Set consolidation

Set consolidation
You are encouraged to solve this task according to the task description, using any language you may know.

Given two sets of items then if any item is common to any set then the result of applying consolidation to those sets is a set of sets whose contents is:

• The two input sets if no common item exists between the two input sets of items.
• The single set that is the union of the two input sets if they share a common item.

Given N sets of items where N>2 then the result is the same as repeatedly replacing all combinations of two sets by their consolidation until no further consolidation between set pairs is possible. If N<2 then consolidation has no strict meaning and the input can be returned.

Example 1:

Given the two sets {A,B} and {C,D} then there is no common element between the sets and the result is the same as the input.

Example 2:

Given the two sets {A,B} and {B,D} then there is a common element B between the sets and the result is the single set {B,D,A}. (Note that order of items in a set is immaterial: {A,B,D} is the same as {B,D,A} and {D,A,B}, etc).

Example 3:

Given the three sets {A,B} and {C,D} and {D,B} then there is no common element between the sets {A,B} and {C,D} but the sets {A,B} and {D,B} do share a common element that consolidates to produce the result {B,D,A}. On examining this result with the remaining set, {C,D}, they share a common element and so consolidate to the final output of the single set {A,B,C,D}

Example 4:

The consolidation of the five sets:

{H,I,K}, {A,B}, {C,D}, {D,B}, and {F,G,H}

Is the two sets:

{A, C, B, D}, and {G, F, I, H, K}

See also

• Connected component (graph theory)

Ada

We start with specifying a generic package Set_Cons that provides the neccessary tools, such as contructing and manipulating sets, truning them, ect.:

<lang Ada>generic

  type Element is (<>);
  with function Image(E: Element) return String;

package Set_Cons is

  type Set is private;

  -- constructor and manipulation functions for type Set
  function "+"(E: Element) return Set;
  function "+"(Left, Right: Element) return Set;
  function "+"(Left: Set; Right: Element) return Set;
  function "-"(Left: Set; Right: Element) return Set;

  -- compare, unite or output a Set
  function Nonempty_Intersection(Left, Right: Set) return Boolean;
  function Union(Left, Right: Set) return Set;
  function Image(S: Set) return String;

  type Set_Vec is array(Positive range <>) of Set;

  -- output a Set_Vec
  function Image(V: Set_Vec) return String;

private

  type Set is array(Element) of Boolean;

end Set_Cons;</lang>

Here is the implementation of Set_Cons:

<lang Ada>package body Set_Cons is

  function "+"(E: Element) return Set is
     S: Set := (others => False);
  begin
     S(E) := True;
     return S;
  end "+";

  function "+"(Left, Right: Element) return Set is
  begin
     return (+Left) + Right;
  end "+";

  function "+"(Left: Set; Right: Element) return Set is
     S: Set := Left;
  begin
     S(Right) := True;
     return S;
  end "+";

  function "-"(Left: Set; Right: Element) return Set is
     S: Set := Left;
  begin
     S(Right) := False;
     return S;
  end "-";

  function Nonempty_Intersection(Left, Right: Set) return Boolean is
  begin
     for E in Element'Range loop
        if Left(E) and then Right(E) then return True;
        end if;
     end loop;
     return False;
  end Nonempty_Intersection;

  function Union(Left, Right: Set) return Set is
     S: Set := Left;
  begin
     for E in Right'Range loop
        if Right(E) then S(E) := True;
        end if;
     end loop;
     return S;
  end Union;

  function Image(S: Set) return String is

     function Image(S: Set; Found: Natural) return String is
     begin
        for E in S'Range loop
           if S(E) then
              if Found = 0 then
                 return Image(E) & Image((S-E), Found+1);
              else
                 return "," & Image(E) & Image((S-E), Found+1);
              end if;
           end if;
        end loop;
        return "";
     end Image;

  begin
     return "{" & Image(S, 0) & "}";
  end Image;

  function Image(V: Set_Vec) return String is
   begin
     if V'Length = 0 then
        return "";
     else
        return Image(V(V'First)) & Image(V(V'First+1 .. V'Last));
     end if;
  end Image;

end Set_Cons;</lang>

Given that package, the task is easy:

<lang Ada>with Ada.Text_IO, Set_Cons;

procedure Set_Consolidation is

  type El_Type is (A, B, C, D, E, F, G, H, I, K);

  function Image(El: El_Type) return String is
  begin
     return El_Type'Image(El);
  end Image;

  package Helper is new Set_Cons(Element => El_Type, Image => Image);
  use Helper;

  function Consolidate(List: Set_Vec) return Set_Vec is
  begin
     for I in List'First .. List'Last - 1 loop
        for J in I+1 .. List'Last loop
           -- if List(I) and List(J) share an element
           -- then recursively consolidate
           --   (List(I) union List(J)) followed by List(K), K not in {I, J}
           if Nonempty_Intersection(List(I), List(J)) then
              return Consolidate
                (Union(List(I), List(J)) 
                   & List(List'First .. I-1) 
                   & List(I+1        .. J-1) 
                   & List(J+1        .. List'Last));
           end if;
        end loop;
     end loop;
     return List;
  end Consolidate;

begin

  Ada.Text_IO.Put_Line(Image(Consolidate((A+B) & (C+D))));
  Ada.Text_IO.Put_Line(Image(Consolidate((A+B) & (B+D))));
  Ada.Text_IO.Put_Line(Image(Consolidate((A+B) & (C+D) & (D+B))));
  Ada.Text_IO.Put_Line
    (Image(Consolidate((H+I+K) & (A+B) & (C+D) & (D+B) & (F+G+H))));

end Set_Consolidation;</lang>

Output:

{A,B}{C,D}
{A,B,D}
{A,B,C,D}
{A,B,C,D}{F,G,H,I,K}

Aime

<lang aime>void display(list l) {

   integer i;
   record r;

   for (i, r in l) {
       text u, v;

       o_text(i ? ", {" : "{");
       for (u in r) {
           o_(v, u);
           v = ", ";
       }
       o_text("}");
   }

   o_text("\n");

}

integer intersect(record r, record u) {

   trap_q(r_vcall, r, r_put, 1, record().copy(u), 0);

}

void merge(record u, record r) {

   r.wcall(r_add, 1, 2, u);

}

list consolidate(list l) {

   integer i;
   record r;

   for (i, r in l) {
       integer j;

       j = i - ~l;
       while (j += 1) {
           record u;

           u = l[j];
           if (intersect(r, u)) {
               merge(u, r);
               l.delete(i);
               i -= 1;
               break;
           }
       }
   }

   l;

}

record R(...) {

   record r;

   ucall(r_put, 1, r, 0);

   return r;

}

integer main(void) {

   display(consolidate(list(R("A", "B"), R("C", "D"))));
   display(consolidate(list(R("A", "B"), R("B", "D"))));
   display(consolidate(list(R("A", "B"), R("C", "D"), R("D", "B"))));
   display(consolidate(list(R("H", "I", "K"), R("A", "B"), R("C", "D"),
                            R("D", "B"), R("F", "G", "K"))));

   0;

}</lang>

Output:

{A, B}, {C, D}
{A, B, D}
{A, B, C, D}
{A, B, C, D}, {F, G, H, I, K}

Bracmat

<lang bracmat>( ( consolidate

 =   a m z mm za zm zz
   .     ( removeNumFactors
         =   a m z
           .     !arg:?a+#%*?m+?z
               & !a+!m+removeNumFactors$!z
             | !arg
         )
       &   !arg
         :   ?a
             %?`m
             ( %?z
             &   !m
               :   ?
                 + ( %@?mm
                   & !z:?za (?+!mm+?:?zm) ?zz
                   )
                 + ?
             )
       & consolidate$(!a removeNumFactors$(!m+!zm) !za !zz)
     | !arg
 )

& (test=.out$(!arg "==>" consolidate$!arg)) & test$(A+B C+D) & test$(A+B B+D) & test$(A+B C+D D+B) & test$(H+I+K A+B C+D D+B F+G+H) );</lang>

Output:

A+B C+D ==> A+B C+D
A+B B+D ==> A+B+D
A+B C+D B+D ==> A+B+C+D
  H+I+K
  A+B
  C+D
  B+D
  F+G+H
  ==>
  F+G+H+I+K
  A+B+C+D

C

<lang c>#include <stdio.h>

1. define s(x) (1U << ((x) - 'A'))

typedef unsigned int bitset;

int consolidate(bitset *x, int len) { int i, j; for (i = len - 2; i >= 0; i--) for (j = len - 1; j > i; j--) if (x[i] & x[j]) x[i] |= x[j], x[j] = x[--len]; return len; }

void show_sets(bitset *x, int len) { bitset b; while(len--) { for (b = 'A'; b <= 'Z'; b++) if (x[len] & s(b)) printf("%c ", b); putchar('\n'); } }

int main(void) { bitset x[] = { s('A') | s('B'), s('C') | s('D'), s('B') | s('D'), s('F') | s('G') | s('H'), s('H') | s('I') | s('K') };

int len = sizeof(x) / sizeof(x[0]);

puts("Before:"); show_sets(x, len); puts("\nAfter:"); show_sets(x, consolidate(x, len)); return 0; }</lang>

The above is O(N²) in terms of number of input sets. If input is large (many sets or huge number of elements), here's an O(N) method, where N is the sum of the sizes of all input sets: <lang c>#include <stdio.h>

1. include <stdlib.h>
2. include <string.h>

struct edge { int to; struct edge *next; }; struct node { int group; struct edge *e; };

int **consolidate(int **x) {

1. define alloc(v, size) v = calloc(size, sizeof(v[0]));

int group, n_groups, n_nodes; int n_edges = 0; struct edge *edges, *ep; struct node *nodes; int pos, *stack, **ret;

void add_edge(int a, int b) { ep->to = b; ep->next = nodes[a].e; nodes[a].e = ep; ep++; }

void traverse(int a) { if (nodes[a].group) return;

nodes[a].group = group; stack[pos++] = a;

for (struct edge *e = nodes[a].e; e; e = e->next) traverse(e->to); }

n_groups = n_nodes = 0; for (int i = 0; x[i]; i++, n_groups++) for (int j = 0; x[i][j]; j++) { n_edges ++; if (x[i][j] >= n_nodes) n_nodes = x[i][j] + 1; }

alloc(ret, n_nodes); alloc(nodes, n_nodes); alloc(stack, n_nodes); ep = alloc(edges, n_edges);

for (int i = 0; x[i]; i++) for (int *s = x[i], j = 0; s[j]; j++) add_edge(s[j], s[j + 1] ? s[j + 1] : s[0]);

group = 0; for (int i = 1; i < n_nodes; i++) { if (nodes[i].group) continue;

group++, pos = 0; traverse(i);

stack[pos++] = 0; ret[group - 1] = malloc(sizeof(int) * pos); memcpy(ret[group - 1], stack, sizeof(int) * pos); }

free(edges); free(stack); free(nodes);

// caller is responsible for freeing ret return realloc(ret, sizeof(ret[0]) * (1 + group));

1. undef alloc

}

void show_sets(int **x) { for (int i = 0; x[i]; i++) { printf("%d: ", i); for (int j = 0; x[i][j]; j++) printf(" %d", x[i][j]); putchar('\n'); } }

int main(void) { int *x[] = { (int[]) {1, 2, 0}, // 0: end of set (int[]) {3, 4, 0}, (int[]) {3, 1, 0}, (int[]) {0}, // empty set (int[]) {5, 6, 0}, (int[]) {7, 6, 0}, (int[]) {3, 9, 10, 0}, 0 // 0: end of sets };

puts("input:"); show_sets(x);

puts("components:"); show_sets(consolidate(x));

return 0; }</lang>

C#

<lang csharp>using System; using System.Linq; using System.Collections.Generic;

public class SetConsolidation {

   public static void Main()
   {
       var setCollection1 = new[] {new[] {"A", "B"}, new[] {"C", "D"}};
       var setCollection2 = new[] {new[] {"A", "B"}, new[] {"B", "D"}};
       var setCollection3 = new[] {new[] {"A", "B"}, new[] {"C", "D"}, new[] {"B", "D"}};
       var setCollection4 = new[] {new[] {"H", "I", "K"}, new[] {"A", "B"}, new[] {"C", "D"},
           new[] {"D", "B"}, new[] {"F", "G", "H"}};
       var input = new[] {setCollection1, setCollection2, setCollection3, setCollection4};
       
       foreach (var sets in input) {
           Console.WriteLine("Start sets:");
           Console.WriteLine(string.Join(", ", sets.Select(s => "{" + string.Join(", ", s) + "}")));
           Console.WriteLine("Sets consolidated using Nodes:");
           Console.WriteLine(string.Join(", ", ConsolidateSets1(sets).Select(s => "{" + string.Join(", ", s) + "}")));
           Console.WriteLine("Sets consolidated using Set operations:");
           Console.WriteLine(string.Join(", ", ConsolidateSets2(sets).Select(s => "{" + string.Join(", ", s) + "}")));
           Console.WriteLine();
       }
   }
   
   /// <summary>
   /// Consolidates sets using a connected-component-finding-algorithm involving Nodes with parent pointers.
   /// The more efficient solution, but more elaborate code.
   /// </summary>
   private static IEnumerable<IEnumerable<T>> ConsolidateSets1<T>(IEnumerable<IEnumerable<T>> sets,
       IEqualityComparer<T> comparer = null)
   {
       if (comparer == null) comparer = EqualityComparer<T>.Default;
       var elements = new Dictionary<T, Node<T>>();
       foreach (var set in sets) {
           Node<T> top = null;
           foreach (T value in set) {
               Node<T> element;
               if (elements.TryGetValue(value, out element)) {
                   if (top != null) {
                       var newTop = element.FindTop();
                       top.Parent = newTop;
                       element.Parent = newTop;
                       top = newTop;
                   } else {
                       top = element.FindTop();
                   }
               } else {
                   elements.Add(value, element = new Node<T>(value));
                   if (top == null) top = element;
                   else element.Parent = top;
               }
           }
       }
       foreach (var g in elements.Values.GroupBy(element => element.FindTop().Value))
           yield return g.Select(e => e.Value);
   }
   
   private class Node<T>
   {
       public Node(T value, Node<T> parent = null) {
           Value = value;
           Parent = parent ?? this;
       }
       
       public T Value { get; }
       public Node<T> Parent { get; set; }
       
       public Node<T> FindTop() {
           var top = this;
           while (top != top.Parent) top = top.Parent;
           //Set all parents to the top element to prevent repeated iteration in the future
           var element = this;
           while (element.Parent != top) {
               var parent = element.Parent;
               element.Parent = top;
               element = parent;
           }
           return top;
       }
   }
   
   /// <summary>
   /// Consolidates sets using operations on the HashSet<T> class.
   /// Less efficient than the other method, but easier to write.
   /// </summary>
   private static IEnumerable<IEnumerable<T>> ConsolidateSets2<T>(IEnumerable<IEnumerable<T>> sets,
       IEqualityComparer<T> comparer = null)
   {
       if (comparer == null) comparer = EqualityComparer<T>.Default;
       var currentSets = sets.Select(s => new HashSet<T>(s)).ToList();
       int previousSize;
       do {
           previousSize = currentSets.Count;
           for (int i = 0; i < currentSets.Count - 1; i++) {
               for (int j = currentSets.Count - 1; j > i; j--) {
                   if (currentSets[i].Overlaps(currentSets[j])) {
                       currentSets[i].UnionWith(currentSets[j]);
                       currentSets.RemoveAt(j);
                   }
               }
           }
       } while (previousSize > currentSets.Count);
       foreach (var set in currentSets) yield return set.Select(value => value);
   }

}</lang>

Output:

Start sets:
{A, B}, {C, D}
Sets consolidated using nodes:
{A, B}, {C, D}
Sets consolidated using Set operations:
{A, B}, {C, D}

Start sets:
{A, B}, {B, D}
Sets consolidated using nodes:
{A, B, D}
Sets consolidated using Set operations:
{A, B, D}

Start sets:
{A, B}, {C, D}, {B, D}
Sets consolidated using nodes:
{A, B, C, D}
Sets consolidated using Set operations:
{A, B, D, C}

Start sets:
{H, I, K}, {A, B}, {C, D}, {D, B}, {F, G, H}
Sets consolidated using nodes:
{H, I, K, F, G}, {A, B, C, D}
Sets consolidated using Set operations:
{H, I, K, F, G}, {A, B, D, C}

Common Lisp

Translation of: Racket

<lang lisp>(defun consolidate (ss)

 (labels ((comb (cs s)
            (cond ((null s) cs)
                  ((null cs) (list s))
                  ((null (intersection s (first cs)))
                   (cons (first cs) (comb (rest cs) s)))
                  ((consolidate (cons (union s (first cs)) (rest cs)))))))
   (reduce #'comb ss :initial-value nil)))</lang>

Output:

> (consolidate '((a b) (c d)))
((A B) (C D))
> (consolidate '((a b) (b c)))
((C A B))
> (consolidate '((a b) (c d) (d b)))
((C D A B))
> (consolidate '((h i k) (a b) (c d) (d b) (f g h)))
((F G H I K) (C D A B))

D

Translation of: Go

<lang d>import std.stdio, std.algorithm, std.array;

dchar[][] consolidate(dchar[][] sets) @safe {

   foreach (set; sets)
       set.sort();

   foreach (i, ref si; sets[0 .. $ - 1]) {
       if (si.empty)
           continue;
       foreach (ref sj; sets[i + 1 .. $])
           if (!sj.empty && !si.setIntersection(sj).empty) {
               sj = si.setUnion(sj).uniq.array;
               si = null;
           }
   }

   return sets.filter!"!a.empty".array;

}

void main() @safe {

   [['A', 'B'], ['C','D']].consolidate.writeln;

   [['A','B'], ['B','D']].consolidate.writeln;

   [['A','B'], ['C','D'], ['D','B']].consolidate.writeln;

   [['H','I','K'], ['A','B'], ['C','D'],
    ['D','B'], ['F','G','H']].consolidate.writeln;

}</lang>

Output:

["AB", "CD"]
["ABD"]
["ABCD"]
["ABCD", "FGHIK"]

Recursive version, as described on talk page. <lang d>import std.stdio, std.algorithm, std.array;

dchar[][] consolidate(dchar[][] sets) @safe {

   foreach (set; sets)
       set.sort();

   dchar[][] consolidateR(dchar[][] s) {
       if (s.length < 2)
           return s;
       auto r = [s[0]];
       foreach (x; consolidateR(s[1 .. $])) {
           if (!r[0].setIntersection(x).empty) {
               r[0] = r[0].setUnion(x).uniq.array;
           } else
               r ~= x;
       }
       return r;
   }

   return consolidateR(sets);

}

void main() @safe {

   [['A', 'B'], ['C','D']].consolidate.writeln;

   [['A','B'], ['B','D']].consolidate.writeln;

   [['A','B'], ['C','D'], ['D','B']].consolidate.writeln;

   [['H','I','K'], ['A','B'], ['C','D'],
    ['D','B'], ['F','G','H']].consolidate.writeln;

}</lang>

["AB", "CD"]
["ABD"]
["ABCD"]
["FGHIK", "ABCD"]

EchoLisp

<lang scheme>

utility

make a set of sets from a list

(define (make-set* s) (or (when (list? s) (make-set (map make-set* s))) s))

union of all sets which intersect - O(n^2)

(define (make-big ss) (make-set (for/list ((u ss)) (for/fold (big u) ((v ss)) #:when (set-intersect? big v) (set-union big v)))))

remove sets which are subset of another one - O(n^2)

(define (remove-small ss) (for/list ((keep ss)) #:when (for/and ((v ss)) #:continue (set-equal? keep v) (not (set-subset? v keep))) keep))

(define (consolidate ss) (make-set (remove-small (make-big ss))))

(define S (make-set* ' ((h i k) ( a b) ( b c) (c d) ( f g h))))

   → { { a b } { b c } { c d } { f g h } { h i k } }

(consolidate S)

   → { { a b c d } { f g h i k } }

</lang>

Egison

<lang egison> (define $consolidate

 (lambda [$xss]
   (match xss (multiset (set char))
     {[<cons <cons $m $xs>
             <cons <cons ,m $ys>
                   $rss>>
       (consolidate {(unique/m char {m @xs @ys}) @rss})]
      [_ xss]})))

(test (consolidate {{'H' 'I' 'K'} {'A' 'B'} {'C' 'D'} {'D' 'B'} {'F' 'G' 'H'}})) </lang>

Output:

<lang egison> {"DBAC" "HIKFG"} </lang>

Ela

This solution emulate sets using linked lists: <lang ela>open list

merge [] ys = ys merge (x::xs) ys | x `elem` ys = merge xs ys

                | else = merge xs (x::ys)

consolidate (_::[])@xs = xs consolidate (x::xs) = conso [x] (consolidate xs)

               where conso xs [] = xs
                     conso (x::xs)@r (y::ys) | intersect x y <> [] = conso ((merge x y)::xs) ys
                                             | else = conso (r ++ [y]) ys</lang>

Usage: <lang ela>open monad io

IO

do

 x <- return $ consolidate [['H','I','K'], ['A','B'], ['C','D'], ['D','B'], ['F','G','H']]
 putLn x
 y <- return $ consolidate [['A','B'], ['B','D']]
 putLn y</lang>

Output:

[['K','I','F','G','H'],['A','C','D','B']]
[['A','B','D']]

Elixir

<lang elixir>defmodule RC do

 def set_consolidate(sets, result\\[])
 def set_consolidate([], result), do: result
 def set_consolidate([h|t], result) do
   case Enum.find(t, fn set -> not MapSet.disjoint?(h, set) end) do
     nil -> set_consolidate(t, [h | result])
     set -> set_consolidate([MapSet.union(h, set) | t -- [set]], result)
   end
 end

end

examples = [[[:A,:B], [:C,:D]],

           [[:A,:B], [:B,:D]],
           [[:A,:B], [:C,:D], [:D,:B]],
           [[:H,:I,:K], [:A,:B], [:C,:D], [:D,:B], [:F,:G,:H]]]
          |> Enum.map(fn sets ->
               Enum.map(sets, fn set -> MapSet.new(set) end)
             end)

Enum.each(examples, fn sets ->

 IO.write "#{inspect sets} =>\n\t"
 IO.inspect RC.set_consolidate(sets)

end)</lang>

Output:

[#MapSet<[:A, :B]>, #MapSet<[:C, :D]>] =>
	[#MapSet<[:C, :D]>, #MapSet<[:A, :B]>]
[#MapSet<[:A, :B]>, #MapSet<[:B, :D]>] =>
	[#MapSet<[:A, :B, :D]>]
[#MapSet<[:A, :B]>, #MapSet<[:C, :D]>, #MapSet<[:B, :D]>] =>
	[#MapSet<[:A, :B, :C, :D]>]
[#MapSet<[:H, :I, :K]>, #MapSet<[:A, :B]>, #MapSet<[:C, :D]>, #MapSet<[:B, :D]>, #MapSet<[:F, :G, :H]>] =>
	[#MapSet<[:A, :B, :C, :D]>, #MapSet<[:F, :G, :H, :I, :K]>]

F#

<lang fsharp>let (|SeqNode|SeqEmpty|) s =

   if Seq.isEmpty s then SeqEmpty
   else SeqNode ((Seq.head s), Seq.skip 1 s)

let SetDisjunct x y = Set.isEmpty (Set.intersect x y)

let rec consolidate s = seq {

   match s with
   | SeqEmpty -> ()
   | SeqNode (this, rest) ->
       let consolidatedRest = consolidate rest
       for that in consolidatedRest do
           if (SetDisjunct this that) then yield that
       yield Seq.fold (fun x y -> if not (SetDisjunct x y) then Set.union x y else x) this consolidatedRest

}

[<EntryPoint>] let main args =

   let makeSeqOfSet listOfList = List.map (fun x -> Set.ofList x) listOfList |> Seq.ofList
   List.iter (fun x -> printfn "%A" (consolidate (makeSeqOfSet x))) [
       [["A";"B"]; ["C";"D"]];
       [["A";"B"]; ["B";"C"]];
       [["A";"B"]; ["C";"D"]; ["D";"B"]];
       [["H";"I";"K"]; ["A";"B"]; ["C";"D"]; ["D";"B"]; ["F";"G";"H"]]
   ]
   0</lang>

Output:

seq [set ["C"; "D"]; set ["A"; "B"]]
seq [set ["A"; "B"; "C"]]
seq [set ["A"; "B"; "C"; "D"]]
seq [set ["A"; "B"; "C"; "D"]; set ["F"; "G"; "H"; "I"; "K"]]

Go

Translation of: Python

<lang go>package main

import "fmt"

type set map[string]bool

var testCase = []set{

   set{"H": true, "I": true, "K": true},
   set{"A": true, "B": true},
   set{"C": true, "D": true},
   set{"D": true, "B": true},
   set{"F": true, "G": true, "H": true},

}

func main() {

   fmt.Println(consolidate(testCase))

}

func consolidate(sets []set) []set {

   setlist := []set{}
   for _, s := range sets {
       if s != nil && len(s) > 0 {
           setlist = append(setlist, s)
       }
   }
   for i, s1 := range setlist {
       if len(s1) > 0 {
           for _, s2 := range setlist[i+1:] {
               if s1.disjoint(s2) {
                   continue
               }
               for e := range s1 {
                   s2[e] = true
                   delete(s1, e)
               }
               s1 = s2
           }
       }
   }
   r := []set{}
   for _, s := range setlist {
       if len(s) > 0 {
           r = append(r, s)
       }
   }
   return r

}

func (s1 set) disjoint(s2 set) bool {

   for e := range s2 {
       if s1[e] {
           return false
       }
   }
   return true

}</lang>

Output:

[map[A:true C:true B:true D:true] map[G:true F:true I:true H:true K:true]]

Haskell

<lang Haskell>import Data.List (intersperse, intercalate) import qualified Data.Set as S

consolidate

 :: Ord a
 => [S.Set a] -> [S.Set a]

consolidate = foldr comb []

 where
   comb s_ [] = [s_]
   comb s_ (s:ss)
     | S.null (s `S.intersection` s_) = s : comb s_ ss
     | otherwise = comb (s `S.union` s_) ss

-- TESTS ------------------------------------------------- main :: IO () main =

 (putStrLn . unlines)
   ((intercalate ", and " . fmap showSet . consolidate) . fmap S.fromList <$>
    [ ["ab", "cd"]
    , ["ab", "bd"]
    , ["ab", "cd", "db"]
    , ["hik", "ab", "cd", "db", "fgh"]
    ])

showSet :: S.Set Char -> String showSet = flip intercalate ["{", "}"] . intersperse ',' . S.elems</lang>

Output:

{c,d}, and {a,b}
{a,b,d}
{a,b,c,d}
{a,b,c,d}, and {f,g,h,i,k}

J

<lang J>consolidate=:4 :0/

 b=. y 1&e.@e.&> x
 (1,-.b)#(~.;x,b#y);y

)</lang>

Examples:

<lang J> consolidate 'ab';'cd' ┌──┬──┐ │ab│cd│ └──┴──┘

  consolidate 'ab';'bd'

┌───┐ │abd│ └───┘

  consolidate 'ab';'cd';'db'

┌────┐ │abcd│ └────┘

  consolidate 'hij';'ab';'cd';'db';'fgh'

┌─────┬────┐ │hijfg│abcd│ └─────┴────┘</lang>

Java

Translation of: D

Works with: Java version 7

<lang java>import java.util.*;

public class SetConsolidation {

   public static void main(String[] args) {
       List<Set<Character>> h1 = hashSetList("AB", "CD");
       System.out.println(consolidate(h1));

       List<Set<Character>> h2 = hashSetList("AB", "BD");
       System.out.println(consolidateR(h2));

       List<Set<Character>> h3 = hashSetList("AB", "CD", "DB");
       System.out.println(consolidate(h3));

       List<Set<Character>> h4 = hashSetList("HIK", "AB", "CD", "DB", "FGH");
       System.out.println(consolidateR(h4));
   }

   // iterative
   private static <E> List<Set<E>>
               consolidate(Collection<? extends Set<E>> sets) {

List<Set<E>> r = new ArrayList<>(); for (Set<E> s : sets) { List<Set<E>> new_r = new ArrayList<>(); new_r.add(s); for (Set<E> x : r) { if (!Collections.disjoint(s, x)) { s.addAll(x); } else { new_r.add(x); } } r = new_r; } return r;

   }

   // recursive
   private static <E> List<Set<E>> consolidateR(List<Set<E>> sets) {
       if (sets.size() < 2)
           return sets;
       List<Set<E>> r = new ArrayList<>();
       r.add(sets.get(0));
       for (Set<E> x : consolidateR(sets.subList(1, sets.size()))) {
           if (!Collections.disjoint(r.get(0), x)) {
               r.get(0).addAll(x);
           } else {
               r.add(x);
           }
       }
       return r;
   }

   private static List<Set<Character>> hashSetList(String... set) {
       List<Set<Character>> r = new ArrayList<>();
       for (int i = 0; i < set.length; i++) {
           r.add(new HashSet<Character>());
           for (int j = 0; j < set[i].length(); j++)
               r.get(i).add(set[i].charAt(j));
       }
       return r;
   }

}</lang>

[A, B] [D, C] 
[D, A, B] 
[D, A, B, C] 
[F, G, H, I, K] [D, A, B, C]

JavaScript

<lang javascript>(() => {

   'use strict';

   // consolidated :: Ord a => [Set a] -> [Set a]
   const consolidated = xs => {
       const go = (s, xs) =>
           0 !== xs.length ? (() => {
               const h = xs[0];
               return 0 === intersection(h, s).size ? (
                   [h].concat(go(s, tail(xs)))
               ) : go(union(h, s), tail(xs));
           })() : [s];
       return foldr(go, [], xs);
   };

   // TESTS -------------------------------------------------
   const main = () =>
       map(xs => intercalate(
               ', and ',
               map(showSet, consolidated(xs))
           ),
           map(x => map(
                   s => new Set(chars(s)),
                   x
               ),
               [
                   ['ab', 'cd'],
                   ['ab', 'bd'],
                   ['ab', 'cd', 'db'],
                   ['hik', 'ab', 'cd', 'db', 'fgh']
               ]
           )
       ).join('\n');

   // GENERIC FUNCTIONS ----------------------------

   // chars :: String -> [Char]
   const chars = s => s.split();

   // concat :: a -> [a]
   // concat :: [String] -> String
   const concat = xs =>
       0 < xs.length ? (() => {
           const unit = 'string' !== typeof xs[0] ? (
               []
           ) : ;
           return unit.concat.apply(unit, xs);
       })() : [];

   // elems :: Dict -> [a]
   // elems :: Set -> [a]
   const elems = x =>
       'Set' !== x.constructor.name ? (
           Object.values(x)
       ) : Array.from(x.values());

   // flip :: (a -> b -> c) -> b -> a -> c
   const flip = f =>
       1 < f.length ? (
           (a, b) => f(b, a)
       ) : (x => y => f(y)(x));

   // Note that that the Haskell signature of foldr is different from that of
   // foldl - the positions of accumulator and current value are reversed

   // foldr :: (a -> b -> b) -> b -> [a] -> b
   const foldr = (f, a, xs) => xs.reduceRight(flip(f), a);

   // intercalate :: [a] -> a -> [a]
   // intercalate :: String -> [String] -> String
   const intercalate = (sep, xs) =>
       0 < xs.length && 'string' === typeof sep &&
       'string' === typeof xs[0] ? (
           xs.join(sep)
       ) : concat(intersperse(sep, xs));

   // intersection :: Ord a => Set a -> Set a -> Set a
   const intersection = (s, s1) =>
       new Set([...s].filter(x => s1.has(x)));

   // intersperse :: a -> [a] -> [a]
   // intersperse :: Char -> String -> String
   const intersperse = (sep, xs) => {
       const bln = 'string' === typeof xs;
       return xs.length > 1 ? (
           (bln ? concat : x => x)(
               (bln ? (
                   xs.split()
               ) : xs)
               .slice(1)
               .reduce((a, x) => a.concat([sep, x]), [xs[0]])
           )) : xs;
   };

   // map :: (a -> b) -> [a] -> [b]
   const map = (f, xs) => xs.map(f);

   // showSet :: Set -> String
   const showSet = s =>
       intercalate(elems(s), ['{', '}']);

   // sort :: Ord a => [a] -> [a]
   const sort = xs => xs.slice()
       .sort((a, b) => a < b ? -1 : (a > b ? 1 : 0));

   // tail :: [a] -> [a]
   const tail = xs => 0 < xs.length ? xs.slice(1) : [];

   // union :: Ord a => Set a -> Set a -> Set a
   const union = (s, s1) =>
       Array.from(s1.values())
       .reduce(
           (a, x) => (a.add(x), a),
           new Set(s)
       );

   // MAIN ---
   return main();

})();</lang>

Output:

{c,d}, and {a,b}
{b,d,a}
{d,b,c,a}
{d,b,c,a}, and {f,g,h,i,k}

jq

Infrastructure:

Currently, jq does not have a "Set" library, so to save space here, we will use simple but inefficient implementations of set-oriented functions as they are fast for sets of moderate size. Nevertheless, we will represent sets as sorted arrays. <lang jq>def to_set: unique;

def union(A; B): (A + B) | unique;

1. boolean

def intersect(A;B):

 reduce A[] as $x (false; if . then . else (B|index($x)) end) | not | not;</lang>

Consolidation:

For clarity, the helper functions are presented as top-level functions, but they could be defined as inner functions of the main function, consolidate/0.

<lang jq># Input: [i, j, sets] with i < j

1. Return [i,j] for a pair that can be combined, else null

def combinable:

  .[0] as $i | .[1] as $j | .[2] as $sets
  | ($sets|length) as $length
  | if intersect($sets[$i]; $sets[$j]) then [$i, $j]
    elif $i < $j - 1      then (.[0] += 1 | combinable)
    elif $j < $length - 1 then [0, $j+1, $sets] | combinable
    else null
    end;

1. Given an array of arrays, remove the i-th and j-th elements,
2. and add their union:

def update(i;j):

 if i > j then update(j;i)
 elif i == j then del(.[i])
 else 
   union(.[i]; .[j]) as $c
   | union(del(.[j]) | del(.[i]); [$c])
 end;

1. Input: a set of sets

def consolidate:

  if length <= 1 then .
  else
    ([0, 1, .] | combinable) as $c
    | if $c then update($c[0]; $c[1]) | consolidate
      else .
      end
  end;

</lang> Examples: <lang jq>def tests:

 [["A", "B"], ["C","D"]],
 [["A","B"], ["B","D"]],
 [["A","B"], ["C","D"], ["D","B"]],
 [["H","I","K"], ["A","B"], ["C","D"], ["D","B"], ["F","G","H"]]

def test:

 tests | to_set | consolidate;

test</lang>

Output:

<lang sh>$ jq -c -n -f Set_consolidation.rc [["A","B"],["C","D"]] "A","B","D" "A","B","C","D" [["A","B","C","D"],["F","G","H","I","K"]]</lang>

Julia

The consolidate Function

Here I assume that the data are contained in a list of sets. Perhaps a recursive solution would be more elegant, but in this case playing games with a stack works well enough. <lang Julia> function consolidate{T}(a::Array{Set{T},1})

   1 < length(a) || return a
   b = copy(a)
   c = Set{T}[]
   while 1 < length(b)
       x = shift!(b)
       cme = true
       for (i, y) in enumerate(b)
           !isempty(intersect(x, y)) || continue
           cme = false
           b[i] = union(x, y)
           break
       end
       !cme || push!(c, x)
   end
   push!(c, b[1])
   return c

end </lang>

Main <lang Julia> p = Set(["A", "B"]) q = Set(["C", "D"]) r = Set(["B", "D"]) s = Set(["H", "I", "K"]) t = Set(["F", "G", "H"])

println("p = ", p) println("q = ", q) println("r = ", r) println("s = ", s) println("t = ", t)

println("consolidate([p, q]) =\n ", consolidate([p, q])) println("consolidate([p, r]) =\n ", consolidate([p, r])) println("consolidate([p, q, r]) =\n ", consolidate([p, q, r])) println("consolidate([p, q, r, s, t]) =\n ",

       consolidate([p, q, r, s, t]))

</lang>

Output:

p = Set{ASCIIString}({"B","A"})
q = Set{ASCIIString}({"C","D"})
r = Set{ASCIIString}({"B","D"})
s = Set{ASCIIString}({"I","K","H"})
t = Set{ASCIIString}({"G","F","H"})
consolidate([p, q]) =
    [Set{ASCIIString}({"B","A"}),Set{ASCIIString}({"C","D"})]
consolidate([p, r]) =
    [Set{ASCIIString}({"B","A","D"})]
consolidate([p, q, r]) =
    [Set{ASCIIString}({"B","A","C","D"})]
consolidate([p, q, r, s, t]) =
    [Set{ASCIIString}({"B","A","C","D"}),Set{ASCIIString}({"I","G","K","H","F"})]

Kotlin

<lang scala>// version 1.0.6

fun<T : Comparable<T>> consolidateSets(sets: Array<Set<T>>): Set<Set<T>> {

   val size = sets.size
   val consolidated = BooleanArray(size) // all false by default
   var i = 0
   while (i < size - 1) {
       if (!consolidated[i]) {
           while (true) {
               var intersects = 0
               for (j in (i + 1) until size) {
                   if (consolidated[j]) continue 
                   if (sets[i].intersect(sets[j]).isNotEmpty()) {
                       sets[i] = sets[i].union(sets[j])
                       consolidated[j] = true
                       intersects++
                   }
               }
               if (intersects == 0) break
           }
       }
       i++
   }
   return (0 until size).filter { !consolidated[it] }.map { sets[it].toSortedSet() }.toSet()                    

}

fun main(args: Array<String>) {

   val unconsolidatedSets = arrayOf(
       arrayOf(setOf('A', 'B'), setOf('C', 'D')),
       arrayOf(setOf('A', 'B'), setOf('B', 'D')),
       arrayOf(setOf('A', 'B'), setOf('C', 'D'), setOf('D', 'B')),
       arrayOf(setOf('H', 'I', 'K'), setOf('A', 'B'), setOf('C', 'D'), setOf('D', 'B'), setOf('F', 'G', 'H')) 
   )
   for (sets in unconsolidatedSets) println(consolidateSets(sets))

}</lang>

Output:

[[A, B], [C, D]]
[[A, B, D]]
[[A, B, C, D]]
[[F, G, H, I, K], [A, B, C, D]]

Mathematica

<lang Mathematica>reduce[x_] :=

Block[{pairs, unique},
 pairs = 
  DeleteCases[
   Subsets[Range@
     Length@x, {2}], _?(Intersection @@ x# == {} &)];
 unique = Complement[Range@Length@x, Flatten@pairs];
 Join[Union[Flatten[x#]] & /@ pairs, xunique]]

consolidate[x__] := FixedPoint[reduce, {x}]</lang>

consolidate[{a, b}, {c, d}]
-> {{a, b}, {c, d}}

consolidate[{a, b}, {b, d}]
-> {{a, b, d}}

consolidate[{a, b}, {c, d}, {d, b}]
-> {{a, b, c, d}}

consolidate[{h, i, k}, {a, b}, {c, d}, {d, b}, {f, g, h}]
-> {{a,b,c,d},{f,g,h,i,k}}

OCaml

<lang ocaml>let join a b =

 List.fold_left (fun acc v ->
   if List.mem v acc then acc else v::acc
 ) b a

let share a b = List.exists (fun x -> List.mem x b) a

let extract p lst =

 let rec aux acc = function
 | x::xs -> if p x then Some (x, List.rev_append acc xs) else aux (x::acc) xs
 | [] -> None
 in
 aux [] lst

let consolidate sets =

 let rec aux acc = function
 | [] -> List.rev acc
 | x::xs ->
     match extract (share x) xs with
     | Some (y, ys) -> aux acc ((join x y) :: ys)
     | None -> aux (x::acc) xs
 in
 aux [] sets

let print_sets sets =

 print_string "{ ";
 List.iter (fun set ->
   print_string "{";
   print_string (String.concat " " set);
   print_string "} "
 ) sets;
 print_endline "}"

let () =

 print_sets (consolidate [["A";"B"]; ["C";"D"]]);
 print_sets (consolidate [["A";"B"]; ["B";"C"]]);
 print_sets (consolidate [["A";"B"]; ["C";"D"]; ["D";"B"]]);
 print_sets (consolidate [["H";"I";"K"]; ["A";"B"]; ["C";"D"]; ["D";"B"];
                          ["F";"G";"H"]]);

</lang>

Output:

{ {A B} {C D} }
{ {A B C} }
{ {B A C D} }
{ {K I F G H} {B A C D} }

ooRexx

<lang oorexx>/* REXX ***************************************************************

• 04.08.2013 Walter Pachl using ooRexx features
• (maybe not in the best way -improvements welcome!)
• but trying to demonstrate the algorithm
  • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • /

s.1=.array~of(.set~of('A','B'),.set~of('C','D')) s.2=.array~of(.set~of('A','B'),.set~of('B','D')) s.3=.array~of(.set~of('A','B'),.set~of('C','D'),.set~of('D','B')) s.4=.array~of(.set~of('H','I','K'),.set~of('A','B'),.set~of('C','D'),,

             .set~of('B','D'),.set~of('F','G','H'))

s.5=.array~of(.set~of('snow','ice','slush','frost','fog'),,

             .set~of('iceburgs','icecubes'),,
             .set~of('rain','fog','sleet'))

s.6=.array~of('one') s.7=.array~new s.8=.array~of() Do si=1 To 8 /* loop through the test data */

 na=s.si                              /* an array of sets           */
 head='Output(s):'
 Say left('Input' si,10) list_as(na)  /* show the input             */
 Do While na~items()>0                /* while the array ain't empty*/
   na=cons(na)                        /* consolidate and get back   */
                                      /*  array of remaining sets   */
   head='          '
   End
 Say '===='                           /* separator line             */
 End

Exit

cons: Procedure Expose head /**********************************************************************

• consolidate the sets in the given array
  • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • /

 Use Arg a
 w=a                                  /* work on a copy             */
 n=w~items()                          /* number of sets in the array*/
 Select
   When n=0 Then                      /* no set in array            */
     Return .array~new                /* retuns an empty array      */
   When n=1 Then Do                   /* one set in array           */
     Say head list(w[1])              /* show its contents          */
     Return .array~new                /* retuns an empty array      */
     End
   Otherwise Do                       /* at least two sets are there*/
     b=.array~new                     /* use for remaining sets     */
     r=w[n]                           /* start with last set        */
     try=1
     Do until changed=0               /* loop until result is stable*/
       changed=0
       new=0
       n=w~items()                    /* number of sets             */
       Do i=1 To n-try                /* loop first through n-1 sets*/
         try=0                        /* then through all of them   */
         is=r~intersection(w[i])
         If is~items>0 Then Do        /* any elements in common     */
           r=r~union(w[i])            /* result is the union        */
           Changed=1                  /* and result is now larger   */
           End
         Else Do                      /* no elemen in common        */
           new=new+1                  /* add the set to the array   */
           b[new]=w[i]                /* of remaining sets          */
           End
         End
       If b~items()=0 Then Do         /* no remaining sets          */
         w=.array~new
         Leave                        /* we are done                */
         End
       w=b                            /* repeat with remaining sets */
       b=.array~new                   /* prepare for next iteration */
       End
     End
   Say head list(r)                   /* show one consolidated set  */
   End
 Return w                             /* return array of remaining  */

list: Procedure /**********************************************************************

• list elements of given set
  • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • /

 Call trace ?O
 Use Arg set
 arr=set~makeArray
 arr~sort()
 ol='('
 Do i=1 To arr~items()
   If i=1 Then
     ol=ol||arr[i]
   Else
     ol=ol||','arr[i]
   End
 Return ol')'

list_as: Procedure /**********************************************************************

• List an array of sets
  • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • /

 Call trace ?O
 Use Arg a
 n=a~items()
 If n=0 Then
   ol='no element in array'
 Else Do
   ol=
   Do i=1 To n
     ol=ol '('
     arr=a[i]~makeArray
     Do j=1 To arr~items()
       If j=1 Then
         ol=ol||arr[j]
       Else
         ol=ol','arr[j]
       End
     ol=ol') '
     End
   End
 Return strip(ol)</lang>

Output:

Input 1    (B,A)  (C,D)
Output(s): (C,D)
           (A,B)
====
Input 2    (B,A)  (B,D)
Output(s): (A,B,D)
====
Input 3    (B,A)  (C,D)  (B,D)
Output(s): (A,B,C,D)
====
Input 4    (H,I,K)  (B,A)  (C,D)  (F,G,H)
Output(s): (F,G,H,I,K)
           (A,B,C,D)
====
Input 5    (snow,fog,ice,frost,slush)  (icecubes,iceburgs)  (fog,sleet,rain)
Output(s): (fog,frost,ice,rain,sleet,slush,snow)
           (iceburgs,icecubes)
====
Input 6    (one)
Output(s): (one)
====
Input 7    no element in array
====
Input 8    ()
Output(s): ()
====

PARI/GP

<lang parigp>cons(V)={

 my(v,u,s);
 for(i=1,#V,
   v=V[i];
   for(j=i+1,#V,
     u=V[j];
     if(#setintersect(u,v),V[i]=v=vecsort(setunion(u,v));V[j]=[];s++)
   )
 );
 V=select(v->#v,V);
 if(s,cons(V),V)

};</lang>

Perl

We implement the key data structure, a set of sets, as an array containing references to arrays of scalars. <lang perl>use strict; use English; use Smart::Comments;

my @ex1 = consolidate( (['A', 'B'], ['C', 'D']) );

1. 1. 1. Example 1: @ex1

my @ex2 = consolidate( (['A', 'B'], ['B', 'D']) );

1. 1. 1. Example 2: @ex2

my @ex3 = consolidate( (['A', 'B'], ['C', 'D'], ['D', 'B']) );

1. 1. 1. Example 3: @ex3

my @ex4 = consolidate( (['H', 'I', 'K'], ['A', 'B'], ['C', 'D'], ['D', 'B'], ['F', 'G', 'H']) );

1. 1. 1. Example 4: @ex4

exit 0;

sub consolidate {

   scalar(@ARG) >= 2 or return @ARG;
   my @result = ( shift(@ARG) );
   my @recursion = consolidate(@ARG);
   foreach my $r (@recursion) {
       if (set_intersection($result[0], $r)) {
           $result[0] = [ set_union($result[0], $r) ];
       }
       else {
           push @result, $r;
       }
   }
   return @result;

}

sub set_union {

   my ($a, $b) = @ARG;
   my %union;
   foreach my $a_elt (@{$a}) { $union{$a_elt}++; }
   foreach my $b_elt (@{$b}) { $union{$b_elt}++; }
   return keys(%union);

}

sub set_intersection {

   my ($a, $b) = @ARG;
   my %a_hash;
   foreach my $a_elt (@{$a}) { $a_hash{$a_elt}++; }
   my @result;
   foreach my $b_elt (@{$b}) {
       push(@result, $b_elt) if exists($a_hash{$b_elt});
   }
   return @result;

}</lang>

Output:

### Example 1: [
###              [
###                'A',
###                'B'
###              ],
###              [
###                'C',
###                'D'
###              ]
###            ]

### Example 2: [
###              [
###                'D',
###                'B',
###                'A'
###              ]
###            ]

### Example 3: [
###              [
###                'A',
###                'C',
###                'D',
###                'B'
###              ]
###            ]

### Example 4: [
###              [
###                'H',
###                'F',
###                'K',
###                'G',
###                'I'
###              ],
###              [
###                'D',
###                'B',
###                'A',
###                'C'
###              ]
###            ]

Perl 6

<lang perl6>multi consolidate() { () } multi consolidate(Set \this is copy, *@those) {

   gather {
       for consolidate |@those -> \that {
           if this ∩ that { this = this ∪ that }
           else           { take that }
       }
       take this;
   }

}

enum Elems <A B C D E F G H I J K>; say $_, "\n ==> ", consolidate |$_

   for [set(A,B), set(C,D)],
       [set(A,B), set(B,D)],
       [set(A,B), set(C,D), set(D,B)],
       [set(H,I,K), set(A,B), set(C,D), set(D,B), set(F,G,H)];</lang>

Output:

set(A, B) set(C, D)
    ==> set(C, D) set(A, B)
set(A, B) set(B, D)
    ==> set(A, B, D)
set(A, B) set(C, D) set(D, B)
    ==> set(A, B, C, D)
set(H, I, K) set(A, B) set(C, D) set(D, B) set(F, G, H)
    ==> set(A, B, C, D) set(H, I, K, F, G)

Phix

Using strings to represent sets of characters <lang Phix>function has_intersection(sequence set1, sequence set2)

   for i=1 to length(set1) do
       if find(set1[i],set2) then
           return true
       end if
   end for
   return false

end function

function union(sequence set1, sequence set2)

   for i=1 to length(set2) do
       if not find(set2[i],set1) then
           set1 = append(set1,set2[i])
       end if
   end for
   return set1

end function

function consolidate(sequence sets)

   for i=length(sets) to 1 by -1 do
       for j=length(sets) to i+1 by -1 do
           if has_intersection(sets[i],sets[j]) then
               sets[i] = union(sets[i],sets[j])
               sets[j..j] = {}
           end if
       end for
   end for
   return sets

end function

?consolidate({"AB","CD"}) ?consolidate({"AB","BD"}) ?consolidate({"AB","CD","DB"}) ?consolidate({"HIK","AB","CD","DB","FGH"})</lang>

Output:

{"AB","CD"}
{"ABD"}
{"ABCD"}
{"HIKFG","ABCD"}

PicoLisp

Translation of: Python

<lang PicoLisp>(de consolidate (S)

  (when S
     (let R (cons (car S))
        (for X (consolidate (cdr S))
           (if (mmeq X (car R))
              (set R (uniq (conc X (car R))))
              (conc R (cons X)) ) )
        R ) ) )</lang>

Test: <lang PicoLisp>: (consolidate '((A B) (C D))) -> ((A B) (C D))

(consolidate '((A B) (B D)))

-> ((B D A))

(consolidate '((A B) (C D) (D B)))

-> ((D B C A))

(consolidate '((H I K) (A B) (C D) (D B) (F G H)))

-> ((F G H I K) (D B C A))</lang>

PL/I

<lang PL/I>Set: procedure options (main); /* 13 November 2013 */

  declare set(20) character (200) varying;
  declare e character (1);
  declare (i, n) fixed binary;

  set = ;
  n = 1;
  do until (e = ']');
     get edit (e) (a(1)); put edit (e) (a(1));
     if e = '}' then n = n + 1; /* end of set. */
     if e ^= '{' & e ^= ',' & e ^= '}' & e ^= ' ' then
        set(n) = set(n) || e;   /* Build set */
  end;
  /* We have read in all sets. */
  n = n - 1; /* we have n sets */
  /* Display the sets: */
  put skip list ('The original sets:');
  do i = 1 to n;
     call print(i);
  end;
  /* Look for sets to combine: */
  do i = 2 to n;
     if length(set(i)) > 0 then
        if search(set(1), set(i)) > 0 then
           /* there's at least one common element */
           do; call combine (1, i); set(i) = ;  end;
  end;

  put skip (2) list ('Results:');
  do i = 1 to n;
     if length(set(i)) > 0 then call print (i);
  end;

combine: procedure (p, q);

  declare (p, q) fixed binary;
  declare e character (1);
  declare i fixed binary;

  do i = 1 to length(set(q));
     e = substr(set(q), i, 1);
     if index(set(p), e) = 0 then set(p) = set(p) || e;
  end;

end combine;

print: procedure(k);

  declare k fixed binary;
  declare i fixed binary;

  put edit ('{') (a);
  do i = 1 to length(set(k));
     put edit (substr(set(k), i, 1)) (a);
     if i < length(set(k)) then put edit (',') (a);
  end;
  put edit ('} ') (a);

end print;

end Set;</lang>

The original sets: {A,B} 

Results: {A,B}

The original sets: {A,B} {C,D} 

Results: {A,B} {C,D}

The original sets: {A,B} {B,C} 

Results: {A,B,C}

The original sets: {A,B} {C,D} {E,B,F,G,H} 

Results: {A,B,E,F,G,H} {C,D} 

Python

Python: Iterative

<lang python>def consolidate(sets):

   setlist = [s for s in sets if s]
   for i, s1 in enumerate(setlist):
       if s1:
           for s2 in setlist[i+1:]:
               intersection = s1.intersection(s2)
               if intersection:
                   s2.update(s1)
                   s1.clear()
                   s1 = s2
   return [s for s in setlist if s]</lang>

Python: Recursive

<lang python>def conso(s): if len(s) < 2: return s

r, b = [s[0]], conso(s[1:]) for x in b: if r[0].intersection(x): r[0].update(x) else: r.append(x) return r</lang>

Python: Testing

The _test function contains solutions to all the examples as well as a check to show the order-independence of the sets given to the consolidate function. <lang python>def _test(consolidate=consolidate):

   def freze(list_of_sets):
       'return a set of frozensets from the list of sets to allow comparison'
       return set(frozenset(s) for s in list_of_sets)
       
   # Define some variables
   A,B,C,D,E,F,G,H,I,J,K = 'A,B,C,D,E,F,G,H,I,J,K'.split(',')
   # Consolidate some lists of sets
   assert (freze(consolidate([{A,B}, {C,D}])) == freze([{'A', 'B'}, {'C', 'D'}]))
   assert (freze(consolidate([{A,B}, {B,D}])) == freze([{'A', 'B', 'D'}]))
   assert (freze(consolidate([{A,B}, {C,D}, {D,B}])) == freze([{'A', 'C', 'B', 'D'}]))
   assert (freze(consolidate([{H,I,K}, {A,B}, {C,D}, {D,B}, {F,G,H}])) ==
            freze([{'A', 'C', 'B', 'D'}, {'G', 'F', 'I', 'H', 'K'}]))
   assert (freze(consolidate([{A,H}, {H,I,K}, {A,B}, {C,D}, {D,B}, {F,G,H}])) ==
            freze([{'A', 'C', 'B', 'D', 'G', 'F', 'I', 'H', 'K'}]))
   assert (freze(consolidate([{H,I,K}, {A,B}, {C,D}, {D,B}, {F,G,H}, {A,H}])) ==
            freze([{'A', 'C', 'B', 'D', 'G', 'F', 'I', 'H', 'K'}]))
   # Confirm order-independence
   from copy import deepcopy
   import itertools
   sets = [{H,I,K}, {A,B}, {C,D}, {D,B}, {F,G,H}, {A,H}]
   answer = consolidate(deepcopy(sets))
   for perm in itertools.permutations(sets):
           assert consolidate(deepcopy(perm)) == answer

   assert (answer == [{'A', 'C', 'B', 'D', 'G', 'F', 'I', 'H', 'K'}])
   assert (len(list(itertools.permutations(sets))) == 720)
   
   print('_test(%s) complete' % consolidate.__name__)

if __name__ == '__main__':

   _test(consolidate)
   _test(conso)</lang>

Output:

_test(consolidate) complete
_test(conso) complete

Python: Functional

Translation of: Haskell

Translation of: JavaScript

As a fold (catamorphism), using union in preference to mutation: <lang python>from functools import (reduce)

1. consolidated :: Ord a => [Set a] -> [Set a]

def consolidated(sets):

   def go(xs, s):
       if xs:
           h = xs[0]
           return go(xs[1:], h.union(s)) if (
               h.intersection(s)
           ) else [h] + go(xs[1:], s)
       else:
           return [s]
   return reduce(go, sets, [])

1. TESTS ----------------------------------------------------

if __name__ == '__main__':

   print(
       *list(map(
           lambda xs: consolidated(map(set, xs)),
           [
               ['ab', 'cd'],
               ['ab', 'bd'],
               ['ab', 'cd', 'db'],
               ['hik', 'ab', 'cd', 'db', 'fgh']
           ]
       )),
       sep='\n'
   )</lang>

Output:

[{'b', 'a'}, {'c', 'd'}]
[{'b', 'a', 'd'}]
[{'a', 'd', 'b', 'c'}]
[{'a', 'd', 'b', 'c'}, {'k', 'h', 'f', 'i', 'g'}]

Racket

<lang racket>

1. lang racket

(define (consolidate ss)

 (define (comb s cs)
   (cond [(set-empty? s) cs]
         [(empty? cs) (list s)]
         [(set-empty? (set-intersect s (first cs)))
          (cons (first cs) (comb s (rest cs)))]
         [(consolidate (cons (set-union s (first cs)) (rest cs)))]))
 (foldl comb '() ss))

(consolidate (list (set 'a 'b) (set 'c 'd))) (consolidate (list (set 'a 'b) (set 'b 'c))) (consolidate (list (set 'a 'b) (set 'c 'd) (set 'd 'b))) (consolidate (list (set 'h 'i 'k) (set 'a 'b) (set 'c 'd) (set 'd 'b) (set 'f 'g 'h))) </lang>

Output:

<lang racket> (list (set 'b 'a) (set 'd 'c)) (list (set 'a 'b 'c)) (list (set 'a 'b 'd 'c)) (list (set 'g 'h 'k 'i 'f) (set 'a 'b 'd 'c)) </lang>

REXX

<lang rexx>/*REXX program demonstrates a method of consolidating some sample sets. */ @.=; @.1 = '{A,B} {C,D}'

        @.2 = "{A,B}     {B,D}"
        @.3 = '{A,B}     {C,D}     {D,B}'
        @.4 = '{H,I,K}   {A,B}     {C,D}     {D,B}     {F,G,H}'
        @.5 = '{snow,ice,slush,frost,fog} {icebergs,icecubes} {rain,fog,sleet}'

              do j=1  while @.j\==            /*traipse through each of sample sets. */
              call SETconsolidate @.j           /*have the function do the heavy work. */
              end   /*j*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ isIn: return wordpos(arg(1), arg(2))\==0 /*is (word) argument 1 in the set arg2?*/ /*──────────────────────────────────────────────────────────────────────────────────────*/ SETconsolidate: procedure; parse arg old; #=words(old); new=

      say ' the old set=' space(old)

        do k=1  for #                           /* [↓]  change all commas to a blank.  */
        !.k=translate(word(old,k), , '},{')     /*create a list of words  (aka, a set).*/
        end   /*k*/                             /* [↑]  ··· and also remove the braces.*/

        do  until \changed;    changed=0        /*consolidate some sets  (well, maybe).*/
             do set=1  for #-1
                 do item=1  for words(!.set);       x=word(!.set, item)
                     do other=set+1  to #
                     if isIn(x, !.other)  then do;  changed=1            /*it's changed*/
                                                    !.set=!.set !.other;   !.other=
                                                    iterate set
                                               end
                     end   /*other*/
                 end       /*item */
             end           /*set  */
        end                /*until ¬changed*/

           do set=1  for #;   $=                                           /*elide dups*/
             do items=1  for words(!.set);   x=word(!.set, items)
             if x==','  then iterate;        if x==  then leave
             $=$ x                                                         /*build new. */
                    do  until  \isIn(x, !.set)
                    _=wordpos(x, !.set)
                    !.set=subword(!.set, 1, _-1)  ','  subword(!.set, _+1) /*purify set*/
                    end   /*until ¬isIn ··· */
             end          /*items*/
           !.set=translate(strip($), ',', " ")
           end            /*set*/

        do i=1  for #; if !.i==  then iterate /*ignore any  set  that is a null set. */
        new=space(new  '{'!.i"}")               /*prepend and append a set identifier. */
        end   /*i*/

      say ' the new set='  new;         say
      return</lang>

output   when using the (internal) default supplied sample sets:

 the old set= {A,B} {C,D}
 the new set= {A,B} {C,D}

 the old set= {A,B} {B,D}
 the new set= {A,B,D}

 the old set= {A,B} {C,D} {D,B}
 the new set= {A,B,D,C}

 the old set= {H,I,K} {A,B} {C,D} {D,B} {F,G,H}
 the new set= {H,I,K,F,G} {A,B,D,C}

 the old set= {snow,ice,slush,frost,fog} {icebergs,icecubes} {rain,fog,sleet}
 the new set= {snow,ice,slush,frost,fog,rain,sleet} {icebergs,icecubes}

Ring

<lang ring>

1. Project : Set consolidation

load "stdlib.ring" test = ["AB","AB,CD","AB,CD,DB","HIK,AB,CD,DB,FGH"] for t in test

    see consolidate(t) + nl

next func consolidate(s) sets = split(s,",") n = len(sets) for i = 1 to n p = i

            ts = ""

for j = i to 1 step -1 if ts = "" p = j ok ts = "" for k = 1 to len(sets[p])

                     if j > 1

if substring(sets[j-1],substr(sets[p],k,1),1) = 0 ts = ts + substr(sets[p],k,1) ok

                     ok

next if len(ts) < len(sets[p])

                   if j > 1

sets[j-1] = sets[j-1] + ts sets[p] = "-" ts = ""

                   ok

else p = i ok next next consolidate = s + " = " + substr(list2str(sets),nl,",")

       return consolidate

</lang> Output:

AB = AB
AB,CD = AB,CD
AB,CD,DB = ABCD,-,-
HIK,AB,CD,DB,FGH = HIKFG,ABCD,-,-,-

Ruby

<lang ruby>require 'set'

tests = [[[:A,:B], [:C,:D]],

        [[:A,:B], [:B,:D]],
        [[:A,:B], [:C,:D], [:D,:B]],
        [[:H,:I,:K], [:A,:B], [:C,:D], [:D,:B], [:F,:G,:H]]]

tests.map!{|sets| sets.map(&:to_set)}

tests.each do |sets|

 until sets.combination(2).none?{|a,b| a.merge(b) && sets.delete(b) if a.intersect?(b)}
 end
 p sets

end</lang>

Output:

[#<Set: {:A, :B}>, #<Set: {:C, :D}>]
[#<Set: {:A, :B, :D}>]
[#<Set: {:A, :B, :D, :C}>]
[#<Set: {:H, :I, :K, :F, :G}>, #<Set: {:A, :B, :D, :C}>]

Note: After execution, the contents of tests are exchanged.

Scala

<lang Scala>object SetConsolidation extends App {

   def consolidate[Type](sets: Set[Set[Type]]): Set[Set[Type]] = {
       var result = sets // each iteration combines two sets and reiterates, else returns
       for (i <- sets; j <- sets - i; k = i.intersect(j);
           if result == sets && k.nonEmpty) result = result - i - j + i.union(j)
       if (result == sets) sets else consolidate(result)
   }

   // Tests:
   def parse(s: String) =
       s.split(",").map(_.split("").toSet).toSet
   def pretty[Type](sets: Set[Set[Type]]) =
       sets.map(_.mkString("{",",","}")).mkString(" ")
   val tests = List(
       parse("AB,CD") -> Set(Set("A", "B"), Set("C", "D")),
       parse("AB,BD") -> Set(Set("A", "B", "D")),
       parse("AB,CD,DB") -> Set(Set("A", "B", "C", "D")),
       parse("HIK,AB,CD,DB,FGH") -> Set(Set("A", "B", "C", "D"), Set("F", "G", "H", "I", "K"))
   )
   require(Set("A", "B", "C", "D") == Set("B", "C", "A", "D"))
   assert(tests.forall{case (test, expect) =>
       val result = consolidate(test)
       println(s"${pretty(test)} -> ${pretty(result)}")
       expect == result
   })

}</lang>

Output:

{A,B} {C,D} -> {A,B} {C,D}
{A,B} {B,D} -> {A,B,D}
{A,B} {C,D} {D,B} -> {C,D,A,B}
{D,B} {F,G,H} {A,B} {C,D} {H,I,K} -> {F,I,G,H,K} {A,B,C,D}

Sidef

Translation of: Perl 6

<lang ruby>func consolidate() { [] } func consolidate(this, *those) {

   gather {
       consolidate(those...).each { |that|
           if (this & that) { this |= that }
           else             { take that }
       }
       take this;
   }

}

enum |A="A", B, C, D, _E, F, G, H, I, _J, K|;

func format(ss) {

   ss.map{ '(' + .join(' ') + ')' }.join(' ')

}

[

   [[A,B], [C,D]],
   [[A,B], [B,D]],
   [[A,B], [C,D], [D,B]],
   [[H,I,K], [A,B], [C,D], [D,B], [F,G,H]]

].each { |ss|

   say (format(ss), "\n\t==> ", format(consolidate(ss...)));

}</lang>

Output:

(A B) (C D)
	==> (C D) (A B)
(A B) (B D)
	==> (A D B)
(A B) (C D) (D B)
	==> (A C D B)
(H I K) (A B) (C D) (D B) (F G H)
	==> (A C D B) (I K F G H)

Tcl

Translation of: Python

Library: Tcllib (Package: struct::set)

This uses just the recursive version, as this is sufficient to handle substantial merges. <lang tcl>package require struct::set

proc consolidate {sets} {

   if {[llength $sets] < 2} {

return $sets

   }

   set r [list {}]
   set r0 [lindex $sets 0]
   foreach x [consolidate [lrange $sets 1 end]] {

if {[struct::set size [struct::set intersect $x $r0]]} { struct::set add r0 $x } else { lappend r $x }

   }
   return [lset r 0 $r0]

}</lang> Demonstrating: <lang tcl>puts 1:[consolidate {{A B} {C D}}] puts 2:[consolidate {{A B} {B D}}] puts 3:[consolidate {{A B} {C D} {D B}}] puts 4:[consolidate {{H I K} {A B} {C D} {D B} {F G H}}]</lang>

Output:

1:{A B} {C D}
2:{D A B}
3:{D A B C}
4:{H I F G K} {D A B C}

TXR

Original solution:

<lang txrlisp>(defun mkset (p x) (set [p x] (or [p x] x)))

(defun fnd (p x) (if (eq [p x] x) x (fnd p [p x])))

(defun uni (p x y)

 (let ((xr (fnd p x)) (yr (fnd p y)))
   (set [p xr] yr)))

(defun consoli (sets)

 (let ((p (hash)))
   (each ((s sets))
     (each ((e s))
       (mkset p e)
       (uni p e (car s))))
   (hash-values 
     [group-by (op fnd p) (hash-keys 
                            [group-by identity (flatten sets)])])))

tests

(each ((test '(((a b) (c d))

              ((a b) (b d))
              ((a b) (c d) (d b))
              ((h i k) (a b) (c d) (d b) (f g h)))))
 (format t "~s -> ~s\n" test (consoli test)))</lang>

Output:

((a b) (c d)) -> ((b a) (d c))
((a b) (b d)) -> ((b a d))
((a b) (c d) (d b)) -> ((b a d c))
((h i k) (a b) (c d) (d b) (f g h)) -> ((g f k i h) (b a d c)

Translation of: Racket

<lang txrlisp>(defun mkset (items) [group-by identity items])

(defun empty-p (set) (zerop (hash-count set)))

(defun consoli (ss)

 (defun combi (cs s)
   (cond ((empty-p s) cs)
         ((null cs) (list s))
         ((empty-p (hash-isec s (first cs)))
          (cons (first cs) (combi (rest cs) s)))
         (t (consoli (cons (hash-uni s (first cs)) (rest cs))))))
 [reduce-left combi ss nil])

tests

(each ((test '(((a b) (c d))

              ((a b) (b d))
              ((a b) (c d) (d b))
              ((h i k) (a b) (c d) (d b) (f g h)))))
 (format t "~s -> ~s\n" test
         [mapcar hash-keys (consoli [mapcar mkset test])]))</lang>

Output:

((a b) (c d)) -> ((b a) (d c))
((a b) (b d)) -> ((d b a))
((a b) (c d) (d b)) -> ((d c b a))
((h i k) (a b) (c d) (d b) (f g h)) -> ((g f k i h) (d c b a))

VBA

Translation of: Phix

This solutions uses collections as sets. The first three coroutines are based on the Phix solution. Two coroutines are written to create the example sets as collections, and another coroutine to show the consolidated set. <lang vb>Private Function has_intersection(set1 As Collection, set2 As Collection) As Boolean

   For Each element In set1
       On Error Resume Next
       tmp = set2(element)
       If tmp = element Then
           has_intersection = True
           Exit Function
       End If
   Next element

End Function Private Sub union(set1 As Collection, set2 As Collection)

   For Each element In set2
       On Error Resume Next
       tmp = set1(element)
       If tmp <> element Then
           set1.Add element, element
       End If
   Next element

End Sub Private Function consolidate(sets As Collection) As Collection

   For i = sets.Count To 1 Step -1
       For j = sets.Count To i + 1 Step -1
           If has_intersection(sets(i), sets(j)) Then
               union sets(i), sets(j)
               sets.Remove j
           End If
       Next j
   Next i
   Set consolidate = sets

End Function Private Function mc(s As Variant) As Collection

   Dim res As New Collection
   For i = 1 To Len(s)
       res.Add Mid(s, i, 1), Mid(s, i, 1)
   Next i
   Set mc = res

End Function Private Function ms(t As Variant) As Collection

   Dim res As New Collection
   Dim element As Collection
   For i = LBound(t) To UBound(t)
       Set element = t(i)
       res.Add t(i)
   Next i
   Set ms = res

End Function Private Sub show(x As Collection)

   Dim t() As String
   Dim u() As String
   ReDim t(1 To x.Count)
   For i = 1 To x.Count
       ReDim u(1 To x(i).Count)
       For j = 1 To x(i).Count
           u(j) = x(i)(j)
       Next j
       t(i) = "{" & Join(u, ", ") & "}"
   Next i
   Debug.Print "{" & Join(t, ", ") & "}"

End Sub Public Sub main()

   show consolidate(ms(Array(mc("AB"), mc("CD"))))
   show consolidate(ms(Array(mc("AB"), mc("BD"))))
   show consolidate(ms(Array(mc("AB"), mc("CD"), mc("DB"))))
   show consolidate(ms(Array(mc("HIK"), mc("AB"), mc("CD"), mc("DB"), mc("FGH"))))

End Sub</lang>

Output:

{{A, B}, {C, D}}
{{A, B, D}}
{{A, B, C, D}}
{{H, I, K, F, G}, {A, B, C, D}}

VBScript

<lang vb> Function consolidate(s) sets = Split(s,",") n = UBound(sets) For i = 1 To n p = i ts = "" For j = i To 1 Step -1 If ts = "" Then p = j End If ts = "" For k = 1 To Len(sets(p)) If InStr(1,sets(j-1),Mid(sets(p),k,1)) = 0 Then ts = ts & Mid(sets(p),k,1) End If Next If Len(ts) < Len(sets(p)) Then sets(j-1) = sets(j-1) & ts sets(p) = "-" ts = "" Else p = i End If Next Next consolidate = s & " = " & Join(sets," , ") End Function

'testing test = Array("AB","AB,CD","AB,CD,DB","HIK,AB,CD,DB,FGH") For Each t In test WScript.StdOut.WriteLine consolidate(t) Next </lang>

Output:

AB = AB
AB,CD = AB , CD
AB,CD,DB = ABCD , - , -
HIK,AB,CD,DB,FGH = HIKFG , ABCD , - , - , -

zkl

Translation of: Tcl

<lang zkl>fcn consolidate(sets){ // set are munged if they are read/write

  if(sets.len()<2) return(sets);
  r,r0 := List(List()),sets[0];
  foreach x in (consolidate(sets[1,*])){
     i,ni:=x.filter22(r0.holds); //-->(intersection, !intersection)
     if(i) r0=r0.extend(ni);
     else  r.append(x);
  }
  r[0]=r0;
  r

}</lang> <lang zkl>fcn prettize(sets){

  sets.apply("concat"," ").pump(String,"(%s),".fmt)[0,-1]

}

foreach sets in (T(

 T(L("A","B")), 
 T(L("A","B"),L("C","D")), 
 T(L("A","B"),L("B","D")),
 T(L("A","B"),L("C","D"),L("D","B")),
 T(L("H","I","K"),L("A","B"),L("C","D"),L("D","B"),L("F","G","H")),
 T(L("A","H"),L("H","I","K"),L("A","B"),L("C","D"),L("D","B"),L("F","G","H")),
 T(L("H","I","K"),L("A","B"),L("C","D"),L("D","B"),L("F","G","H"), L("A","H")),

)){

  prettize(sets).print(" --> ");
  consolidate(sets) : prettize(_).println();

}</lang>

Output:

(A B) --> (A B)
(A B),(C D) --> (A B),(C D)
(A B),(B D) --> (A B D)
(A B),(C D),(D B) --> (A B C D)
(H I K),(A B),(C D),(D B),(F G H) --> (H I K F G),(A B C D)
(A H),(H I K),(A B),(C D),(D B),(F G H) --> (A H I K F G B C D)
(H I K),(A B),(C D),(D B),(F G H),(A H) --> (H I K A B C D F G)