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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 us to efficiently make reachability queries
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like "is event <code>A</code> in the auth chain of event <code>B</code>?". We could do this with an index
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that tracks all pairs <code>(A, B)</code> such that <code>A</code> is in the auth chain of <code>B</code>. However, this
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would be prohibitively large, scaling poorly as the room accumulates more state
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events.</p>
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<p>Instead, we break down the graph into <em>chains</em>. A chain is a subset of a DAG
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with the following property: for any pair of events <code>E</code> and <code>F</code> in the chain,
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the chain contains a path <code>E -> F</code> or a path <code>F -> E</code>. This forces a chain to be
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linear (without forks), e.g. <code>E -> F -> G -> ... -> H</code>. Each event in the chain
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is given a <em>sequence number</em> local to that chain. The oldest event <code>E</code> in the
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chain has sequence number 1. If <code>E</code> has a child <code>F</code> in the chain, then <code>F</code> has
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sequence number 2. If <code>E</code> has a grandchild <code>G</code> in the chain, then <code>G</code> has
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sequence number 3; and so on.</p>
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<p>Synapse ensures that each persisted event belongs to exactly one chain, and
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tracks how the chains are connected to one another. This allows us to
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efficiently answer reachability queries. Doing so uses less storage than
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tracking reachability on an event-by-event basis, particularly when we have
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fewer and longer chains. See</p>
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<blockquote>
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<p>Jagadish, H. (1990). <a href="https://doi.org/10.1145/99935.99944">A compression technique to materialize transitive closure</a>.
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<em>ACM Transactions on Database Systems (TODS)</em>, 15*(4)*, 558-598.</p>
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</blockquote>
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<p>for the original idea or</p>
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<blockquote>
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<p>Y. Chen, Y. Chen, <a href="https://doi.org/10.1109/ICDE.2008.4497498">An efficient algorithm for answering graph
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reachability queries</a>,
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in: 2008 IEEE 24th International Conference on Data Engineering, April 2008,
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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>
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</blockquote>
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<p>for a more modern take.</p>
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<p>In practical terms, the chain cover assigns every event a
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<em>chain ID</em> and <em>sequence number</em> (e.g. <code>(5,3)</code>), and maintains a map of <em>links</em>
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between events in chains (e.g. <code>(5,3) -> (2,4)</code>) such that <code>A</code> is reachable by <code>B</code>
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(i.e. <code>A</code> is in the auth chain of <code>B</code>) if and only if either:</p>
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<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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