Tag Archives: algorithm

Project Euler in F#: Problem 8

I’ve been trying to teach myself F# using the Project Euler problems, and I’m starting to feel I’m getting somewhere with the language. The few Euler problems I’ve solved so far have had very straightforward and natural solutions.

Problem 8 is as follows: Find the subsequence of 5 consecutive digits that yield the greatest product when multiplied together, in the 1000-digit number:

73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450

I was able to come up with an F# solution that is one line, plus a helper line to convert the string into a sequence of digits:

let str = "731<...>"

let digits = Seq.map (fun x -> int (Char.GetNumericValue x)) str

let maxproduct num list =
 Seq.max (Seq.map (fun x -> Seq.reduce (*) x) (Seq.windowed num list))

The value digits is just a list of the digits in the string, converted into integers. The function maxproduct works the obvious way: take every subsequence of five digits (Seq.windowed), multiply them together (Seq.reduce, applied to each element of the sequence with Seq.map) and then find the maximum (Seq.max).

The only reason this needs quite so little work is the existence of Seq.windowed in the standard library, which does exactly the right thing in turning a 1000-element list into 996 arrays of subsequences of consecutive digits.

I’m not sure I like ramming all the functions into one line, and I’m sure there must be a way to combine map and reduce without the lambda, which adds a lot of clutter. If this was real code, it would need quite a lot of work to make it readable. However, the standard library is a big win, because the process of ‘windowing’ a sequence is nicely separated from the code. It’s also nice (for toy problems like this, at any rate) that the program is pretty much a definition of the problem, with little thought being necessary as to how to do the processing.

Implementing the Graham Scan for a convex hull in Clojure

I saw implementation of the Graham scan mentioned in Real World Haskell as an exercise. I figured I’d have a go at doing it in Clojure, as it might prove useful for another project I was doing in a combination of Java and Clojure.

I’m mentioning this here as it really illustrated to me the power of Clojure basing itself on the JVM. During the development of this inherently graphical algorithm, I was able to quickly cobble together a Swing UI that showed what was going on. Working with the UI was very nearly as responsive as working at a REPL (indeed, UI elements were being drawn and redrawn as a result of commands from the REPL) and illustrated what was going on with the algorithm far better. For the first time ever I didn’t feel the dichotomy between dynamic and graphical ways of working with code.

I was also able to follow a very test-driven approach with test-is. This works so well at the REPL that it starts to make my work with C++ seem laughable.

The main body of the algorithm itself was something that seemed to fit well into Clojure’s loop construct as it’s inherently iterative rather than recursive. This was the first time I’ve done something where the loop seemed more natural than both the recursive solution and a for / while loop in C++. I tend to like declarations of intent in code, and listing the variables that are going to vary at the top of the loop seems like a sensible piece of discipline.

For what it’s worth, here’s the code of the main algorithm:

(defn make-convex-hull
  "Make a convex hull for a set of points, all of which are assumed to have positive x and y
coordinates. Returns a vector of lines, each of which is a sequence of two points"
  [points]
  (let [starting-point (find-starting-point points)
        remaining-points (remove #(= % starting-point) points)
        working-set (sort angle-comparator remaining-points)]
    (loop [remaining-points working-set
           p1 starting-point
           p2 (first working-set)
           p3 (second working-set)
           hull []]
      (if (= p3 (last working-set))
        ; We have reached the end of the set, return the full hull
        (if (left-turn? p1 p2 p3)
          (concat hull [[p1 p2] [p2 p3] [p3 starting-point]])
          (concat hull [[p1 p3] [p3 starting-point]]))

        ; Does this form a left turn?
        (if (left-turn? p1 p2 p3)
          (recur (rest remaining-points) p2 p3 (nth remaining-points 2) (conj hull [p1 p2]))
          (recur (rest remaining-points) p1 p3 (nth remaining-points 2) hull))))))

You can also download a complete tarball. Note that this is a tarball of a darcs repository, so if you’re someone who might want to employ me you can see a full history of how I worked on the project. And if you’re a potential employer who knows how to use darcs, I might be interested in hearing from you. 🙂