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How to Monitor with PostgreSQL Indexes

BTree

My issue here is that the BTree index will be huge since afaict it will store duplicate values (it has too, since it can't assume the table is physically sorted). If the BTree is huge I end up having to read both the index and the parts of the table that the index points too...
Not necessarily — Having a btree index that is 'covering' will be the fastest read time, and if that is all you want (ie if you can afford the extra storage), then it is your best bet.

BRIN

My understanding is that I can have a small index here at the expense of reading useless pages. Using a small pages_per_range means that the index is bigger (which is a problem with BRIN since I need to read the whole index), having a big pages_per_range means that I'll read a lot of useless pages.
If you can't afford the storage overhead of a covering btree index, BRIN is ideal for you, because you have clustering already in place (this is crucial for BRIN to be useful). BRIN indexes are tiny, so all the pages are likely to be in memory if you choose a suitable value of pages_per_range.
Is there a magic formula to find a good value of pages_per_range that takes into account those trade offs?
No magic formula, but start with pages_per_range somewhat less than the average size (in pages) occupied by the average a value. You are probably trying to minimize: (number of BRIN pages scanned)+(number of heap pages scanned) for a typical query. Look for Heap Blocks: lossy=n in the execution plan for pages_per_range=1 and compare with other values for pages_per_range — i.e. see how many unnecessary heap blocks are being scanned.

GIN/GiST

Not sure those are relevant here since they're mostly used for full text search, but I also hear that they're good at dealing with duplicate keys. Would either a GIN/GiST index help here?
GIN may be worth considering, but probably not GiST — however if the natural clustering really is good, then BRIN will probably be a better bet.
Here is a sample comparison between the different index types for dummy data a bit like yours:
table and indexes:
create table foo(a,b,c) as
select *, lpad('',20)
from (select chr(g) a from generate_series(97,122) g) a
     cross join (select generate_series(1,100000) b) b
order by a;
create index foo_btree_covering on foo(a,b);
create index foo_btree on foo(a);
create index foo_gin on foo using gin(a);
create index foo_brin_2 on foo using brin(a) with (pages_per_range=2);
create index foo_brin_4 on foo using brin(a) with (pages_per_range=4);
vacuum analyze;
relation sizes:
select relname "name", pg_size_pretty(siz) "size", siz/8192 pages, (select count(*) from foo)*8192/siz "rows/page"
from( select relname, pg_relation_size(C.oid) siz
      from pg_class c join pg_namespace n on n.oid = c.relnamespace
      where nspname = current_schema ) z;
name               | size    | pages | rows/page
:----------------- | :------ | ----: | --------:
foo                | 149 MB  | 19118 |       135
foo_btree_covering | 56 MB   |  7132 |       364
foo_btree          | 56 MB   |  7132 |       364
foo_gin            | 2928 kB |   366 |      7103
foo_brin_2         | 264 kB  |    33 |     78787
foo_brin_4         | 136 kB  |    17 |    152941
covering btree:
explain analyze select sum(b) from foo where a='a';
| QUERY PLAN                                                                                                                                      |
| :---------------------------------------------------------------------------------------------------------------------------------------------- |
| Aggregate  (cost=3282.57..3282.58 rows=1 width=8) (actual time=45.942..45.942 rows=1 loops=1)                                                   |
|   ->  Index Only Scan using foo_btree_covering on foo  (cost=0.43..3017.80 rows=105907 width=4) (actual time=0.038..27.286 rows=100000 loops=1) |
|         Index Cond: (a = 'a'::text)                                                                                                             |
|         Heap Fetches: 0                                                                                                                         |
| Planning time: 0.099 ms                                                                                                                         |
| Execution time: 45.968 ms                                                                                                                       |
plain btree:
drop index foo_btree_covering;
explain analyze select sum(b) from foo where a='a';
| QUERY PLAN                                                                                                                        |
| :-------------------------------------------------------------------------------------------------------------------------------- |
| Aggregate  (cost=4064.57..4064.58 rows=1 width=8) (actual time=54.242..54.242 rows=1 loops=1)                                     |
|   ->  Index Scan using foo_btree on foo  (cost=0.43..3799.80 rows=105907 width=4) (actual time=0.037..33.084 rows=100000 loops=1) |
|         Index Cond: (a = 'a'::text)                                                                                               |
| Planning time: 0.135 ms                                                                                                           |
| Execution time: 54.280 ms                                                                                                         |
BRIN pages_per_range=4:
drop index foo_btree;
explain analyze select sum(b) from foo where a='a';
| QUERY PLAN                                                                                                                        |
| :-------------------------------------------------------------------------------------------------------------------------------- |
| Aggregate  (cost=21595.38..21595.39 rows=1 width=8) (actual time=52.455..52.455 rows=1 loops=1)                                   |
|   ->  Bitmap Heap Scan on foo  (cost=888.78..21330.61 rows=105907 width=4) (actual time=2.738..31.967 rows=100000 loops=1)        |
|         Recheck Cond: (a = 'a'::text)                                                                                             |
|         Rows Removed by Index Recheck: 96                                                                                         |
|         Heap Blocks: lossy=736                                                                                                    |
|         ->  Bitmap Index Scan on foo_brin_4  (cost=0.00..862.30 rows=105907 width=0) (actual time=2.720..2.720 rows=7360 loops=1) |
|               Index Cond: (a = 'a'::text)                                                                                         |
| Planning time: 0.101 ms                                                                                                           |
| Execution time: 52.501 ms                                                                                                         |
BRIN pages_per_range=2:
drop index foo_brin_4;
explain analyze select sum(b) from foo where a='a';
| QUERY PLAN                                                                                                                        |
| :-------------------------------------------------------------------------------------------------------------------------------- |
| Aggregate  (cost=21659.38..21659.39 rows=1 width=8) (actual time=53.971..53.971 rows=1 loops=1)                                   |
|   ->  Bitmap Heap Scan on foo  (cost=952.78..21394.61 rows=105907 width=4) (actual time=5.286..33.492 rows=100000 loops=1)        |
|         Recheck Cond: (a = 'a'::text)                                                                                             |
|         Rows Removed by Index Recheck: 96                                                                                         |
|         Heap Blocks: lossy=736                                                                                                    |
|         ->  Bitmap Index Scan on foo_brin_2  (cost=0.00..926.30 rows=105907 width=0) (actual time=5.275..5.275 rows=7360 loops=1) |
|               Index Cond: (a = 'a'::text)                                                                                         |
| Planning time: 0.095 ms                                                                                                           |
| Execution time: 54.016 ms                                                                                                         |
GIN:
drop index foo_brin_2;
explain analyze select sum(b) from foo where a='a';
| QUERY PLAN                                                                                                                         |
| :--------------------------------------------------------------------------------------------------------------------------------- |
| Aggregate  (cost=21687.38..21687.39 rows=1 width=8) (actual time=55.331..55.331 rows=1 loops=1)                                    |
|   ->  Bitmap Heap Scan on foo  (cost=980.78..21422.61 rows=105907 width=4) (actual time=12.377..33.956 rows=100000 loops=1)        |
|         Recheck Cond: (a = 'a'::text)                                                                                              |
|         Heap Blocks: exact=736                                                                                                     |
|         ->  Bitmap Index Scan on foo_gin  (cost=0.00..954.30 rows=105907 width=0) (actual time=12.271..12.271 rows=100000 loops=1) |
|               Index Cond: (a = 'a'::text)                                                                                          |
| Planning time: 0.118 ms                                                                                                            |
| Execution time: 55.366 ms                                                                

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