The Problem
I need to create a function that, when given a finite sequence of potentially infinite sequences, it produces the sequence that is their "cartesian product".
i.e. given the sequence
'((1 2) (3 4))
the function produces (some ordering of):
'((1 3) (1 4) (2 3) (2 4)
Importantly, for every p in the list of cartesian products ps, there must be some natural number n such that (= p (last (take n ps))). Or, informally, you only need to iterate through the sequence a finite amount to reach any element in it.
This condition becomes important when dealing with infinite lists.
Solution in Haskell
In Haskell, this is how I would have done it:
interleave :: [a] -> [a] -> [a]
interleave [] ys = ys
interleave (x:xs) ys = x : interleave ys xs
combine :: [[a]] -> [[a]]
combine = foldr prod [[]]
where
prod xs css = foldr1 interleave [ [x:cs | cs <- css] | x <- xs]
And calling it you get the following:
combine [[0..] [0..]] = [[0,0,0],[1,0,0],[,1,0],[2,0,0],[0,0,1],[1,1,0],...
Solution in Clojure
And so I attempted to replicate this in Clojure, like so, (It's pretty much a direct translation):
(defn interleave
"Returns a lazy sequence of the interleavings of sequences `xs` and `ys`
(both potentially infinite), leaving no elements discarded."
[xs ys]
(lazy-seq
(if-let [[x & xs*] (seq xs)]
(cons x (interleave ys xs*))
ys)))
(defn interleave*
"Converts a sequence of potentially infinite sequences into its lazy
interleaving."
[xss]
(lazy-seq
(when-let [[xs & xss*] (seq xss)]
(interleave xs (interleave* xss*)))))
(defn combine
"Takes a finite sequence of potentially infinite sequences, and combines
them to produce a possibly infinite sequence of their cartesian product."
[xss]
(if-let [[xs & xss*] (seq xss)]
(interleave*
(for [x xs]
(for [cs (combine xss*)]
(lazy-seq (cons x cs)))))
'(()) ))
But when I run:
(take 1 (combine [(range) (range)]))
I get:
StackOverflowError cfg.util/combine/iter--3464--3468/fn--3469/fn--3470/iter--3471--3475/fn--3476
So, how do I make it lazy enough, so as to avoid the stack overflow? Really, I don't understand how Clojure's lazy sequence model works which is the main problem.
I think your solution may be algorithmically intractable, reconstructing the sub-sequences time and again, much as the simple Fibonacci function:
(defn fib [n]
(case n
(0N 1N) n
(+ (fib (- n 1)) (fib (- n 2)))))
... recomputes its precedents.
In any event, the search for [100 10] in the cartesian product of (range) and (range):
(first (filter #{[100 10]} (combine [(range) (range)])))
... does not return in a reasonable time.
I can offer you a faster though far less elegant solution.
First, a couple of utilities:
Something from #amalloy to compute the Cartesian product of finite sequences:
(defn cart [colls]
(if (empty? colls)
'(())
(for [x (first colls)
more (cart (rest colls))]
(cons x more))))
A function adapted from the Clojure Cookbook to map the values of a map:
(defn map-vals [f m] (zipmap (keys m) (map f (vals m))))
Now for the function we want, which I've called enum-cart, as it enumerates the Cartesian product even of infinite sequences:
(defn enum-cart [colls]
(let [ind-colls (into (sorted-map) (map-indexed (fn [n s] [n (seq s)]) colls))
entries ((fn tins [ss] (let [ss (select-keys ss (map key (filter val ss)))]
(lazy-seq
(if (seq ss)
(concat
(map-vals first ss)
(tins (map-vals next ss)))))))
ind-colls)
seens (reductions
(fn [a [n x]] (update-in a [n] conj x))
(vec (repeat (count colls) []))
entries)]
(mapcat
(fn [sv [n x]] (cart (assoc sv n [x])))
seens entries)))
The idea is to generate an indexed sequence of entries, going round the non-exhausted sequences. From this we generate a companion sequence of what we have already seen from each sequence. We pairwise combine these two, generating the free cartesian product of the new element with what we have of the other sequences. The answer is the concatenation of these free products.
For example
(enum-cart [(range 3) (range 10 15)])
... produces
((0 10)
(1 10)
(0 11)
(1 11)
(2 10)
(2 11)
(0 12)
(1 12)
(2 12)
(0 13)
(1 13)
(2 13)
(0 14)
(1 14)
(2 14))
And
(first (filter #{[100 10]} (enum-cart [(range) (range)])))
;(100 10)
... returns more or less instantly.
Notes
Is this better done in Knuth or elsewhere? I don't have access to
it.
The last non-exhausted sequence need not be kept, as there is nothing
else to use it.
So, I figured it out. And the issue is a subtle, but frustrating one. The problem stems from the destructuring I perform, in basically every function: I use this sort of idiom: [x & xs*] (seq xs), however, this realizes the first element of xs*, as well as realizing x. This behaviour is similar to what you would see if you were to use first and next to get the head and tail of the list respectively.
Using first/rest instead of destructuring in this way fixed the stack overflow:
(defn interleave
"Returns a lazy sequence of the interleavings of sequences `xs` and `ys`
(both potentially infinite), leaving no elements discarded."
[xs ys]
(lazy-seq
(if-let [xs* (seq xs)]
(cons (first xs*) (interleave ys (rest xs*)))
ys)))
(defn interleave*
"Converts a sequence of potentially infinite sequences into its lazy
interleaving."
[xss]
(lazy-seq
(when-let [xss* (seq xss)]
(interleave (first xss*)
(interleave* (rest xss*))))))
(defn combine
"Takes a finite sequence of potentially infinite sequences, and combines
them to produce a possibly infinite sequence of their cartesian product."
[xss]
(if-let [xss* (seq xss)]
(interleave*
(for [x (first xss*)]
(for [cs (combine (rest xss*))]
(lazy-seq (cons x cs)))))
'(()) ))
And running it, we get:
(= (take 5 (combine [(range) (range) (range)]))
'((0 0 0) (1 0 0) (0 1 0) (2 0 0) (0 0 1)))
Related
I'd like to find out concise, functional and tail-recursive (if possible) way of implementing the below specified function:
(define (make-domain digits dimension)
;; Implementation)
;; Usage
(make-domain '(0 1) 0) => (())
(make-domain '(0 1) 1) => ((0) (1))
(make-domain '(0 1) 2) => ((0 0) (0 1) (1 0) (1 1))
(make-domain '(0 1) 3) => ((0 0 0) (0 0 1) (0 1 0) (0 1 1) (1 0 0) (1 0 1) (1 1 0) (1 1 1))
I'd prefer Scheme implementation with as few helper or library functions as possible, but SML or Haskell will do as well. I'm trying to find a tail-recursive solution possibly using mutual or nested recursion, but with no luck at the moment.
Thank you very much!
That one, in Haskell, is at least “functional” and concise (I think):
makeDomain :: [α] -> Int -> [[α]]
makeDomain xs 0 = [[]]
makeDomain xs n = let mdn1 = makeDomain xs (n-1)
fn x = map (x:) mdn1
in concat (map fn xs)
Trying it:
λ>
λ> makeDomain [0,1] 2
[[0,0],[0,1],[1,0],[1,1]]
λ>
λ> makeDomain [0,1] 3
[[0,0,0],[0,0,1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,1,0],[1,1,1]]
λ>
As mentioned in the comments, going tail-recursive might be a not so good idea, in Haskell at least.
Addendum re: memory efficiency:
You did not list performance concerns in your requirements (was it because you think tail-recursive functions tend to perform better ?).
The above version of makeDomain, as hinted in the comments by amalloy suffers from exponential memory consumption, at least for some compiler versions / optimization levels. This is because the compiler can see makeDomain xs (n-1) as a loop-invariant value to be kept around.
So this is one of these situations where you have to pick a trade-off between elegance and efficiency. The problem has been discussed recently in this related SO question in the context of the very similar replicateM library function; drawing on the answer by K. A. Buhr, one can come up with a version of makeDomain that runs in constant memory, leveraging the Haskell list comprehension construct.
makeDomain1 :: [α] -> Int -> [[α]]
makeDomain1 xs n =
map reverse (helper xs n)
where
helper xs 0 = [[]]
helper xs n = [ x:ys | ys <- helper xs (n-1), x <- xs ]
Testing: running with an OS-enforced memory hard limit of 1200 MB.
λ>
λ> import Control.Monad (replicateM)
λ> replicateM 3 [0,1]
[[0,0,0],[0,0,1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,1,0],[1,1,1]]
λ>
λ> makeDomain1 [0,1] 3
[[0,0,0],[0,0,1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,1,0],[1,1,1]]
λ>
λ> length $ replicateM 30 [0,1]
<interactive>: internal error: Unable to commit 1048576 bytes of memory
...
λ>
λ> length $ makeDomain [0,1] 30
<interactive>: internal error: Unable to commit 1048576 bytes of memory
...
λ>
λ> length $ makeDomain1 [0,1] 30
1073741824
λ>
Using GHC v8.6.5 with -O2 option, that last version never takes more than 150 MB memory, and runs at a speed of about 30 nsec per output list on a vanilla Intel x86-64 PC. This is perfectly reasonable.
Your answer can be made tail-recursive by the usual trick of using an accumulator. The following is Racket not Scheme, but perhaps only because it uses append* which can be defined, I think, as
(define (append* . args)
(apply append args))
Here is a tail-recursive version, therefore:
(define (make-domain digits dimension)
(let mdl ([d dimension] [r '(())])
(if (zero? d)
r
(mdl (- d 1)
(append* (map (λ (d)
(map (λ (sd)
(cons d sd))
r))
digits))))))
Here is my constructive take on solving the above described problem.
The solution is functional, concise, recursive (but not tail-recursive) implementation in Scheme.
The idea is that the domain has an inductive (recursive) definition: each combination in the domain (first map) is a pair of a digit that is taken one in one from the initial digits dictionary and all combination for a smaller by one dimension (second map)
(define (make-domain digits dimension)
"Builds all combinations of digits for a dimension"
;; There is an empty combination for a dimension 0
(if [zero? dimension] '(())
;; Combine all combinations
(apply append
;; For each digit from digits
(map (lambda (d)
;; Prepend the digit to each combination
;; for a smaller by one dimension
(map (lambda (sd) (cons d sd))
(make-domain digits (1- dimension))))
digits))))
For completeness, here is the Haskell solution translated to Standard ML:
fun curry f x y = f (x, y)
fun concatMap f xs = List.concat (List.map f xs)
fun makeDomain _ 0 = [[]]
| makeDomain ys n =
let val res = makeDomain ys (n-1)
in concatMap (fn x => map (curry op:: x) res) ys
end
One could apply the usual trick of an accumulator to avoid the n stack frames that tfb demonstrates. But as amalloy points out, this is hardly the bottleneck of this function with its memory use an exponential factor of n. In the Standard ML variant, the excessive list concatenation will cost more.
So depending on what you intend to do with this list, you may want to consider, in Standard ML, generating its elements and process them one at a time (like lazy streams allow you to); for example, rather than generate a long list and filter it, you could generate the filtered list. Here's an example: Translation of Pythagorean Triplets from Haskell to Standard ML.
This problem takes many forms. For example, given the input '(1 2 3 4 5 6), we might want to swap the values between even and odd pairs. The output would be '(2 1 4 3 6 5).
In Haskell, this is rather easy:
helper [] = []
helper [x] = [x]
helper (x : y : ys) = y : x : helper ys
I wrote some Clojure code to accomplish the same task, but I feel that there is probably a cleaner way. Any suggestions on how to improve this?
(defn helper [[x y & ys]]
(cond
(nil? x) (list)
(nil? y) (list x)
:else (lazy-seq (cons y (cons x (helper ys))))))
Ideally the list would be consumed and produced lazily. Thanks.
(for [[a b] (partition 2 '(1 2 3 4 5 6))
i [b a]]
i)
OR something resembling the haskell version:
(defn helper
([] (list))
([x] (list x))
([x y & r] (concat [y x] (apply helper r))))
(apply helper '(1 2 3 4 5 6))
Avoiding intermediate object creation (vectors / seqs to be concatenated) and in direct correspondence to the Haskell original while handling nil items in the input (which the approach from the question text doesn't):
(defn helper [[x & [y & zs :as ys] :as xs]]
(if xs
(lazy-seq
(if ys
(cons y (cons x (helper zs)))
(list x)))))
Normally I'd use something like tom's answer though, only with mapcat rather than flatten:
(defn helper [xs]
(mapcat reverse (partition-all 2 xs)))
You need to use partition-all rather than partition to avoid dropping the final element from lists of odd length.
This is one lazy way to do it:
user=> (mapcat reverse (partition 2 '(1 2 3 4 5 6)))
(2 1 4 3 6 5)
I know that quicksort has O(n log n) average time complexity. A pseudo-quicksort (which is only a quicksort when you look at it from far enough away, with a suitably high level of abstraction) that is often used to demonstrate the conciseness of functional languages is as follows (given in Haskell):
quicksort :: Ord a => [a] -> [a]
quicksort [] = []
quicksort (p:xs) = quicksort [y | y<-xs, y<p] ++ [p] ++ quicksort [y | y<-xs, y>=p]
Okay, so I know this thing has problems. The biggest problem with this is that it does not sort in place, which is normally a big advantage of quicksort. Even if that didn't matter, it would still take longer than a typical quicksort because it has to do two passes of the list when it partitions it, and it does costly append operations to splice it back together afterwards. Further, the choice of the first element as the pivot is not the best choice.
But even considering all of that, isn't the average time complexity of this quicksort the same as the standard quicksort? Namely, O(n log n)? Because the appends and the partition still have linear time complexity, even if they are inefficient.
This "quicksort" is actually deforested tree sort:
http://www.reddit.com/r/programming/comments/2h0j2/real_quicksort_in_haskell
data Tree a = Leaf | Node a (Tree a) (Tree a)
mkTree [] = Leaf
mkTree (x:xs) = Node x (mkTree (filter (<= x) xs)) (mkTree (filter (x <) xs))
Binary tree is unbalanced, so O(N^2) worst-case and O(N*Log N) average-case complexity for building search tree.
foldTree f g Leaf = g
foldTree f g (Node x l r) = f x (foldTree f g l) (foldTree f g r)
treeSort l = foldTree (\x lft rht -> lft++[x]++rht) [] (mkTree l)
Retrieval algorithm have O(N^2) worst-case and O(N*Log N) average-case complexity.
Well-balanced:
Prelude> let rnds = iterate step where step x = (75*x) `mod` 65537
Prelude> length . quicksort . take 4000 . rnds $ 1
4000
(0.08 secs, 10859016 bytes)
Prelude> length . quicksort . take 8000 . rnds $ 1
8000
(0.12 secs, 21183208 bytes)
Prelude> length . quicksort . take 16000 . rnds $ 1
16000
(0.25 secs, 42322744 bytes)
Not-so-well-balanced:
Prelude> length . quicksort . map (`mod` 10) $ [1..4000]
4000
(0.62 secs, 65024528 bytes)
Prelude> length . quicksort . map (`mod` 10) $ [1..8000]
8000
(2.45 secs, 241906856 bytes)
Prelude> length . quicksort . map (`mod` 10) $ [1..16000]
16000
(9.52 secs, 941667704 bytes)
I agree with your assumption that the average time complexity still is O(n log n). I'm not an expert and 100% sure, but these are my thoughts:
This is a pseudo code of the in-place quicksort: (call quicksort with l=1 and r=length of the array)
Quicksort(l,r)
--------------
IF r-l>=1 THEN
choose pivot element x of {x_l,x_l+1,...,x_r-1,x_r}
order the array-segment x_l,...x_r in such a way that
all elements < x are on the left side of x // line 6
all elements > x are on the right side of x // line 7
let m be the position of x in the 'sorted' array (as said in the two lines above)
Quicksort(l,m-1);
Quicksort(m+1,r)
FI
The average time complexity analysis then reasons by selecting the "<"-comparisons in line 6 and 7 as the dominant operation in this algorithm and finally comes to the conclusion that the average time complexity is O(n log n). As the cost of line "order the array-segment x_l,...x_r in such a way that..." are not considered (only the dominant operation is important in time complexity analysis if you want to find bounds), I think "because it has to do two passes of the list when it partitions it" is not a problem, also as your Haskell version would just take approximately twice as long in this step. The same holds true for the appendix-operation and I agree with on that this adds nothing to the asymptotic costs:
Because the appends and the partition still have linear time complexity, even if they are inefficient.
For the sake of convenience lets assume that this adds up "n" to our time complexity costs, so that we have "O(n log n+n)". As there exists a natural number o for that n log n > n for all natural numbers greater than o holds true, you can estimate n log n +n to the top by 2 n log n and to the bottom by n log n, therefore n log n+n = O(n log n).
Further, the choice of the first element as the pivot is not the best choice.
I think the choice of the pivot element is irrelevant here, because in the average case analysis you assume uniform distribution of the elements in the array. You can't know from which place in the array you should select it, and you therefore have to consider all these cases in which your pivot-element (independently from which place of the list you take it) is the i-st smallest element of your list, for i=1...r.
I can offer you a run time test on Ideone.com which seems to show more or less linearithmic run-times for both (++) based versions and the one using accumulator technique from the Landei's answer, as well as another one, using one-pass three-way partitioning. On ordered data this turns quadratic or worse for all of them.
-- random: 100k 200k 400k 800k
-- _O 0.35s-11MB 0.85s-29MB 1.80s-53MB 3.71s-87MB n^1.3 1.1 1.0
-- _P 0.36s-12MB 0.80s-20MB 1.66s-45MB 3.76s-67MB n^1.2 1.1 1.2
-- _A 0.31s-14MB 0.62s-20MB 1.58s-54MB 3.22s-95MB n^1.0 1.3 1.0
-- _3 0.20s- 9MB 0.41s-14MB 0.88s-24MB 1.92s-49MB n^1.0 1.1 1.1
-- ordered: 230 460 900 1800
-- _P 0.09s 0.33s 1.43s 6.89s n^1.9 2.1 2.3
-- _A 0.09s 0.33s 1.44s 6.90s n^1.9 2.1 2.3
-- _3 0.05s 0.15s 0.63s 3.14s n^1.6 2.1 2.3
quicksortO xs = go xs where
go [] = []
go (x:xs) = go [y | y<-xs, y<x] ++ [x] ++ go [y | y<-xs, y>=x]
quicksortP xs = go xs where
go [] = []
go (x:xs) = go [y | y<-xs, y<x] ++ (x : go [y | y<-xs, y>=x])
quicksortA xs = go xs [] where
go [] acc = acc
go (x:xs) acc = go [y | y<-xs, y<x] (x : go [y | y<-xs, y>=x] acc)
quicksort3 xs = go xs [] where
go (x:xs) zs = part x xs zs [] [] []
go [] zs = zs
part x [] zs a b c = go a ((x : b) ++ go c zs)
part x (y:ys) zs a b c =
case compare y x of
LT -> part x ys zs (y:a) b c
EQ -> part x ys zs a (y:b) c
GT -> part x ys zs a b (y:c)
The empirical run-time complexities are estimated here as O(n^a) where a = log( t2/t1 ) / log( n2/n1 ). The timings are very approximate as ideone aren't very reliable with occasional far outlyers, but for checking the time complexity it's enough.
Thus these data seem to indicate that one-pass partition is faster by 1.5x-2x than two-pass schemes, and that using (++) is in no way slowing things down - at all. I.e. the "append operations" are not "costly" at all. The quadratic behaviour or (++)/append seems to be an urban myth — in Haskell context of course (edit: ... i.e. in the context of guarded recursion/tail recursion modulo cons; cf. this answer) (update: as user:AndrewC explains, it really is quadratic with the left folding; linear when (++) is used with the right folding; more about this here and here).
later addition: To be stable, the three-way partitioning quicksort version should too build its parts in the top-down manner:
q3s xs = go xs [] where
go (x:xs) z = part x xs go (x:) (`go` z)
go [] z = z
part x [] a b c = a [] (b (c []))
part x (y:ys) a b c =
case compare y x of
LT -> part x ys (a . (y:)) b c
EQ -> part x ys a (b . (y:)) c
GT -> part x ys a b (c . (y:))
(performance not tested).
I don't know how much this improves the runtime complexity, but by using an accumulator you can avoid the expensive (++):
quicksort xs = go xs [] where
go [] acc = acc
go (x:xs) acc = go [y | y<-xs, y<x] (x : go [y | y<-xs, y>=x] acc)
Look here for a true O(n log n) quicksort that will work on both arrays and lists :
http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.23.4398&rep=rep1&type=pdf
It is quite easy to implement in Common Lisp, and it outperforms the sort implementation of many commercial lisps.
Yes, this version has the same asymptotic complexity as the classic version -- you replace the linear-time partition with: two passes (< and >=), and you have the additional linear-time ++ (which includes linear re-allocing/copying). So it's a hefty constant-factor worse than an in-place partition, but it's still linear. All the other aspects of the algorithm are the same, so the same analysis that gives O(n log n) average-case for "true" (i.e. in-place) quicksort still holds here.
How to define a rotates function that generates all rotations of the given list?
For example: rotates [1,2,3,4] =[[1,2,3,4],[2,3,4,1],[3,4,1,2],[4,1,2,3]]
I wrote a shift function that can rearrange the order
shift ::[Int]->[Int]
shift x=tail ++ take 1 x
but I don't how to generate these new arrays and append them together.
Another way to calculate all rotations of a list is to use the predefined functions tails and inits. The function tails yields a list of all final segments of a list while inits yields a list of all initial segments. For example,
tails [1,2,3] = [[1,2,3], [2,3], [3], []]
inits [1,2,3] = [[], [1], [1,2], [1,2,3]]
That is, if we concatenate these lists pointwise as indicated by the indentation we get all rotations. We only get the original list twice, namely, once by appending the empty initial segment at the end of original list and once by appending the empty final segment to the front of the original list. Therefore, we use the function init to drop the last element of the result of applying zipWith to the tails and inits of a list. The function zipWith applies its first argument pointwise to the provided lists.
allRotations :: [a] -> [[a]]
allRotations l = init (zipWith (++) (tails l) (inits l))
This solution has an advantage over the other solutions as it does not use length. The function length is quite strict in the sense that it does not yield a result before it has evaluated the list structure of its argument completely. For example, if we evaluate the application
allRotations [1..]
that is, we calculate all rotations of the infinite list of natural numbers, ghci happily starts printing the infinite list as first result. In contrast, an implementation that is based on length like suggested here does not terminate as it calculates the length of the infinite list.
shift (x:xs) = xs ++ [x]
rotates xs = take (length xs) $ iterate shift xs
iterate f x returns the stream ("infinite list") [x, f x, f (f x), ...]. There are n rotations of an n-element list, so we take the first n of them.
The following
shift :: [a] -> Int -> [a]
shift l n = drop n l ++ take n l
allRotations :: [a] -> [[a]]
allRotations l = [ shift l i | i <- [0 .. (length l) -1]]
yields
> ghci
Prelude> :l test.hs
[1 of 1] Compiling Main ( test.hs, interpreted )
Ok, modules loaded: Main.
*Main> allRotations [1,2,3,4]
[[1,2,3,4],[2,3,4,1],[3,4,1,2],[4,1,2,3]]
which is as you expect.
I think this is fairly readable, although not particularly efficient (no memoisation of previous shifts occurs).
If you care about efficiency, then
shift :: [a] -> [a]
shift [] = []
shift (x:xs) = xs ++ [x]
allRotations :: [a] -> [[a]]
allRotations l = take (length l) (iterate shift l)
will allow you to reuse the results of previous shifts, and avoid recomputing them.
Note that iterate returns an infinite list, and due to lazy evaluation, we only ever evaluate it up to length l into the list.
Note that in the first part, I've extended your shift function to ask how much to shift, and I've then a list comprehension for allRotations.
The answers given so far work fine for finite lists, but will eventually error out when given an infinite list. (They all call length on the list.)
shift :: [a] -> [a]
shift xs = drop 1 xs ++ take 1 xs
rotations :: [a] -> [[a]]
rotations xs = zipWith const (iterate shift xs) xs
My solution uses zipWith const instead. zipWith const foos bars might appear at first glance to be identical to foos (recall that const x y = x). But the list returned from zipWith terminates when either of the input lists terminates.
So when xs is finite, the returned list is the same length as xs, as we want; and when xs is infinite, the returned list will not be truncated, so will be infinite, again as we want.
(In your particular application it may not make sense to try to rotate an infinite list. On the other hand, it might. I submit this answer for completeness only.)
I would prefer the following solutions, using the built-in functions cycle and tails:
rotations xs = take len $ map (take len) $ tails $ cycle xs where
len = length xs
For your example [1,2,3,4] the function cycle produces an infinite list [1,2,3,4,1,2,3,4,1,2...]. The function tails generates all possible tails from a given list, here [[1,2,3,4,1,2...],[2,3,4,1,2,3...],[3,4,1,2,3,4...],...]. Now all we need to do is cutting down the "tails"-lists to length 4, and cutting the overall list to length 4, which is done using take. The alias len was introduced to avoid to recalculate length xs several times.
I think it will be something like this (I don't have ghc right now, so I couldn't try it)
shift (x:xs) = xs ++ [x]
rotateHelper xs 0 = []
rotateHelper xs n = xs : (rotateHelper (shift xs) (n - 1))
rotate xs = rotateHelper xs (length xs)
myRotate lst = lst : myRotateiter lst lst
where myRotateiter (x:xs) orig
|temp == orig = []
|otherwise = temp : myRotateiter temp orig
where temp = xs ++ [x]
I suggest:
rotate l = l : rotate (drop 1 l ++ take 1 l)
distinctRotations l = take (length l) (rotate l)
I am going over this haskell lecture on count down game, i don't know any haskell but i am intrested in the problem, i am trying to port his code to clojure.
this is the part i got stuck must be something i don't get in haskell,
split :: [a] -> [([a],[a])]
split [] = [([],[])]
split (x:xs) = ([],x:xs) : [(x:ls,rs) | (ls,rs) [([a],[a])]
nesplit = filter ne . split
ne :: ([a],[b]) -> Bool
ne (xs,ys) = not (null xs || null ys)
exprs :: [Int] -> [Expr]
exprs [] = []
exprs [n] = [Val n]
exprs ns = [e | (ls,rs)
I have my own split given 1 2 3 4 it spits out,
(((1) (2 3 4)) ((1 2) (3 4)) ((1 2 3) (4)))
(defn split [v]
(if (= (count v) 1)
(list (first v))
(map #(list (take % v) (drop % v)) (range 1 (count v)))))
(defn exprs [v]
(if (= (count v) 1)
v
(map #(concat (exprs (first %)) (exprs (second %))) v)))
(exprs (split [1 2 3 4]))
that gives me,
java.lang.IllegalArgumentException: Don't know how to create ISeq from: java.lang.Integer
Can anyone tell me what am i missing from the haskell code?
His full code listing is available here.
This is closely following the Haskell implementation as far as my limited Haskell fu allows me to do....
(defn split
[s]
(map #(split-at % s) (range 1 (count s))))
(defn ne
[s]
(every? (complement empty?) s))
(defn nesplit
[s]
(filter ne (split s)))
(declare combine)
(defn exprs
[s]
(when-let [s (seq s)]
(if (next s)
(for [[ls rs] (nesplit s)
l (exprs ls)
r (exprs rs)
e (combine l r)]
e)
s)))
Haven't tested it though.
As for your error message: I think the problem is, that you don't call split recursively in exprs. Then you get 1 were a sequence is expected...
Random other note: count is linear in time for sequences. Since we just need to know, whether we have more than one element, we can check the value of (next s) against nil.
the exception results from exprs being called recursively and eventually being invoked with a list of integers. your code only handles a list of lists or a list of length one.
(exprs '(2 3 4))
leads to the else branch of the if statement which expands out to:
(map #(concat (exprs (first %)) (exprs (second %))) '(2 3 4))))
which comes out to:
(concat (exprs (first 2)) (exprs (second 2)))
(concat (exprs (first 3)) (exprs (second 3)))
(concat (exprs (first 4)) (exprs (second 4)))
and (first 2) throws:
java.lang.IllegalArgumentException: Don't know how to create ISeq from: java.lang.Integer