Slow query: Select all buildings that have >1 pharmacies and >1 schools within 1000m

Hi

I have an interesting query to be optimized related to this one [1].

The query definition is: Select all buildings that have more than 1
pharmacies and more than 1 schools within a radius of 1000m.

The problem is that I think that this query is inherently O(n^2). In
fact the solution I propose below takes forever...

My questions:

1. Any comments about the nature of this problem?

2. ... on how to speed it up ?

3. In the original query [1] there's a count which contains a
subquery. According to my tests PostgreSQL does not allow this despite
the documentation which says "count(expression)".

Remarks: I know that "count(*)" could be faster on PostgreSQL but
"count(osm_id)" does not change the query plan and this does not seem
to be the bottleneck here anyway.

Yours, S.

[1] http://gis.stackexchange.com/questions/11445/selecting-pois-around-specific-buildings-using-postgis

Here's my query:

-- Select all buildings that have >1 pharmacies and >1 schools within 1000m:
SELECT osm_id AS building_id
FROM
(SELECT osm_id, way
FROM osm_polygon
WHERE tags @> hstore('building','yes')
) AS b
WHERE
(SELECT count(*) > 1
FROM osm_poi AS p
WHERE p.tags @> hstore('amenity','pharmacy')
AND ST_DWithin(b.way,p.way,1000)
)
AND
(SELECT count(*) > 1
FROM osm_poi AS p
WHERE p.tags @> hstore('amenity','school')
AND ST_DWithin(b.way,p.way,1000)
)
-- Total query runtime: 4308488 ms. 66345 rows retrieved.

Here's the query plan (from EXPLAIN):
"Index Scan using osm_polygon_tags_idx on osm_polygon
(cost=0.00..406812.81 rows=188 width=901)"
" Index Cond: (tags @> '"building"=>"yes"'::hstore)"
" Filter: ((SubPlan 1) AND (SubPlan 2))"
" SubPlan 1"
" -> Aggregate (cost=269.19..269.20 rows=1 width=0)"
" -> Bitmap Heap Scan on osm_poi p (cost=7.76..269.19
rows=1 width=0)"
" Recheck Cond: (way && st_expand(osm_polygon.way,
1000::double precision))"
" Filter: ((tags @> '"amenity"=>"pharmacy"'::hstore)
AND (osm_polygon.way && st_expand(way, 1000::double precision)) AND
_st_dwithin(osm_polygon.way, way, 1000::double precision))"
" -> Bitmap Index Scan on osm_poi_way_idx
(cost=0.00..7.76 rows=62 width=0)"
" Index Cond: (way && st_expand(osm_polygon.way,
1000::double precision))"
" SubPlan 2"
" -> Aggregate (cost=269.19..269.20 rows=1 width=0)"
" -> Bitmap Heap Scan on osm_poi p (cost=7.76..269.19
rows=1 width=0)"
" Recheck Cond: (way && st_expand(osm_polygon.way,
1000::double precision))"
" Filter: ((tags @> '"amenity"=>"school"'::hstore) AND
(osm_polygon.way && st_expand(way, 1000::double precision)) AND
_st_dwithin(osm_polygon.way, way, 1000::double precision))"
" -> Bitmap Index Scan on osm_poi_way_idx
(cost=0.00..7.76 rows=62 width=0)"
" Index Cond: (way && st_expand(osm_polygon.way,
1000::double precision))"

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