Note a more efficient computation of the auth difference. (#1119)

changelogs/room_versions/newsfragments/1119.clarification

@@ -0,0 +1 @@
+For room version 2 through 10: Note a more efficient way to compute the auth difference during state resolution.

content/rooms/fragments/v2-state-res.md

@@ -47,13 +47,20 @@ all of *their* auth events, and so on recursively, stretching back to the
 start of the room. Put differently, these are the events reachable by walking
 the graph induced by an event's `auth_events` links.
 
+If *S* is a collection of
+events, the *full auth chain of S* is the union of the auth chains of every
+event *E* in *S*.
+
 **Auth difference.**
-The *auth difference* is calculated by first calculating the full auth
+For each state *S<sub>i</sub>*, compute its full auth chain.
-chain for each state *S*<sub>*i*</sub>, that is the union of the auth
+The *auth difference* is the set of events which belong to some, but not all,
-chains for each event in *S*<sub>*i*</sub>, and then taking every event
+of these full auth chains. In symbols: if *F*(*S*) is the full auth
-that doesn't appear in every auth chain. If *C*<sub>*i*</sub> is the
+chain of a collection of events *S*, the auth difference is
-full auth chain of *S*<sub>*i*</sub>, then the auth difference is
+    ∪<sub>*i*</sub> *F*(*S<sub>i</sub>*) - ∩<sub>*i*</sub> *F*(*S<sub>i</sub>*).
- ∪ *C*<sub>*i*</sub> −  ∩ *C*<sub>*i*</sub>.
+
+This can be computed more efficiently as
+    *F*(*C*) - ∩<sub>*i*</sub> *F*(*S*<sub>*i* - *C*</sub>) ,
+where *C* is the conflicted state set.
 
 **Full conflicted set.**
 The *full conflicted set* is the union of the conflicted state set and