and decide what to do with it. There’s nothing we can possibly do with a value of type a
unless we have a way to connect it back to the IEnumeration a
it came from.
Here’s a simple example of using the functionOf
combinator to enumerate all Bool -> Bool
functions, and then locating the index of not
:
>>> bbs = functionOf (boundedEnum @Bool) (boundedEnum @Bool)
>>> card bbs
Finite 4
>>> locate bbs not
2
>>> map (select bbs 2) [False, True]
[True,False]
And here’s an example of enumerating recursive trees, which is parallel to an example given in my previous post. Note, however, how we can no longer use combinators like <$>
, <*>
, and <|>
, but must explicitly use <+>
(disjoint sum of enumerations) and ><
(enumeration product) in combination with mapE
. In return, though, we can find the index of any given tree in addition to selecting trees by index.
data Tree = L | B Tree Tree
deriving Show
toTree :: Either () (Tree, Tree) -> Tree
toTree = either (const L) (uncurry B)
fromTree :: Tree -> Either () (Tree, Tree)
fromTree L = Left ()
fromTree (B l r) = Right (l,r)
trees :: IEnumeration Tree
trees = infinite $ mapE toTree fromTree (unit <+> (trees >< trees))
>>> locate trees (B (B L (B L L)) (B (B L (B L L)) (B L (B L L))))
123
>>> select trees 123
B (B L (B L L)) (B (B L (B L L)) (B L (B L L)))
Of course, the original Data.Enumeration
module remains available; there is clearly an inherent tradeoff to invertibility, and you are free to choose either style depending on your needs. Other than the tradeoffs outlined above and a couple other minor exceptions, the two modules export largely identical APIs.
lines
and words
. There is another common class of problems, however, which follow this pattern:
The first line of the input consists of an integer . Each of the next lines consists of…
That is, the input contains integers which are not input data per se but just tell you how many things are to follow. This is really easy to process in an imperative language like Java or C++. For example, in Java we might write code like this:
Scanner in = new Scanner(System.in);
int T = in.nextInt();
for (int i = 0; i < T; i++) {
// process each line
}
Occasionally, we can get away with completely ignoring the extra information in Haskell. For example, if the input consists of a number followed by lines, each of which contains a number followed by a list of numbers, we can just write
main = interact $
lines >>> drop 1 >>> map (words >>> drop 1 >>> map read) >>> ...
That is, we can ignore the first line containing since the end-of-file will tell us how many lines there are; and we can ignore the at the beginning of each line, since the newline character tells us when the list on that line is done.
Sometimes, however, this isn’t possible, especially when there are multiple test cases, or when a single test case has multiple parts, each of which can have a variable length. For example, consider Popular Vote, which describes its input as follows:
The first line of input contains a single positive integer indicating the number of test cases. The first line of each test case also contains a single positive integer indicating the number of candidates in the election. This is followed by lines, with the th line containing a single nonnegative integer indicating the number of votes candidate received.
How would we parse this? We could still ignore —just keep reading until the end of the file—but there’s no way we can ignore the values. Since the values for each test case are all on separate lines instead of on one line, there’s otherwise no way to know when one test case ends and the next begins.
Once upon a time, I would have done this using splitAt
and explicit recursion, like so:
type Election = [Int]
readInput :: String -> [Election]
readInput = lines >>> drop 1 {- ignore T -} >>> map read >>> go
where
go :: [Int] -> [Election]
go [] = []
go (n:xs) = votes : go rest
where (votes,rest) = splitAt n xs
However, this is really annoying to write and easy to get wrong. There are way too many variable names to keep track of (n
, xs
, votes
, rest
, go
) and for more complex inputs it becomes simply unmanageable. You might think we should switch to using a real parser combinator library—parsec
is indeed installed in the environment Kattis uses to run Haskell solutions—and although sometimes a full-blown parser combinator library is needed, in this case it’s quite a bit more heavyweight than we would like. I can never remember which modules I have to import to get parsec
set up; there’s a bunch of boilerplate needed to set up a lexer; and so on. Using parsec
is only worth it if we’re parsing something really complex.
The heart of the issue is that we want to be able to specify a high-level description of the sequence of things we expect to see in the input, without worrying about managing the stream of tokens explicitly. Another key insight is that 99% of the time, we don’t need the ability to deal with parse failure or the ability to parse multiple alternatives. With these insights in mind, we can create a very simple Scanner
abstraction, which is just a State
ful computation over a list of tokens:
type Scanner = State [String]
runScanner :: Scanner a -> String -> a
runScanner s = evalState s . words
To run a scanner, we just feed it the entire input as a String
, which gets chopped into tokens using words
. (Of course in some scenarios we might want to use lines
instead of words
, or even do more complex tokenization.)
Note since Scanner
is just a type synonym for State [String]
, it is automatically an instance of Functor
, Applicative
, and Monad
(but not Alternative
).
So let’s develop a little Scanner
DSL. The most fundamental thing we can do is read the next token.
str :: Scanner String
str = get >>= \case { s:ss -> put ss >> return s }
(This uses the LambdaCase
extension, though we could easily rewrite it without.) str
gets the current list of tokens, puts it back without the first token, and returns the first token. Note that I purposely didn’t include a case for the empty list. You might think we want to include a case for the empty token list and have it return the empty string or something like that. But since the input will always be properly formatted, if this scenario ever happens it means my program has a bug—e.g. perhaps I misunderstood the description of the input format. In this scenario I want it to crash loudly, as soon as possible, rather than continuing on with some bogus data.
We can now add some scanners for reading specific token types other than String
, simply by mapping the read
function over the output of str
:
int :: Scanner Int
int = read <$> str
integer :: Scanner Integer
integer = read <$> str
double :: Scanner Double
double = read <$> str
Again, these will crash if they see a token in an unexpected format, and that is a very deliberate choice.
Now, as I explained earlier, a very common pattern is to have an integer followed by copies of something. So let’s make a combinator to encapsulate that pattern:
numberOf :: Scanner a -> Scanner [a]
numberOf s = int >>= flip replicateM s
numberOf s
expects to first see an Int
value , and then it runs the provided scanner times, returning a list of the results.
It’s also sometimes useful to have a way to repeat a Scanner
some unknown number of times until encountering EOF (for example, the input for some problems doesn’t specify the number of test cases up front the way that Popular Vote does). This is similar to the many
combinator from Alternative
.
many :: Scanner a -> Scanner [a]
many s = get >>= \case { [] -> return []; _ -> (:) <$> s <*> many s }
many s
repeats the scanner s
as many times as it can, returning a list of the results. In particular it first peeks at the current token list to see if it is empty. If so, it returns the empty list of results; if there are more tokens, it runs s
once and then recursively calls many s
, consing the results together.
Finally, it’s quite common to want to parse a specific small number of something, e.g. two double values representing a 2D coordinate pair. We could just write replicateM 2 double
, but this is common enough that I find it helpful to define dedicated combinators with short names:
two, three, four :: Scanner a -> Scanner [a]
[two, three, four] = map replicateM [2..4]
The complete file can be found on GitHub. As I continue this series I’ll be putting more code into that repository. Note I do not intend to make this into a Hackage package, since that wouldn’t be useful: you can’t tell Kattis to go download a package from Hackage before running your submission. However, it is possible to submit multiple files at once, so you can include Scanner.hs
in your submission and just import Scanner
at the top of your main module.
So what have we gained? Writing the parser for Popular Vote is now almost trivial:
type Election = [Int]
main = interact $ runScanner elections >>> ...
elections :: Scanner [Election]
elections = numberOf (numberOf int)
In practice I would probably just inline the definition of elections
directly: interact $ runScanner (numberOf (numberOf int)) >>> ...
As a slightly more involved example, chosen almost at random, consider Board Wrapping:
On the first line of input there is one integer, , giving the number of test cases (moulds) in the input. After this line, test cases follow. Each test case starts with a line containing one integer , which is the number of boards in the mould. Then lines follow, each with five floating point numbers where and . The and are the coordinates of the center of the board and and are the width and height of the board, respectively. is the angle between the height axis of the board to the -axis in degrees, positive clockwise.
Here’s how I would set up the input, using Scanner
and a custom data type to represent boards.
import Scanner
type V = [Double] -- 2D vectors/points
newtype A = A Double -- angle (radians)
-- newtype helps avoid conversion errors
fromDeg :: Double -> A
fromDeg d = A (d * pi / 180)
data Board = Board { boardLoc :: V, boardDims :: V, boardAngle :: A }
board :: Scanner Board
board = Board
<$> two double
<*> two double
<*> ((fromDeg . negate) <$> double)
main = interact $
runScanner (numberOf (numberOf board)) >>> ...
]]>universe
(and related packages) is very nice, but it’s focused on enumerating values of Haskell data types, not arbitrary sets: since it uses type classes, you have to make a new Haskell type for each thing you want to enumerate. It also uses actual Haskell lists of values, which doesn’t play nicely with sampling.enumerable
has not been updated in a long time and seems to be superseded by universe
.enumerate
is likewise focused on generating values of Haskell data types, with accompanying generic deriving machinery.size-based
is used as the basis for the venerable testing-feat
library, but these are again focused on generating values of Haskell data types. I’m also not sure I need the added complexity of size-indexed enumerations.enumeration
looks super interesting, and I might be able to use it for what I want, but (a) I’m not sure whether it’s maintained anymore, and (b) it seems rather more complex than I need.I really want something like Racket’s nice data/enumerate
package, but nothing like that seems to exist in Haskell. So, of course, I made my own! For now you can find it on GitHub.^{1} Here’s the package in a nutshell:
Enumeration
, which is an instance of Functor
, Applicative
, and Alternative
(but not Monad
).I wrote about something similar a few years ago. The main difference is that in that post I limited myself to only finite enumerations. There’s a lot more I could say but for now I think I will just show some examples:
>>> enumerate empty
[]
>>> enumerate unit
[()]
>>> enumerate $ empty <|> unit <|> unit
[(),()]
>>> enumerate $ finite 4 >< finiteList [27,84,17]
[(0,27),(0,84),(0,17),(1,27),(1,84),(1,17),(2,27),(2,84),(2,17),(3,27),(3,84),(3,17)]
>>> select (finite 4000000000000 >< finite 123456789) 0
(0,0)
>>> select (finite 4000000000000 >< finite 123456789) 196598723084073
(1592449,82897812)
>>> card (finite 4000000000000 >< finite 123456789)
Finite 493827156000000000000
>>> :set -XTypeApplications
>>> enumerate $ takeE 26 . dropE 65 $ boundedEnum @Char
"ABCDEFGHIJKLMNOPQRSTUVWXYZ"
>>> take 10 . enumerate $ nat >< nat
[(0,0),(0,1),(1,0),(0,2),(1,1),(2,0),(0,3),(1,2),(2,1),(3,0)]
>>> take 10 . enumerate $ cw
[1 % 1,1 % 2,2 % 1,1 % 3,3 % 2,2 % 3,3 % 1,1 % 4,4 % 3,3 % 5]
>>> take 15 . enumerate $ listOf nat
[[],[0],[0,0],[1],[0,0,0],[1,0],[2],[0,1],[1,0,0],[2,0],[3],[0,0,0,0],[1,1],[2,0,0],[3,0]]
data Tree = L | B Tree Tree
deriving (Eq, Show)
trees :: Enumeration Tree
trees = infinite $ singleton L <|> B <$> trees <*> trees
>>> take 3 . enumerate $ trees
[L,B L L,B L (B L L)]
>>> select trees 87239862967296
B (B (B (B (B L L) (B (B (B L L) L) L)) (B L (B L (B L L)))) (B (B (B L (B L (B L L))) (B (B L L) (B L L))) (B (B L (B L (B L L))) L))) (B (B L (B (B (B L (B L L)) (B L L)) L)) (B (B (B L (B L L)) L) L))
treesOfDepthUpTo :: Int -> Enumeration Tree
treesOfDepthUpTo 0 = singleton L
treesOfDepthUpTo n = singleton L <|> B <$> t' <*> t'
where t' = treesOfDepthUpTo (n-1)
>>> card (treesOfDepthUpTo 0)
Finite 1
>>> card (treesOfDepthUpTo 1)
Finite 2
>>> card (treesOfDepthUpTo 3)
Finite 26
>>> card (treesOfDepthUpTo 10)
Finite
14378219780015246281818710879551167697596193767663736497089725524386087657390556152293078723153293423353330879856663164406809615688082297859526620035327291442156498380795040822304677
>>> select (treesOfDepthUpTo 10) (2^50)
B L (B L (B L (B (B L (B (B L (B (B L L) L)) (B (B (B (B L L) (B L L)) (B L (B L L))) (B (B (B L L) L) (B (B L L) L))))) (B (B (B (B (B (B L L) L) (B (B L L) L)) (B L L)) (B (B (B (B L L) L) (B L (B L L))) (B (B (B L L) (B L L)) L))) (B (B (B (B L L) (B L L)) (B (B (B L L) L) L)) (B (B L L) (B (B (B L L) L) (B (B L L) L))))))))
Comments, questions, suggestions for additional features, etc. are all very welcome!
What is good code style? You probably have some opinions about this. In fact, I’m willing to bet you might even have some very strong opinions about this; I know I do. Whether consciously or not, we tend to frame good coding practices as a moral issue. Following good coding practices makes us feel virtuous; ignoring them makes us feel guilty. I can guess that this is why Yom said “I don’t think I could bring myself to be satisfied with partial functions” [emphasis added]. And this is why we say “good code style”, not “optimal” or “rational” or “best practice” code style.
Why is this? Partly, it is just human: we like to have right and wrong ways to do everything (load the dishwasher, enforce grammar “rules”, use a text editor, etc.), and we naturally create and enforce community standards via subtle and not-so-subtle social cues. In the case of coding practices, I think we also sometimes do it consciously and explicitly, because the benefits can be unintuitive or only manifest in the long term. So the only way to get our students—or ourselves—to follow practices that are in our rational self-interest is by framing them in moral terms; rational arguments do not work in and of themselves. For example, I cannot get my students to write good comments by explaining to them how it will be beneficial to them in the future. It seems obvious to them that they will remember perfectly how their code works in the future, so any argument claiming the opposite falls on deaf ears. The only way to get them to write comments is to make it a moral issue: they should feel bad (i.e. lose points, lose respect, feel like they are “taking shortcuts”) if they don’t. Of course I do this “for their own good”: I trust that in the future they will come to appreciate this ingrained behavior on its own merits.
The problem is that things framed in moral terms become absolutes, and it is then difficult for us to assess them rationally. My students will never be able to write a [function without comments, partial function, goto
statement, …] without feeling bad about it, and they probably won’t stop to think about why.
I ask again: what is good code style—and why? I have identified a few reasons for various “good” coding practices. Ultimately, we want our code to have properties such as:
String
vs Text
or ByteString
).Even in scenarios where one might initially think these properties are not needed (e.g. writing a one-off script for some sysadmin or data processing task), they often end up being important anyway (e.g. that one-off script gets copied and mutated until it becomes a key piece of some production system). And this is exactly one of the reasons for framing good coding style in moral terms! I won’t write comments or use good function decomposition in my one-off script just because I know, rationally, that it might end up in a production system someday. (I “know” that this particular script really is just a one-off script!) But I just might follow good coding practices anyway if I feel bad about not doing it (e.g. I would feel ashamed if other people saw it).
It seems to me that most things we would typically think of as good code style are geared towards producing code with some or all of the above properties (and perhaps some other properties as well), and most scenarios in which code is being written really do benefit from these properties.
But what if there was a scenario where these properties are actually, concretely of no benefit? As you can probably guess, I would argue that competitive programming is one such scenario:
So what do we care about?
The combination of optimizing for speed and not caring about things like robustness, maintainability, and efficiency leads to a number of “best practices” for competitive programming that fly in the face of typical standards. For example:
read
, head
, tail
, fromJust
, and so on, even though I would almost never use these functions in other contexts. This is also why I used a partial function that was only defined on lists of length two in my previous post (though as I argue in a comment, perhaps it’s not so much that the function is partial as that its type is too big).String
for text processing, even though something like Text
or ByteString
(depending on the scenario) would be faster or more robust. (The exception is problems with a large amount of I/O, when the overhead of String
really does become a problem; more on this in a future post.)foldr
, foldl'
, and scanl
, I don’t bother with generic recursion schemes; I tend to just write lots of explicit recursion, which I find quicker to write and easier to debug.There are similar things I do in Java as well. It has taken me quite a while to become comfortable with these things and stop feeling bad about them, and I think I finally understand why.
I’m not sure I really have a main point, other than to encourage you to consider your coding practices, and why you consider certain practices to be good or bad (and whether it depends on the context!).
Next time, back to your regularly scheduled competitive programming tips!
]]>After solving so many problems in Haskell, by now I’ve figured out some patterns that work well, identified some common pitfalls, developed some nice little libraries, and so forth. I thought it would be fun to write a series of blog posts sharing my experience for the benefit of others—and because I expect I will also learn things from the ensuing discussion!
As a basic running example I’ll use the same example problem that Kattis uses in its help section, namely, A Different Problem. In this problem, we are told that the input will consist of a number of pairs of integers between and , one pair per line, and we should output the absolute value of the difference between each pair. The given example is that if the input looks like this:
10 12
71293781758123 72784
1 12345677654321
then our program should produce output that looks like this:
2
71293781685339
12345677654320
Kattis problems are always set up this way, with input of a specified format provided on standard input, and output to be written to standard output. To do this in Haskell, one might think we will need to use things like getLine
and putStrLn
to read and write the input. But wait! There is a much better way. Haskell’s standard Prelude
has a function
interact :: (String -> String) -> IO ()
It takes a pure String -> String
function, and creates an IO
action which reads from standard input, feeds the input to the function, and then writes the function’s output to standard output. It uses lazy IO, so the reading and writing can be interleaved with computation of the function—a bit controversial and dangerous in general, but absolutely perfect for our use case! Every single Kattis problem I have ever solved begins with
main = interact $ ...
(or the equivalent for ByteString
, more on that in a future post) and that is the only bit of IO
in the entire program. Yay!
So now we need to write a pure function which transforms the input into the output. Of course, in true Haskell fashion, we will do this by constructing a chained pipeline of functions to do the job incrementally. The general plan of attack (for any Kattis problem) is as follows:
String
input into some more semantically meaningful representation—typically using a combination of functions like lines
, words
, read
, map
, and so on (or more sophisticated tools—see a later post).show
, unwords
, unlines
, and so on.Idiomatic Haskell uses the composition operator (.)
to combine functions. However, when solving competitive programming problems, I much prefer to use the reverse composition operator, (>>>)
from Control.Arrow
(that is, (>>>) = flip (.)
). The reason is that since I often end up constructing long function pipelines, I want to be able to think about the process of transforming input to output and type from left to right at the same time; having to add functions from right to left would be tedious.
So here’s my solution to A Different Problem:
main = interact $
lines >>> map (words >>> map read >>> solve >>> show) >>> unlines
solve :: [Integer] -> Integer
solve [a,b] = abs (a - b)
A few notes:
map
.solve
function in this case, but I prefer to split it out explicitly in order to specify its type, which both prevents problems with read
/show
ambiguity and also serves as a sanity check on the parsing and formatting code.Int
instead of Integer
(maxBound :: Int64
is a bit more than , plenty big enough for inputs up to ), but there would be no benefit to doing so. If we use Integer
we don’t even have to consider potential problems with overflow.And one last thing: I said we were going to parse the input into a “semantically meaningful representation”, but I lied a teensy bit: the problem says we are going to get a pair of integers but I wrote my solve
function as though it takes a list of integers. And even worse, my solve
function is partial! Why did I do that?
The fact is that I almost never use actual Haskell tuples in my solutions, because they are too awkward and inconvenient. Representing homogeneous tuples as Haskell lists of a certain known length allows us to read and process “tuples” using standard functions like words
and map
, to combine them using zipWith
, and so on. And since we get to assume that the input always precisely follows the specification—which will never change—this is one of the few situations where, in my opinion, we are fully justified in writing partial functions like this if it makes the code easier to write. So I always represent homogeneous tuples as lists and just pattern match on lists of the appropriate (known) length. (If I need heterogeneous tuples, on the other hand, I create an appropriate data
type.)
Of course I’ve only scratched the surface here—I’ll have a lot more to say in future posts—but this should be enough to get you started! I’ll leave you with a few very easy problems, which can each be done with just a few lines of Haskell:
Of course you can also try solving any of the other problems (as of this writing, over 2400 of them!) on Kattis as well.
This is a 2D physics-based puzzle/obstacle game where you control a ball (aka circle). The twist that distinguishes it from similar games I’ve seen is that you have only two ways to control the ball:
Pushing the left or right arrow keys changes the ball’s angular velocity, that is, its rate of spin. If the ball is sitting on a surface, this will cause it to roll due to friction, but if the ball is in the air, it will just change its spin rate without changing its trajectory at all.
The down arrow key increases the ball’s velocity in the downwards direction. If the ball is sitting on a surface this will cause it to bounce upwards a bit. If the ball is in the air you can cause it to either bounce higher, by adding to its downward velocity while it is already falling, or you can dampen a bounce by pushing the down arrow while the ball is travelling upwards.
Those are the key mechanics. My intuition is that controlling the ball would be challenging but doable, and there would be lots of opportunities for interesting obstacles to navigate. For example, to get out of a deep pit you have to keep bouncing higher and then once you’re high enough, you impart a bit of spin so the next time you bounce you travel sideways over the lip of the pit. Or there could be a ledge so that you have to bounce once or twice while travelling towards it to get high enough to clear it, but then immediately control your subsequent bounce so you don’t bounce too high and hit some sort of hazard on the ceiling. And so on.
Finally, of course there could be various power-ups (e.g. to make the ball faster, or sticky, or to alter gravity in various ways). Various puzzles might be based on figuring out which power-ups to use or how to use them to overcome various obstacles.
So, does this game already exist? Or does someone want to make it? (Preferably in Haskell? =)
]]>stack
and nix
and cabal-v2
, seems like much less of a big deal than it used to be), and use the hmatrix
package to find the eigenvalues of the companion matrix, which are exactly the roots.
So I tried that, and it seems to work great! The only problem is that I still don’t know how to write a reasonable test suite. I started by making a QuickCheck property expressing the fact that if we evaluate a polynomial at the returned roots, we should get something close to zero. I evaluate the polynomial using Horner’s method, which as far as I understand has good numerical stability in addition to being efficient.
polyRoots :: [Double] -> [Double]
polyRoots = ... stuff using hmatrix, see code at end of post ...
horner :: [Double] -> Double -> Double
horner as x = foldl' (\r a -> r*x + a) 0 as
_polyRoots_prop :: [Double] -> Property
_polyRoots_prop as = (length as > 1) ==>
all ((< 1e-10) . abs . horner as) (polyRoots as)
This property passes 100 tests for quadratic polynomials, but for cubic I get failures; here’s an example. Consider the polynomial
Finding its roots via hmatrix
yields three:
[-1.077801388041068, 0.5106483227001805, 150.6238979010295]
Notice that the third root is much bigger in magnitude than the other two, and indeed, that third root is the problematic one. Evaluating the polynomial at these roots via Horner’s method yields
[1.2434497875801753e-14, 1.7763568394002505e-15, -1.1008971512183052e-10]
the third of which is bigger than 1e-10
which I had (very arbitrarily!) chosen as the cutoff for “close enough to zero”. But here’s the thing: after playing around with it a bit, it seems like this is the most accurate possible value for the root that can be represented using Double
. That is, if I evaluate the polynomial at any value other than the root that was returned—even if I just change the very last digit by 1 in either direction—I get a result which is farther from zero.
If I make the magic cutoff value bigger—say, 1e-8
instead of 1e-10
—then I still get similar counterexamples, but for larger-degree polynomials. I never liked the arbitrary choice of a tolerance anyway, and now it seems to me that saying “evaluating the polynomial at the computed roots should be within this absolute distance from zero” is fundamentally the wrong thing to say; depending on the polynomial, we might have to take what we can get. Some other things I could imagine saying instead include:
…but, first of all, I don’t know if these are reasonable properties to expect; and even if they were, I’m not sure I know how to express them in Haskell! Any advice is most welcome. Are there any best practices for expressing desirable test properties for floating-point computations?
For completeness, here is the actual code I came up with for finding roots via hmatrix
. Notice there is another annoying arbitrary value in there, for deciding when a complex root is close enough to being real that we call it a real root. I’m not really sure what to do about this either.
-- Compute the roots of a polynomial as the eigenvalues of its companion matrix,
polyRoots :: [Double] -> [Double]
polyRoots [] = []
polyRoots (0:as) = polyRoots as
polyRoots (a:as) = mapMaybe toReal eigVals
where
n = length as'
as' = map (/a) as
companion = (konst 0 (1,n-1) === ident (n-1)) ||| col (map negate . reverse $ as')
eigVals = toList . fst . eig $ companion
toReal (a :+ b)
| abs b < 1e-10 = Just a -- This arbitrary value is annoying too!
| otherwise = Nothing
]]>