Optimize how we calculate `likely_domains` during backfill (#13575)

Optimize how we calculate `likely_domains` during backfill because I've seen this take 17s in production just to `get_current_state` which is used to `get_domains_from_state` (see case [*2. Loading tons of events* in the `/messages` investigation issue](https://github.com/matrix-org/synapse/issues/13356)).

There are 3 ways we currently calculate hosts that are in the room:

 1. `get_current_state` -> `get_domains_from_state`
    - Used in `backfill` to calculate `likely_domains` and `/timestamp_to_event` because it was cargo-culted from `backfill`
    - This one is being eliminated in favor of `get_current_hosts_in_room` in this PR 🕳
 1. `get_current_hosts_in_room`
    - Used for other federation things like sending read receipts and typing indicators
 1. `get_hosts_in_room_at_events`
    - Used when pushing out events over federation to other servers in the `_process_event_queue_loop`

Fix https://github.com/matrix-org/synapse/issues/13626

Part of https://github.com/matrix-org/synapse/issues/13356

Mentioned in [internal doc](https://docs.google.com/document/d/1lvUoVfYUiy6UaHB6Rb4HicjaJAU40-APue9Q4vzuW3c/edit#bookmark=id.2tvwz3yhcafh)


### Query performance

#### Before

The query from `get_current_state` sucks just because we have to get all 80k events. And we see almost the exact same performance locally trying to get all of these events (16s vs 17s):
```
synapse=# SELECT type, state_key, event_id FROM current_state_events WHERE room_id = '!OGEhHVWSdvArJzumhm:matrix.org';
Time: 16035.612 ms (00:16.036)

synapse=# SELECT type, state_key, event_id FROM current_state_events WHERE room_id = '!OGEhHVWSdvArJzumhm:matrix.org';
Time: 4243.237 ms (00:04.243)
```

But what about `get_current_hosts_in_room`: When there is 8M rows in the `current_state_events` table, the previous query in `get_current_hosts_in_room` took 13s from complete freshness (when the events were first added). But takes 930ms after a Postgres restart or 390ms if running back to back to back.

```sh
$ psql synapse
synapse=# \timing on
synapse=# SELECT COUNT(DISTINCT substring(state_key FROM '@[^:]*:(.*)$'))
FROM current_state_events
WHERE
    type = 'm.room.member'
    AND membership = 'join'
    AND room_id = '!OGEhHVWSdvArJzumhm:matrix.org';
 count
-------
  4130
(1 row)

Time: 13181.598 ms (00:13.182)

synapse=# SELECT COUNT(*) from current_state_events where room_id = '!OGEhHVWSdvArJzumhm:matrix.org';
 count
-------
 80814

synapse=# SELECT COUNT(*) from current_state_events;
  count
---------
 8162847

synapse=# SELECT pg_size_pretty( pg_total_relation_size('current_state_events') );
 pg_size_pretty
----------------
 4702 MB
```

#### After

I'm not sure how long it takes from complete freshness as I only really get that opportunity once (maybe restarting computer but that's cumbersome) and it's not really relevant to normal operating times. Maybe you get closer to the fresh times the more access variability there is so that Postgres caches aren't as exact. Update: The longest I've seen this run for is 6.4s and 4.5s after a computer restart.

After a Postgres restart, it takes 330ms and running back to back takes 260ms.

```sh
$ psql synapse
synapse=# \timing on
Timing is on.
synapse=# SELECT
    substring(c.state_key FROM '@[^:]*:(.*)$') as host
FROM current_state_events c
/* Get the depth of the event from the events table */
INNER JOIN events AS e USING (event_id)
WHERE
    c.type = 'm.room.member'
    AND c.membership = 'join'
    AND c.room_id = '!OGEhHVWSdvArJzumhm:matrix.org'
GROUP BY host
ORDER BY min(e.depth) ASC;
Time: 333.800 ms
```

#### Going further

To improve things further we could add a `limit` parameter to `get_current_hosts_in_room`. Realistically, we don't need 4k domains to choose from because there is no way we're going to query that many before we a) probably get an answer or b) we give up. 

Another thing we can do is optimize the query to use a index skip scan:

 - https://wiki.postgresql.org/wiki/Loose_indexscan
 - Index Skip Scan, https://commitfest.postgresql.org/37/1741/
 - https://www.timescale.com/blog/how-we-made-distinct-queries-up-to-8000x-faster-on-postgresql/

2 files changed, 16 insertions, 51 deletions

diff --git a/synapse/handlers/federation.py b/synapse/handlers/federation.py
index e151962055..dd4b9f66d1 100644
--- a/synapse/handlers/federation.py
+++ b/synapse/handlers/federation.py
@@ -70,7 +70,7 @@ from synapse.replication.http.federation import (
 from synapse.storage.databases.main.events import PartialStateConflictError
 from synapse.storage.databases.main.events_worker import EventRedactBehaviour
 from synapse.storage.state import StateFilter
-from synapse.types import JsonDict, StateMap, get_domain_from_id
+from synapse.types import JsonDict, get_domain_from_id
 from synapse.util.async_helpers import Linearizer
 from synapse.util.retryutils import NotRetryingDestination
 from synapse.visibility import filter_events_for_server
@@ -104,37 +104,6 @@ backfill_processing_before_timer = Histogram(
 )
 
 
-def get_domains_from_state(state: StateMap[EventBase]) -> List[Tuple[str, int]]:
-    """Get joined domains from state
-
-    Args:
-        state: State map from type/state key to event.
-
-    Returns:
-        Returns a list of servers with the lowest depth of their joins.
-            Sorted by lowest depth first.
-    """
-    joined_users = [
-        (state_key, int(event.depth))
-        for (e_type, state_key), event in state.items()
-        if e_type == EventTypes.Member and event.membership == Membership.JOIN
-    ]
-
-    joined_domains: Dict[str, int] = {}
-    for u, d in joined_users:
-        try:
-            dom = get_domain_from_id(u)
-            old_d = joined_domains.get(dom)
-            if old_d:
-                joined_domains[dom] = min(d, old_d)
-            else:
-                joined_domains[dom] = d
-        except Exception:
-            pass
-
-    return sorted(joined_domains.items(), key=lambda d: d[1])
-
-
 class _BackfillPointType(Enum):
     # a regular backwards extremity (ie, an event which we don't yet have, but which
     # is referred to by other events in the DAG)
@@ -432,21 +401,19 @@ class FederationHandler:
         )
 
         # Now we need to decide which hosts to hit first.
-
-        # First we try hosts that are already in the room
+        # First we try hosts that are already in the room.
         # TODO: HEURISTIC ALERT.
+        likely_domains = (
+            await self._storage_controllers.state.get_current_hosts_in_room(room_id)
+        )
 
-        curr_state = await self._storage_controllers.state.get_current_state(room_id)
-
-        curr_domains = get_domains_from_state(curr_state)
-
-        likely_domains = [
-            domain for domain, depth in curr_domains if domain != self.server_name
-        ]
-
-        async def try_backfill(domains: List[str]) -> bool:
+        async def try_backfill(domains: Collection[str]) -> bool:
             # TODO: Should we try multiple of these at a time?
             for dom in domains:
+                # We don't want to ask our own server for information we don't have
+                if dom == self.server_name:
+                    continue
+
                 try:
                     await self._federation_event_handler.backfill(
                         dom, room_id, limit=100, extremities=extremities_to_request
diff --git a/synapse/handlers/room.py b/synapse/handlers/room.py
index 2fc8264858..f64a8690a5 100644
--- a/synapse/handlers/room.py
+++ b/synapse/handlers/room.py
@@ -60,7 +60,6 @@ from synapse.event_auth import validate_event_for_room_version
 from synapse.events import EventBase
 from synapse.events.utils import copy_and_fixup_power_levels_contents
 from synapse.federation.federation_client import InvalidResponseError
-from synapse.handlers.federation import get_domains_from_state
 from synapse.handlers.relations import BundledAggregations
 from synapse.module_api import NOT_SPAM
 from synapse.rest.admin._base import assert_user_is_admin
@@ -1462,17 +1461,16 @@ class TimestampLookupHandler:
                 timestamp,
             )
 
-            # Find other homeservers from the given state in the room
-            curr_state = await self._storage_controllers.state.get_current_state(
-                room_id
+            likely_domains = (
+                await self._storage_controllers.state.get_current_hosts_in_room(room_id)
             )
-            curr_domains = get_domains_from_state(curr_state)
-            likely_domains = [
-                domain for domain, depth in curr_domains if domain != self.server_name
-            ]
 
             # Loop through each homeserver candidate until we get a succesful response
             for domain in likely_domains:
+                # We don't want to ask our own server for information we don't have
+                if domain == self.server_name:
+                    continue
+
                 try:
                     remote_response = await self.federation_client.timestamp_to_event(
                         domain, room_id, timestamp, direction