F Sharp Programming/Sets and Maps

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In addition to lists and sequences, F# provides two related immutable data structures called sets and maps. Unlike lists, sets and maps are unordered data structures, meaning that these collections do not preserve the order of elements as they are inserted, nor do they permit duplicates.

[edit] Sets

A set in F# is just a container for items. Sets do not preserver the order in which items are inserted, nor do they allow duplicate entries to be inserted into the collection.

Sets can be created in a variety of ways:

Adding an item to an empty set The Set module contains a useful function Set.empty which returns an empty set to start with.

Conveniently, all instances of sets have an Add function with the type val Add : 'a -> Set<'a>. In other words, our Add returns a set containing our new item, which makes it easy to add items together in this fashion:

> Set.empty.Add(1).Add(2).Add(7);;
val it : Set<int> = seq [1; 2; 7]

Converting lists and sequences into sets Additionally, the we can use Set.ofList and Set.ofSeq to convert an entire collection into a set:

> Set.ofList ["Mercury"; "Venus"; "Earth"; "Mars"; "Jupiter"; "Saturn"; "Uranus"; "Neptune"];;
val it : Set<string> = seq ["Earth"; "Jupiter"; "Mars"; "Mercury"; ...]

The example above demonstrates the unordered nature of sets.

[edit] The Set Module

The Microsoft.FSharp.Collections.Set module contains a variety of useful methods for working with sets.

val add : 'a -> Set<'a> -> Set<'a>

Return a new set with an element added to the set. No exception is raised if the set already contains the given element.

val compare : Set<'a> -> Set<'a> -> int

Compare two sets. Places sets into a total order.

val count : Set<'a> -> int

Return the number of elements in the set. Same as "size".

val diff : Set<'a> -> Set<'a> -> Set<'a>

Return a new set with the elements of the second set removed from the first.

> let a = Set.ofSeq [ 1 .. 10 ]
let b = Set.ofSeq [ 5 .. 15 ];;
 
val a : Set<int>
val b : Set<int>
 
> Set.diff a b;;
val it : Set<int> = seq [1; 2; 3; 4]
 
> a - b;; (* The '-' operators is equivalent to Set.diff *)
val it : Set<int> = seq [1; 2; 3; 4]

val exists : ('a -> bool) -> Set<'a> -> bool

Test if any element of the collection satisfies the given predicate.

val filter : ('a -> bool) -> Set<'a> -> Set<'a>

Return a new collection containing only the elements of the collection for which the given predicate returns "true".

val intersect : Set<'a> -> Set<'a> -> Set<'a>

Compute the intersection, or overlap, of the two sets.

> let a = Set.ofSeq [ 1 .. 10 ]
let b = Set.ofSeq [ 5 .. 15 ];;
 
val a : Set<int>
val b : Set<int>
 
> Set.iter (fun x -> printf "%O " x) (Set.intersect a b);;
5 6 7 8 9 10

val map : ('a -> 'b) -> Set<'a> -> Set<'b>

Return a new collection containing the results of applying the given function to each element of the input set.

val mem : 'a -> Set<'a> -> bool

Evaluates to true if the given element is in the given set.

val remove : 'a -> Set<'a> -> Set<'a>

Return a new set with the given element removed. No exception is raised in the set doesn't contain the given element.

val size : Set<'a> -> int

Return the number of elements in the set.

val subset : Set<'a> -> Set<'a> -> bool

Evaluates to "true" if all elements of the first set are in the second.

> let a = Set.ofSeq [ 1 .. 10 ]
let b = Set.ofSeq [ 5 .. 15 ]
let c = Set.ofSeq [ 2; 4; 5; 9 ];;
 
val a : Set<int>
val b : Set<int>
val c : Set<int>
 
> Set.subset c a;; (* All elements of 'c' exist in 'a' *)
val it : bool = true
 
> Set.subset c b;; (* Not all of the elements of 'c' exist in 'b' *);;
val it : bool = false

val union : Set<'a> -> Set<'a> -> Set<'a>

Compute the union of the two sets.

> let a = Set.ofSeq [ 1 .. 10 ]
let b = Set.ofSeq [ 5 .. 15 ];;
 
val a : Set<int>
val b : Set<int>
 
> Set.iter (fun x -> printf "%O " x) (Set.union a b);;
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 val it : unit = ()
 
> Set.iter (fun x -> printf "%O " x) (a + b);; (* '+' computes union *)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

[edit] Examples

open System
 
let shakespeare = "O Romeo, Romeo! wherefore art thou Romeo?"
let shakespeareArray =
    shakespeare.Split([| ' '; ','; '!'; '?' |], StringSplitOptions.RemoveEmptyEntries)
let shakespeareSet = shakespeareArray |> Set.ofSeq
 
let main() =
    printfn "shakespeare: %A" shakespeare
 
    let printCollection msg coll =
        printfn "%s:" msg
        Seq.iteri (fun index item -> printfn "  %i: %O" index item) coll
 
    printCollection "shakespeareArray" shakespeareArray
    printCollection "shakespeareSet" shakespeareSet
 
    Console.ReadKey(true) |> ignore
 
main()

shakespeare: "O Romeo, Romeo! wherefore art thou Romeo?"
shakespeareArray:
  0: O
  1: Romeo
  2: Romeo
  3: wherefore
  4: art
  5: thou
  6: Romeo
shakespeareSet:
  0: O
  1: Romeo
  2: art
  3: thou
  4: wherefore

[edit] Maps

A map is a special kind of set: it is associate keys with values. A map is created in a similar way to sets:

> let holidays =
    Map.empty.
        Add("Christmas", "Dec. 25").
        Add("Halloween", "Oct. 31").
        Add("Darwin Day", "Feb. 12").
        Add("World Vegan Day", "Nov. 1");; (* Start with empty Map *)
 
val holidays : Map<string,string>
 
> let monkeys =
    [ "Squirrel Monkey", "Simia sciureus";
        "Marmoset", "Callithrix jacchus";
        "Macaque", "Macaca mulatta";
        "Gibbon", "Hylobates lar";
        "Gorilla", "Gorilla gorilla";
        "Humans", "Homo sapiens";
        "Chimpanzee", "Pan troglodytes" ]
    |> Map.ofList;; (* Convert list to Map *)
 
val monkeys : Map<string,string>

You can use the .[key] to access elements in the map:

> holidays.["Christmas"];;
val it : string = "Dec. 25"
 
> monkeys.["Marmoset"];;
val it : string = "Callithrix jacchus"

[edit] The Map Module

The Microsoft.FSharp.Collections.Map module handles map operations.

val add : 'key -> 'a -> Map<'key,'a> -> Map<'key,'a>

Return a new map with the binding added to the given map.

val empty<'key,'a> : Map<'key,'a>

Returns an empty map.

val exists : ('key -> 'a -> bool) -> Map<'key,'a> -> bool

Return true if the given predicate returns true for one of the bindings in the map.

val filter : ('key -> 'a -> bool) -> Map<'key,'a> -> Map<'key,'a>

Build a new map containing only the bindings for which the given predicate returns true.

val find : 'key -> Map<'key,'a> -> 'a

Lookup an element in the map, raising KeyNotFoundException if no binding exists in the map.

val mem : 'key -> Map<'key,'a> -> bool

Test is an element is in the domain of the map.

val remove : 'key -> Map<'key,'a> -> Map<'key,'a>

Remove an element from the domain of the map. No exception is raised if the element is not present.

val tryfind : 'key -> Map<'key,'a> -> 'a option

Lookup an element in the map, returning a Some value if the element is in the domain of the map and None if not.

[edit] Examples

open System
 
let capitals =
    [("Australia", "Canberra"); ("Canada", "Ottawa"); ("China", "Beijing");
        ("Denmark", "Copenhagen"); ("Egypt", "Cairo"); ("Finland", "Helsinki");
        ("France", "Paris"); ("Germany", "Berlin"); ("India", "New Delhi");
        ("Japan", "Tokyo"); ("Mexico", "Mexico City"); ("Russia", "Moscow");
        ("Slovenia", "Ljubljana"); ("Spain", "Madrid"); ("Sweden", "Stockholm");
        ("Taiwan", "Taipei"); ("USA", "Washington D.C.")]
    |> Map.ofList
 
let rec main() =
    Console.Write("Find a capital by country (type 'q' to quit): ")
    match Console.ReadLine() with
    | "q" -> Console.WriteLine("Bye bye")
    | country ->
        match capitals.TryFind(country) with
        | Some(capital) -> Console.WriteLine("The capital of {0} is {1}\n", country, capital)
        | None -> Console.WriteLine("Country not found.\n")
        main() (* loop again *)
 
main()

Find a capital by country (type 'q' to quit): Egypt
The capital of Egypt is Cairo

Find a capital by country (type 'q' to quit): Slovenia
The capital of Slovenia is Ljubljana

Find a capital by country (type 'q' to quit): Latveria
Country not found.

Find a capital by country (type 'q' to quit): USA
The capital of USA is Washington D.C.

Find a capital by country (type 'q' to quit): q
Bye bye

[edit] Set and Map Performance

F# sets and maps are implemented as immutable red-black trees, an efficient data structure which forms a self-balancing binary tree. Red-black trees are well-known for their efficiency, in which they can search, insert, and delete elements in the tree in O(log n) time, where n is the number of elements in the tree.