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<h1 id="auth-chain-difference-algorithm"><a class="header" href="#auth-chain-difference-algorithm">Auth Chain Difference Algorithm</a></h1>
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<p>The auth chain difference algorithm is used by V2 state resolution, where a
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naive implementation can be a significant source of CPU and DB usage.</p>
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<h3 id="definitions"><a class="header" href="#definitions">Definitions</a></h3>
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<p>A <em>state set</em> is a set of state events; e.g. the input of a state resolution
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algorithm is a collection of state sets.</p>
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<p>The <em>auth chain</em> of a set of events are all the events' auth events and <em>their</em>
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auth events, recursively (i.e. the events reachable by walking the graph induced
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by an event's auth events links).</p>
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<p>The <em>auth chain difference</em> of a collection of state sets is the union minus the
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intersection of the sets of auth chains corresponding to the state sets, i.e an
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event is in the auth chain difference if it is reachable by walking the auth
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event graph from at least one of the state sets but not from <em>all</em> of the state
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sets.</p>
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<h2 id="breadth-first-walk-algorithm"><a class="header" href="#breadth-first-walk-algorithm">Breadth First Walk Algorithm</a></h2>
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<p>A way of calculating the auth chain difference without calculating the full auth
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chains for each state set is to do a parallel breadth first walk (ordered by
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depth) of each state set's auth chain. By tracking which events are reachable
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from each state set we can finish early if every pending event is reachable from
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every state set.</p>
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<p>This can work well for state sets that have a small auth chain difference, but
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can be very inefficient for larger differences. However, this algorithm is still
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used if we don't have a chain cover index for the room (e.g. because we're in
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the process of indexing it).</p>
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<h2 id="chain-cover-index"><a class="header" href="#chain-cover-index">Chain Cover Index</a></h2>
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<p>Synapse computes auth chain differences by pre-computing a "chain cover" index
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for the auth chain in a room, allowing efficient reachability queries like "is
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event A in the auth chain of event B". This is done by assigning every event a
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<em>chain ID</em> and <em>sequence number</em> (e.g. <code>(5,3)</code>), and having a map of <em>links</em>
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between chains (e.g. <code>(5,3) -> (2,4)</code>) such that A is reachable by B (i.e. <code>A</code>
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is in the auth chain of <code>B</code>) if and only if either:</p>
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<ol>
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<li>A and B have the same chain ID and <code>A</code>'s sequence number is less than <code>B</code>'s
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sequence number; or</li>
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<li>there is a link <code>L</code> between <code>B</code>'s chain ID and <code>A</code>'s chain ID such that
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<code>L.start_seq_no</code> <= <code>B.seq_no</code> and <code>A.seq_no</code> <= <code>L.end_seq_no</code>.</li>
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</ol>
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<p>There are actually two potential implementations, one where we store links from
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each chain to every other reachable chain (the transitive closure of the links
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graph), and one where we remove redundant links (the transitive reduction of the
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links graph) e.g. if we have chains <code>C3 -> C2 -> C1</code> then the link <code>C3 -> C1</code>
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would not be stored. Synapse uses the former implementations so that it doesn't
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need to recurse to test reachability between chains.</p>
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<h3 id="example"><a class="header" href="#example">Example</a></h3>
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<p>An example auth graph would look like the following, where chains have been
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formed based on type/state_key and are denoted by colour and are labelled with
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<code>(chain ID, sequence number)</code>. Links are denoted by the arrows (links in grey
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are those that would be remove in the second implementation described above).</p>
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<p><img src="auth_chain_diff.dot.png" alt="Example" /></p>
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<p>Note that we don't include all links between events and their auth events, as
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most of those links would be redundant. For example, all events point to the
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create event, but each chain only needs the one link from it's base to the
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create event.</p>
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<h2 id="using-the-index"><a class="header" href="#using-the-index">Using the Index</a></h2>
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<p>This index can be used to calculate the auth chain difference of the state sets
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by looking at the chain ID and sequence numbers reachable from each state set:</p>
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<ol>
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<li>For every state set lookup the chain ID/sequence numbers of each state event</li>
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<li>Use the index to find all chains and the maximum sequence number reachable
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from each state set.</li>
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<li>The auth chain difference is then all events in each chain that have sequence
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numbers between the maximum sequence number reachable from <em>any</em> state set and
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the minimum reachable by <em>all</em> state sets (if any).</li>
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</ol>
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<p>Note that steps 2 is effectively calculating the auth chain for each state set
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(in terms of chain IDs and sequence numbers), and step 3 is calculating the
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difference between the union and intersection of the auth chains.</p>
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<h3 id="worked-example"><a class="header" href="#worked-example">Worked Example</a></h3>
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<p>For example, given the above graph, we can calculate the difference between
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state sets consisting of:</p>
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<ol>
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<li><code>S1</code>: Alice's invite <code>(4,1)</code> and Bob's second join <code>(2,2)</code>; and</li>
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<li><code>S2</code>: Alice's second join <code>(4,3)</code> and Bob's first join <code>(2,1)</code>.</li>
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</ol>
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<p>Using the index we see that the following auth chains are reachable from each
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state set:</p>
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<ol>
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<li><code>S1</code>: <code>(1,1)</code>, <code>(2,2)</code>, <code>(3,1)</code> & <code>(4,1)</code></li>
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<li><code>S2</code>: <code>(1,1)</code>, <code>(2,1)</code>, <code>(3,2)</code> & <code>(4,3)</code></li>
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</ol>
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<p>And so, for each the ranges that are in the auth chain difference:</p>
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<ol>
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<li>Chain 1: None, (since everything can reach the create event).</li>
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<li>Chain 2: The range <code>(1, 2]</code> (i.e. just <code>2</code>), as <code>1</code> is reachable by all state
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sets and the maximum reachable is <code>2</code> (corresponding to Bob's second join).</li>
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<li>Chain 3: Similarly the range <code>(1, 2]</code> (corresponding to the second power
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level).</li>
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<li>Chain 4: The range <code>(1, 3]</code> (corresponding to both of Alice's joins).</li>
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</ol>
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<p>So the final result is: Bob's second join <code>(2,2)</code>, the second power level
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<code>(3,2)</code> and both of Alice's joins <code>(4,2)</code> & <code>(4,3)</code>.</p>
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</main>
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