Finding the number of connected components in a graph

Question

I don't know haow many people have seen this problem posted on c.d.o.m, but here is the essence
given an equivalence relation, e.g.
x y
1 1
2 2
2 1
1 2
3 3
find the number of equivalence classes. In this example equivalence classes are {1,2} and {3}.

Answer 1

The idea: Label all the graph nodes with distinct primes. Find all simple paths in the graph weighted with product of nodes. Find the skyline of all maximal products. Count them.
Implementation (swithced to boolean vectors instead of primes)
create table input as (
select 1 x, 2 y from dual
union
select 2, 3 from dual
union
select 2, 4 from dual
union
select 6, 7 from dual
union
select 8, 9 from dual);
with TC as (
select connect_by_root(x) x, y
from (
select x,y from input
union
select y,x from input
) connect by nocycle x = prior y
), weightedNodes as (
select node, power(2,rownum-1) code
from (select x node from input
union select y node from input )
select count(distinct code) from (
select x,y,sum(code) code from (
select distinct pre.x, pre.y mid, post.y from TC pre, TC post
where pre.y=post.x
), weightedNodes
where node = mid
group by x,y
drop table input;