Select Group with Limit 1 Speed

createDummyData.sql

-- CREATE EXTENSION IF NOT EXISTS "uuid-ossp";

DROP TABLE IF EXISTS products;
DROP TABLE IF EXISTS categories;

CREATE TABLE categories (
  id UUID primary key DEFAULT uuid_generate_v4(),
  title text
);

CREATE TABLE products (
  id UUID primary key DEFAULT uuid_generate_v4(),
  category_id UUID,
  title text,
  foreign key (category_id)
    REFERENCES categories (id)
);

INSERT INTO categories (
  title
) SELECT
  md5(random()::text)
FROM generate_series(1, 100) s(i);



INSERT INTO products (
  category_id, title
)
WITH expanded AS (
  SELECT RANDOM(), seq, c.id AS category_id
  FROM generate_series(1, 100000) seq, categories c
), shuffled AS (
  SELECT e.*
  FROM expanded e
  INNER JOIN (
    SELECT ei.seq, MIN(ei.random) FROM expanded ei GROUP BY ei.seq
  ) em ON (e.seq = em.seq AND e.random = em.min)
  ORDER BY e.seq
)
SELECT
  s.category_id,
  md5(random()::text)
FROM shuffled s; 

GravyQeury.sql

-- slitely adapted to PostgresQL
SELECT
  id,
  category_title,
  (array_agg(product_title))[1]  
FROM
    (SELECT c.id, c.title AS category_title, p.id AS product_id, p.title AS product_title
    FROM categories AS c
    JOIN products AS p ON c.id = p.category_id
    ORDER BY c.id ASC) AS a 
GROUP BY id, category_title;

KostanosQuery.sql

SELECT
 c.id,
 c.title,
 p.id AS product_id,
 p.title AS product_title
FROM categories AS c
JOIN products AS p ON
 p.id = (                                 --- the PRIMARY KEY
  SELECT p1.id FROM products AS p1
  WHERE c.id=p1.category_id
  ORDER BY p1.id LIMIT 1
 )