Postgresql Group by array elements in common - arrays

Question

I have a table like this:
CREATE TABLE preferences (name varchar, preferences varchar[]);
INSERT INTO preferences (name, preferences)
VALUES
('John','{pizza, spaghetti}'),
('Charlie','{spaghetti, rice}'),
('Lucy','{rice, potatoes}'),
('Beth','{bread, cheese}'),
('Trudy','{rice, milk}');
So from the table
John {pizza, spaghetti}
Charlie {spaghetti, rice}
Lucy {rice, potatoes}
Beth {bread, cheese}
Trudy {rice, milk}
I would like group all rows that have elements in common (even if it is through other people).
So in this case I would like to end up with:
{John,Charlie,Lucy,Trudy} {pizza,spaghetti,rice,potatoes,milk}
{Beth} {bread, cheese}
because Johns preferences intersect with those of Charlie, and those of Charlie intersect with those of Lucy and with those of Trudy.
I already haven an array_intersection function like this:
CREATE OR REPLACE FUNCTION array_intersection(anyarray, anyarray)
RETURNS anyarray
language sql
as $FUNCTION$
SELECT ARRAY(
SELECT UNNEST($1)
INTERSECT
SELECT UNNEST($2)
);
$FUNCTION$;
and know the array_agg function to aggregate arrays, but how to turn those into a grouping like I want is the step I am missing.

Answer 1

This is a typical task for recursion. You need an auxiliary function to merge and sort two arrays:
create or replace function public.array_merge(arr1 anyarray, arr2 anyarray)
returns anyarray
language sql immutable
as $function$
select array_agg(distinct elem order by elem)
from (
select unnest(arr1) elem
union
select unnest(arr2)
) s
$function$;
Use the function in the recursive query:
with recursive cte(name, preferences) as (
select *
from preferences
union
select p.name, array_merge(c.preferences, p.preferences)
from cte c
join preferences p
on c.preferences && p.preferences
and c.name <> p.name
)
select array_agg(name) as names, preferences
from (
select distinct on(name) *
from cte
order by name, cardinality(preferences) desc
) s
group by preferences;
names | preferences
---------------------------+--------------------------------------
{Charlie,John,Lucy,Trudy} | {milk,pizza,potatoes,rice,spaghetti}
{Beth} | {bread,cheese}
(2 rows)