Ninety-Nine F# Problems - Problems 21 - 28 - Lists again

These are F# solutions of Ninety-Nine Haskell Problems which are themselves translations of Ninety-Nine Lisp Problems and Ninety-Nine Prolog Problems. The solutions are hidden so you can try to solve them yourself.

Ninety-Nine F# Problems - Problems 21 - 28 - Lists again

 1: /// These are F# solutions of Ninety-Nine Haskell Problems 
 2: /// (http://www.haskell.org/haskellwiki/H-99:_Ninety-Nine_Haskell_Problems), 
 3: /// which are themselves translations of Ninety-Nine Lisp Problems
 4: /// (http://www.ic.unicamp.br/~meidanis/courses/mc336/2006s2/funcional/L-99_Ninety-Nine_Lisp_Problems.html)
 5: /// and Ninety-Nine Prolog Problems
 6: /// (https://sites.google.com/site/prologsite/prolog-problems).
 7: ///
 8: /// If you would like to contribute a solution or fix any bugs, send 
 9: /// an email to paks at kitiara dot org with the subject "99 F# problems". 
10: /// I'll try to update the problem as soon as possible.
11: ///
12: /// The problems have different levels of difficulty. Those marked with a single asterisk (*) 
13: /// are easy. If you have successfully solved the preceeding problems you should be able to 
14: /// solve them within a few (say 15) minutes. Problems marked with two asterisks (**) are of 
15: /// intermediate difficulty. If you are a skilled F# programmer it shouldn't take you more than 
16: /// 30-90 minutes to solve them. Problems marked with three asterisks (***) are more difficult. 
17: /// You may need more time (i.e. a few hours or more) to find a good solution
18: ///
19: /// Though the problems number from 1 to 99, there are some gaps and some additions marked with 
20: /// letters. There are actually only 88 problems.

(*) Problem 21 : Insert an element at a given position into a list.

 1: /// Example: 
 2: /// * (insert-at 'alfa '(a b c d) 2)
 3: /// (A ALFA B C D)
 4: ///  
 5: /// Example in F#: 
 6: /// 
 7: /// > insertAt 'X' (List.ofSeq "abcd") 2;;
 8: /// val it : char list = ['a'; 'X'; 'b'; 'c'; 'd']
 9: 
10: (Solution 1)
11: 
12: (Solution 2)

(*) Problem 22 : Create a list containing all integers within a given range.

 1: /// Example: 
 2: /// * (range 4 9)
 3: /// (4 5 6 7 8 9)
 4: ///  
 5: /// Example in F#: 
 6: /// 
 7: /// > range 4 9;;
 8: /// val it : int list = [4; 5; 6; 7; 8; 9]
 9: 
10: (Solution)

(**) Problem 23 : Extract a given number of randomly selected elements from a list.

 1: /// Example: 
 2: /// * (rnd-select '(a b c d e f g h) 3)
 3: /// (E D A)
 4: ///  
 5: /// Example in F#: 
 6: /// 
 7: /// > rnd_select "abcdefgh" 3;;
 8: /// val it : seq<char> = seq ['e'; 'a'; 'h']
 9: 
10: (Solution)

(*) Problem 24 : Lotto: Draw N different random numbers from the set 1..M.

 1: /// Example: 
 2: /// * (rnd-select 6 49)
 3: /// (23 1 17 33 21 37)
 4: ///  
 5: /// Example in F#: 
 6: /// 
 7: /// > diff_select 6 49;;
 8: /// val it : int list = [27; 20; 22; 9; 15; 29]
 9: 
10: // using problem 23
11: (Solution)
12: 
13: (Solution)

(*) Problem 25 : Generate a random permutation of the elements of a list.

 1: /// Example: 
 2: /// * (rnd-permu '(a b c d e f))
 3: /// (B A D C E F)
 4: ///  
 5: /// Example in F#: 
 6: /// 
 7: /// > rnd_permu <| List.ofSeq "abcdef";;
 8: /// val it : char list = ['b'; 'c'; 'd'; 'f'; 'e'; 'a']
 9: 
10: (Solution 1)
11: 
12: (Solution 2)

(**) Problem 26 : Generate the combinations of K distinct objects chosen from the N elements of a list.

 1: /// In how many ways can a committee of 3 be chosen from a group of 12 people? We all know that 
 2: /// there are C(12,3) = 220 possibilities (C(N,K) denotes the well-known binomial coefficients). For 
 3: /// pure mathematicians, this result may be great. But we want to really generate all the 
 4: /// possibilities in a list.
 5: ///  
 6: /// Example: 
 7: /// * (combinations 3 '(a b c d e f))
 8: /// ((A B C) (A B D) (A B E) ... )
 9: ///  
10: /// Example in F#: 
11: /// 
12: /// > combinations 3 ['a' .. 'f'];;
13: /// val it : char list list =
14: ///   [['a'; 'b'; 'c']; ['a'; 'b'; 'd']; ['a'; 'b'; 'e']; ['a'; 'b'; 'f'];
15: ///    ['a'; 'c'; 'd']; ['a'; 'c'; 'e']; ['a'; 'c'; 'f']; ['a'; 'd'; 'e'];
16: ///    ['a'; 'd'; 'f']; ['a'; 'e'; 'f']; ['b'; 'c'; 'd']; ['b'; 'c'; 'e'];
17: ///    ['b'; 'c'; 'f']; ['b'; 'd'; 'e']; ['b'; 'd'; 'f']; ['b'; 'e'; 'f'];
18: ///    ['c'; 'd'; 'e']; ['c'; 'd'; 'f']; ['c'; 'e'; 'f']; ['d'; 'e'; 'f']] 
19: 
20: (Solution 1)
21: 
22: (Solution 2)

(**) Problem 27 : Group the elements of a set into disjoint subsets.

 1: /// a) In how many ways can a group of 9 people work in 3 disjoint subgroups of 2, 3 
 2: /// and 4 persons? Write a function that generates all the possibilities and returns them 
 3: /// in a list.
 4: ///  
 5: /// Example: 
 6: /// * (group3 '(aldo beat carla david evi flip gary hugo ida))
 7: /// ( ( (ALDO BEAT) (CARLA DAVID EVI) (FLIP GARY HUGO IDA) )
 8: /// ... )
 9: ///  
10: /// b) Generalize the above predicate in a way that we can specify a list of group sizes 
11: /// and the predicate will return a list of groups.
12: ///  
13: /// Example: 
14: /// * (group '(aldo beat carla david evi flip gary hugo ida) '(2 2 5))
15: /// ( ( (ALDO BEAT) (CARLA DAVID) (EVI FLIP GARY HUGO IDA) )
16: /// ... )
17: ///  
18: /// Note that we do not want permutations of the group members; i.e. ((ALDO BEAT) ...) 
19: /// is the same solution as ((BEAT ALDO) ...). However, we make a difference between 
20: /// ((ALDO BEAT) (CARLA DAVID) ...) and ((CARLA DAVID) (ALDO BEAT) ...).
21: ///  
22: /// You may find more about this combinatorial problem in a good book on discrete 
23: /// mathematics under the term "multinomial coefficients".
24: ///  
25: /// Example in F#: 
26: /// 
27: /// > group [2;3;4] ["aldo";"beat";"carla";"david";"evi";"flip";"gary";"hugo";"ida"];;
28: /// val it : string list list list =
29: ///   [[["aldo"; "beat"]; ["carla"; "david"; "evi"];
30: ///     ["flip"; "gary"; "hugo"; "ida"]];...]
31: /// (altogether 1260 solutions)
32: ///  
33: /// > group [2;2;5] ["aldo";"beat";"carla";"david";"evi";"flip";"gary";"hugo";"ida"];;
34: /// val it : string list list list =
35: ///   [[["aldo"; "beat"]; ["carla"; "david"];
36: ///     ["evi"; "flip"; "gary"; "hugo"; "ida"]];...]
37: /// (altogether 756 solutions)
38: 
39: (Solution)

(**) Problem 28 : Sorting a list of lists according to length of sublists

 1: /// a) We suppose that a list contains elements that are lists themselves. The objective 
 2: /// is to sort the elements of this list according to their length. E.g. short lists first,
 3: /// longer lists later, or vice versa.
 4: ///  
 5: /// Example: 
 6: /// * (lsort '((a b c) (d e) (f g h) (d e) (i j k l) (m n) (o)))
 7: /// ((O) (D E) (D E) (M N) (A B C) (F G H) (I J K L))
 8: ///  
 9: /// Example in F#: 
10: /// 
11: /// > lsort ["abc";"de";"fgh";"de";"ijkl";"mn";"o"];;
12: /// val it : string list = ["o"; "de"; "de"; "mn"; "abc"; "fgh"; "ijkl"]
13: ///
14: /// b) Again; we suppose that a list contains elements that are lists themselves. But this 
15: /// time the objective is to sort the elements of this list according to their length 
16: /// frequency; i.e.; in the default; where sorting is done ascendingly; lists with rare 
17: /// lengths are placed first; others with a more frequent length come later.
18: ///  
19: /// Example: 
20: /// * (lfsort '((a b c) (d e) (f g h) (d e) (i j k l) (m n) (o)))
21: /// ((i j k l) (o) (a b c) (f g h) (d e) (d e) (m n))
22: ///  
23: /// Example in F#: 
24: /// 
25: /// > lfsort ["abc"; "de"; "fgh"; "de"; "ijkl"; "mn"; "o"];;
26: /// val it : string list = ["ijkl"; "o"; "abc"; "fgh"; "de"; "de"; "mn"]
27: 
28: (Solution)

let insertAt x xs n =
    let rec insAt acc xs n =
        match xs, n with
            | [], 1 -> List.rev (x::acc)
            | [], _ -> failwith "Empty list you fool!"
            | xs, 1 -> List.rev (x::acc) @ xs
            | x::xs, n -> insAt (x::acc) xs (n - 1)
    insAt [] xs n

let rec insertAt' x xs n =
    match xs, n with
        | [], 1 -> [x]
        | [], _ -> failwith "Empty list you fool!"
        | _, 1 -> x::xs
        | y::ys, n -> y::insertAt' x ys (n - 1)

let range a b = [a..b]

let rnd_select xs n =
let rndSeq = let r = new System.Random() in seq { while true do yield r.Next() }
xs |> Seq.zip rndSeq |> Seq.sortBy fst |> Seq.map snd |> Seq.take n

let diff_select n m = rnd_select (seq { 1 .. m }) n |> List.ofSeq

let diff_select' n m =
let rndSeq = let r = new System.Random() in Seq.initInfinite(ignore >> r.Next )
seq { 1 .. m } |> Seq.zip rndSeq |> Seq.sortBy fst |> Seq.map snd |> Seq.take n |> List.ofSeq

// using problem 23
let rnd_permu xs = List.length xs |> rnd_select xs

let rnd_permu' xs =
    let rec rndSeq =
        let r = new System.Random()
        seq { while true do yield r.Next() }
    xs |> Seq.zip rndSeq |> Seq.sortBy fst |> Seq.map snd |> List.ofSeq

// as a bonus you get the powerset of xs with combinations 0 xs
let rec combinations n xs =
    match xs, n with
        | [],_ -> [[]]
        | xs, 1 -> [for x in xs do yield [x]]
        | x::xs, n ->
            [for ys in combinations (n-1) xs do
                yield x::ys
             if List.length xs > n then
                yield! combinations n xs
             else
                yield xs]

let rec combinations' n xs =
    let rec tails = function
        | [] -> [[]]
        | _::ys as xs -> xs::tails ys
    match xs, n with
        | _, 0 -> [[]]
        | xs, n ->
            [ for tail in tails xs do
                match tail with
                    | [] -> ()
                    | y::xs' ->
                        for ys in combinations' (n - 1) xs' do
                            yield y::ys ]

let rec group ns xs =
    let rec combination n xs =
        match n,xs with
            | 0, xs -> [([], xs)]
            | _, [] -> []
            | n, x::xs ->
                let ts = [ for ys, zs in combination (n-1) xs do yield (x::ys, zs)]
                let ds = [ for ys, zs in combination n xs do yield (ys, x::zs)]
                ts @ ds
    match ns,xs with
        | [], _ -> [[]]
        | n::ns, xs ->
            [ for g, rs in combination n xs do
                for gs in group ns rs do
                    yield g::gs ]

More information

Link:	http://fssnip.net/ap
Posted:	1 years ago
Author:	Cesar Mendoza (website)
Tags:	Ninety-Nine F# Problems, tutorial, F#, lists