计算一组所有子集(功率集) [英] Computing a set of all subsets (power set)

问题描述

我正在尝试获取一个函数(作为集合的参数给定)以返回一个集合，该集合的元素都是从主集合形成的子集. 例如:{1; 2; 3}-> {{1}，{2}，{3}，{1,2}，{1,3}，{2,3}，{1,2,3}}

I am trying to get a function (given as a parameter a set) to return a set whose elements are all the subset formed from the main set. Ex: {1;2;3} -> { {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }

但是我不完全知道如何制作一个可以让我使用一组集合的模块.我应该将其定义为哪种类型?

But I don't exactly know how to make a module that lets me work with a set of sets. What type should I define it as?

推荐答案

一组所有子集称为电源设置.要实现算法，您实际上不需要特殊的数据结构，因为您可以用列表表示一个集合.相应地，一组集合将是列表的列表.例如，

A set of all subsets is called a power set. To implement an algorithm you don't really need a special data structure, as you can represent a set with a list. Correspondingly a set of sets would be a list of lists. E.g.,

let rec superset = function | [] -> [[]] | x :: xs -> let ps = superset xs in ps @ List.map (fun ss -> x :: ss) ps

let rec superset = function | [] -> [[]] | x :: xs -> let ps = superset xs in ps @ List.map (fun ss -> x :: ss) ps

这是用法示例:

superset [1;2;3];;
- : int list list = [[]; [1]; [2]; [2; 1]; [3]; [3; 1]; [3; 2]; [3; 2; 1]

如果要使用实数集，如Jeffrey建议的那样.然后，我们将需要对算法进行一些调整，因为Set模块提供了一些不同的接口.

If you want to use real sets, like Jeffrey suggested. Then we will need to adapt the algorithm a little bit, since Set module provides a little bit different interface.

module S = Set.Make(String)
module SS = Set.Make(S) let superset xs = S.fold (fun x ps -> SS.fold (fun ss -> SS.add (S.add x ss)) ps ps) xs (SS.singleton S.empty)

module S = Set.Make(String)
module SS = Set.Make(S) let superset xs = S.fold (fun x ps -> SS.fold (fun ss -> SS.add (S.add x ss)) ps ps) xs (SS.singleton S.empty)

要运行该算法并打印结果，我们需要将其转换回列表，因为OCaml顶层不知道如何打印集:

To run the algorithm and to print the result we need to transform it back to a list, since OCaml toplevel doesn't know how to print sets:

# superset (S.of_list ["1"; "2"; "3"]) |> SS.elements |> List.map S.elements;;
- : S.elt list list =
[[]; ["1"]; ["1"; "2"]; ["1"; "2"; "3"]; ["1"; "3"]; ["2"]; ["2"; "3"];
 ["3"]]

现在，我们可以泛化算法，以便它可以在任何有序类型上使用.

Now, we can generalize algorithm, so that it will work on any ordered type.

module Superset(T : Set.OrderedType) = struct module S = Set.Make(T)
module SS = Set.Make(S) let of_set xs = S.fold (fun x ps -> SS.fold (fun ss -> SS.add (S.add x ss)) ps ps) xs (SS.singleton S.empty) end

module Superset(T : Set.OrderedType) = struct module S = Set.Make(T)
module SS = Set.Make(S) let of_set xs = S.fold (fun x ps -> SS.fold (fun ss -> SS.add (S.add x ss)) ps ps) xs (SS.singleton S.empty) end

现在我们可以运行它:

# module Superset_of_strings = Superset(String);;
# open Superset_of_string;;  
# of_set (S.of_list ["1"; "2"; "3"]) |> SS.elements |> List.map S.elements;;
- : S.elt list list =
[[]; ["1"]; ["1"; "2"]; ["1"; "2"; "3"]; ["1"; "3"]; ["2"]; ["2"; "3"];
 ["3"]]

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