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- <h1 id="auth-chain-difference-algorithm"><a class="header" href="#auth-chain-difference-algorithm">Auth Chain Difference Algorithm</a></h1>
- <p>The auth chain difference algorithm is used by V2 state resolution, where a
- naive implementation can be a significant source of CPU and DB usage.</p>
- <h3 id="definitions"><a class="header" href="#definitions">Definitions</a></h3>
- <p>A <em>state set</em> is a set of state events; e.g. the input of a state resolution
- algorithm is a collection of state sets.</p>
- <p>The <em>auth chain</em> of a set of events are all the events' auth events and <em>their</em>
- auth events, recursively (i.e. the events reachable by walking the graph induced
- by an event's auth events links).</p>
- <p>The <em>auth chain difference</em> of a collection of state sets is the union minus the
- intersection of the sets of auth chains corresponding to the state sets, i.e an
- event is in the auth chain difference if it is reachable by walking the auth
- event graph from at least one of the state sets but not from <em>all</em> of the state
- sets.</p>
- <h2 id="breadth-first-walk-algorithm"><a class="header" href="#breadth-first-walk-algorithm">Breadth First Walk Algorithm</a></h2>
- <p>A way of calculating the auth chain difference without calculating the full auth
- chains for each state set is to do a parallel breadth first walk (ordered by
- depth) of each state set's auth chain. By tracking which events are reachable
- from each state set we can finish early if every pending event is reachable from
- every state set.</p>
- <p>This can work well for state sets that have a small auth chain difference, but
- can be very inefficient for larger differences. However, this algorithm is still
- used if we don't have a chain cover index for the room (e.g. because we're in
- the process of indexing it).</p>
- <h2 id="chain-cover-index"><a class="header" href="#chain-cover-index">Chain Cover Index</a></h2>
- <p>Synapse computes auth chain differences by pre-computing a "chain cover" index
- for the auth chain in a room, allowing us to efficiently make reachability queries
- like "is event <code>A</code> in the auth chain of event <code>B</code>?". We could do this with an index
- that tracks all pairs <code>(A, B)</code> such that <code>A</code> is in the auth chain of <code>B</code>. However, this
- would be prohibitively large, scaling poorly as the room accumulates more state
- events.</p>
- <p>Instead, we break down the graph into <em>chains</em>. A chain is a subset of a DAG
- with the following property: for any pair of events <code>E</code> and <code>F</code> in the chain,
- the chain contains a path <code>E -> F</code> or a path <code>F -> E</code>. This forces a chain to be
- linear (without forks), e.g. <code>E -> F -> G -> ... -> H</code>. Each event in the chain
- is given a <em>sequence number</em> local to that chain. The oldest event <code>E</code> in the
- chain has sequence number 1. If <code>E</code> has a child <code>F</code> in the chain, then <code>F</code> has
- sequence number 2. If <code>E</code> has a grandchild <code>G</code> in the chain, then <code>G</code> has
- sequence number 3; and so on.</p>
- <p>Synapse ensures that each persisted event belongs to exactly one chain, and
- tracks how the chains are connected to one another. This allows us to
- efficiently answer reachability queries. Doing so uses less storage than
- tracking reachability on an event-by-event basis, particularly when we have
- fewer and longer chains. See</p>
- <blockquote>
- <p>Jagadish, H. (1990). <a href="https://doi.org/10.1145/99935.99944">A compression technique to materialize transitive closure</a>.
- <em>ACM Transactions on Database Systems (TODS)</em>, 15*(4)*, 558-598.</p>
- </blockquote>
- <p>for the original idea or</p>
- <blockquote>
- <p>Y. Chen, Y. Chen, <a href="https://doi.org/10.1109/ICDE.2008.4497498">An efficient algorithm for answering graph
- reachability queries</a>,
- in: 2008 IEEE 24th International Conference on Data Engineering, April 2008,
- pp. 893–902. (PDF available via <a href="https://scholar.google.com/scholar?q=Y.%20Chen,%20Y.%20Chen,%20An%20efficient%20algorithm%20for%20answering%20graph%20reachability%20queries,%20in:%202008%20IEEE%2024th%20International%20Conference%20on%20Data%20Engineering,%20April%202008,%20pp.%20893902.">Google Scholar</a>.)</p>
- </blockquote>
- <p>for a more modern take.</p>
- <p>In practical terms, the chain cover assigns every event a
- <em>chain ID</em> and <em>sequence number</em> (e.g. <code>(5,3)</code>), and maintains a map of <em>links</em>
- between events in chains (e.g. <code>(5,3) -> (2,4)</code>) such that <code>A</code> is reachable by <code>B</code>
- (i.e. <code>A</code> is in the auth chain of <code>B</code>) if and only if either:</p>
- <ol>
- <li><code>A</code> and <code>B</code> have the same chain ID and <code>A</code>'s sequence number is less than <code>B</code>'s
- sequence number; or</li>
- <li>there is a link <code>L</code> between <code>B</code>'s chain ID and <code>A</code>'s chain ID such that
- <code>L.start_seq_no</code> <= <code>B.seq_no</code> and <code>A.seq_no</code> <= <code>L.end_seq_no</code>.</li>
- </ol>
- <p>There are actually two potential implementations, one where we store links from
- each chain to every other reachable chain (the transitive closure of the links
- graph), and one where we remove redundant links (the transitive reduction of the
- links graph) e.g. if we have chains <code>C3 -> C2 -> C1</code> then the link <code>C3 -> C1</code>
- would not be stored. Synapse uses the former implementation so that it doesn't
- need to recurse to test reachability between chains. This trades-off extra storage
- in order to save CPU cycles and DB queries.</p>
- <h3 id="example"><a class="header" href="#example">Example</a></h3>
- <p>An example auth graph would look like the following, where chains have been
- formed based on type/state_key and are denoted by colour and are labelled with
- <code>(chain ID, sequence number)</code>. Links are denoted by the arrows (links in grey
- are those that would be remove in the second implementation described above).</p>
- <p><img src="auth_chain_diff.dot.png" alt="Example" /></p>
- <p>Note that we don't include all links between events and their auth events, as
- most of those links would be redundant. For example, all events point to the
- create event, but each chain only needs the one link from it's base to the
- create event.</p>
- <h2 id="using-the-index"><a class="header" href="#using-the-index">Using the Index</a></h2>
- <p>This index can be used to calculate the auth chain difference of the state sets
- by looking at the chain ID and sequence numbers reachable from each state set:</p>
- <ol>
- <li>For every state set lookup the chain ID/sequence numbers of each state event</li>
- <li>Use the index to find all chains and the maximum sequence number reachable
- from each state set.</li>
- <li>The auth chain difference is then all events in each chain that have sequence
- numbers between the maximum sequence number reachable from <em>any</em> state set and
- the minimum reachable by <em>all</em> state sets (if any).</li>
- </ol>
- <p>Note that steps 2 is effectively calculating the auth chain for each state set
- (in terms of chain IDs and sequence numbers), and step 3 is calculating the
- difference between the union and intersection of the auth chains.</p>
- <h3 id="worked-example"><a class="header" href="#worked-example">Worked Example</a></h3>
- <p>For example, given the above graph, we can calculate the difference between
- state sets consisting of:</p>
- <ol>
- <li><code>S1</code>: Alice's invite <code>(4,1)</code> and Bob's second join <code>(2,2)</code>; and</li>
- <li><code>S2</code>: Alice's second join <code>(4,3)</code> and Bob's first join <code>(2,1)</code>.</li>
- </ol>
- <p>Using the index we see that the following auth chains are reachable from each
- state set:</p>
- <ol>
- <li><code>S1</code>: <code>(1,1)</code>, <code>(2,2)</code>, <code>(3,1)</code> & <code>(4,1)</code></li>
- <li><code>S2</code>: <code>(1,1)</code>, <code>(2,1)</code>, <code>(3,2)</code> & <code>(4,3)</code></li>
- </ol>
- <p>And so, for each the ranges that are in the auth chain difference:</p>
- <ol>
- <li>Chain 1: None, (since everything can reach the create event).</li>
- <li>Chain 2: The range <code>(1, 2]</code> (i.e. just <code>2</code>), as <code>1</code> is reachable by all state
- sets and the maximum reachable is <code>2</code> (corresponding to Bob's second join).</li>
- <li>Chain 3: Similarly the range <code>(1, 2]</code> (corresponding to the second power
- level).</li>
- <li>Chain 4: The range <code>(1, 3]</code> (corresponding to both of Alice's joins).</li>
- </ol>
- <p>So the final result is: Bob's second join <code>(2,2)</code>, the second power level
- <code>(3,2)</code> and both of Alice's joins <code>(4,2)</code> & <code>(4,3)</code>.</p>
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