Cache is_partial_state_room (#13693)

Fixes #13613.

changelog.d/13693.misc

@@ -0,0 +1 @@
+Add cache to `is_partial_state_room`.

synapse/storage/databases/main/room.py

@@ -1185,8 +1185,9 @@ async def clear_partial_state_room(self, room_id: str) -> bool:
             )
             return False
 
-    @staticmethod
-    def _clear_partial_state_room_txn(txn: LoggingTransaction, room_id: str) -> None:
+    def _clear_partial_state_room_txn(
+        self, txn: LoggingTransaction, room_id: str
+    ) -> None:
         DatabasePool.simple_delete_txn(
             txn,
             table="partial_state_rooms_servers",
@@ -1197,7 +1198,9 @@ def _clear_partial_state_room_txn(txn: LoggingTransaction, room_id: str) -> None
             table="partial_state_rooms",
             keyvalues={"room_id": room_id},
         )
+        self._invalidate_cache_and_stream(txn, self.is_partial_state_room, (room_id,))
 
+    @cached()
     async def is_partial_state_room(self, room_id: str) -> bool:
         """Checks if this room has partial state.
 
@@ -1771,9 +1774,8 @@ async def store_partial_state_room(
             servers,
         )
 
-    @staticmethod
     def _store_partial_state_room_txn(
-        txn: LoggingTransaction, room_id: str, servers: Collection[str]
+        self, txn: LoggingTransaction, room_id: str, servers: Collection[str]
     ) -> None:
         DatabasePool.simple_insert_txn(
             txn,
@@ -1788,6 +1790,7 @@ def _store_partial_state_room_txn(
             keys=("room_id", "server_name"),
             values=((room_id, s) for s in servers),
         )
+        self._invalidate_cache_and_stream(txn, self.is_partial_state_room, (room_id,))
 
     async def maybe_store_room_on_outlier_membership(
         self, room_id: str, room_version: RoomVersion