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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> + + </main> + + <nav class="nav-wrapper" aria-label="Page navigation"> + <!-- Mobile navigation buttons --> + <a rel="prev" href="development/room-dag-concepts.html" class="mobile-nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left"> + <i class="fa fa-angle-left"></i> + </a> + <a rel="next" href="media_repository.html" class="mobile-nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right"> + <i class="fa fa-angle-right"></i> + </a> + <div style="clear: both"></div> + </nav> + </div> + </div> + + <nav class="nav-wide-wrapper" aria-label="Page navigation"> + <a rel="prev" href="development/room-dag-concepts.html" class="nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left"> + <i class="fa fa-angle-left"></i> + </a> + <a rel="next" href="media_repository.html" class="nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right"> + <i class="fa fa-angle-right"></i> + </a> + </nav> + + </div> + + <script type="text/javascript"> + window.playground_copyable = true; + </script> + <script src="elasticlunr.min.js" type="text/javascript" charset="utf-8"></script> + <script src="mark.min.js" type="text/javascript" charset="utf-8"></script> + <script src="searcher.js" type="text/javascript" charset="utf-8"></script> + <script src="clipboard.min.js" type="text/javascript" charset="utf-8"></script> + <script src="highlight.js" type="text/javascript" charset="utf-8"></script> + <script src="book.js" type="text/javascript" charset="utf-8"></script> + + <!-- Custom JS scripts --> + <script type="text/javascript" src="docs/website_files/table-of-contents.js"></script> + </body> +</html> \ No newline at end of file |