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+<!DOCTYPE html>
+<html>
+<head>
+<title>RDF Graph Normalization</title>
+<meta http-equiv="content-type" content="text/html; charset=UTF-8">
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+ For the three scripts below, if your spec resides on dev.w3 you can check them
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+ // extend the bibliography entries
+ berjon.biblio["MICRODATA"] = "Ian Hickson; et al. <a href=\"http://www.w3.org/TR/microdata/\"><cite>Microdata</cite></a> 04 March 2010. W3C Working Draft. URL: <a href=\"http://www.w3.org/TR/microdata/\">http://www.w3.org/TR/microdata/</a> ";
+ berjon.biblio["HTML-RDFA"] = "Manu Sporny; et al. <a href=\"http://www.w3.org/TR/rdfa-in-html/\"><cite>HTML+RDFa</cite></a> 04 March 2010. W3C Working Draft. URL: <a href=\"http://www.w3.org/TR/rdfa-in-html/\">http://www.w3.org/TR/rdfa-in-html/</a> ";
+ berjon.biblio["BCP47"] = "A. Phillips, M. Davis. <a href=\"http://tools.ietf.org/rfc/bcp/bcp47.txt\"><cite>Tags for Identifying Languages</cite></a> September 2009. IETF Best Current Practice. URL: <a href=\"http://tools.ietf.org/rfc/bcp/bcp47.txt\">http://tools.ietf.org/rfc/bcp/bcp47.txt</a>";
+ berjon.biblio["JSON-LD"] = "Manu Sporny, Gregg Kellogg, et al. <a href=\"http://json-ld.org/spec/latest/json-ld-syntax/\"><cite>The JSON-LD Syntax</cite></a> Latest. W3C Editor's Draft. URL: <a href=\"http://json-ld.org/spec/latest/json-ld-syntax/\">http://json-ld.org/spec/latest/json-ld-syntax/</a>";
+ berjon.biblio["RDF-API"] = "Manu Sporny, Benjamin Adrian, Nathan Rixham; et al. <a href=\"http://www.w3.org/2010/02/rdfa/sources/rdf-api/\"><cite>RDF API</cite></a> Latest. W3C Editor's Draft. URL: <a href=\"http://www.w3.org/2010/02/rdfa/sources/rdf-api/\">http://www.w3.org/2010/02/rdfa/sources/rdf-api/</a>";
+ berjon.biblio["RDF-INTERFACES"] = "Nathan Rixham, Manu Sporny, Benjamin Adrian; et al. <a href=\"http://www.w3.org/2010/02/rdfa/sources/rdf-interfaces/\"><cite>RDF Interfaces</cite></a> Latest. W3C Editor's Draft. URL: <a href=\"http://www.w3.org/2010/02/rdfa/sources/rdf-interfaces/\">http://www.w3.org/2010/02/rdfa/sources/rdf-interfaces/</a>";
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+ var respecConfig = {
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+ copyrightStart: "2010",
+
+ // the specification's short name, as in http://www.w3.org/TR/short-name/
+ shortName: "rdf-graph-normalization",
+ subtitle: "A Standard RDF Graph Normalization Algorithm",
+ // if you wish the publication date to be other than today, set this
+ // publishDate: "2009-08-06",
+
+ // if there is a previously published draft, uncomment this and set its YYYY-MM-DD date
+ // and its maturity status
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+
+ // editors, add as many as you like
+ // only "name" is required
+ editors: [
+ { name: "Manu Sporny", url: "http://manu.sporny.org/",
+ company: "Digital Bazaar", companyURL: "http://digitalbazaar.com/" },
+ { name: "Dave Longley", url: "http://digitalbazaar.com/",
+ company: "Digital Bazaar", companyURL: "http://digitalbazaar.com/"}
+ ],
+
+ // authors, add as many as you like.
+ // This is optional, uncomment if you have authors as well as editors.
+ // only "name" is required. Same format as editors.
+
+ authors: [
+ { name: "Dave Longley", url: "http://digitalbazaar.com/",
+ company: "Digital Bazaar", companyURL: "http://digitalbazaar.com/"},
+ { name: "Manu Sporny", url: "http://digitalbazaar.com/",
+ company: "Digital Bazaar", companyURL: "http://digitalbazaar.com/" },
+ ],
+
+ // name of the WG
+ wg: "Linking Data in JSON Community Group",
+
+ // URI of the public WG page
+ wgURI: "http://json-ld.org/",
+
+ // name (with the @w3c.org) of the public mailing to which comments are due
+ wgPublicList: "public-linked-json@w3c.org",
+
+ // URI of the patent status for this WG, for Rec-track documents
+ // !!!! IMPORTANT !!!!
+ // This is important for Rec-track documents, do not copy a patent URI from a random
+ // document unless you know what you're doing. If in doubt ask your friendly neighbourhood
+ // Team Contact.
+ wgPatentURI: "",
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+ function updateTTL(doc, content) {
+ // perform transformations to
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+ </script>
+<style>
+.diff { font-weight:bold; color:#0a3; }
+ol.algorithm.update { margin-left: 2em; }
+ol.algorithm.update>li { list-style-type: none; }
+ol.algorithm.update>li>span.list-number {
+ display:block;
+ float: left;
+ margin-left: -3.5em;
+}
+</style>
+</head>
+
+<body>
+<section id="abstract">
+<p>
+RDF [[RDF-SYNTAX]] describes a graph-based data model for making claims about
+the world and provides the foundation for reasoning upon that graph of
+information. At times, it becomes necessary to compare the differences between
+graphs, digitally sign graphs, or generate short identifiers for graphs via
+hashing algorithms. This document outlines an algorithm for normalizing RDF
+graphs such that these operations can be performed on the normalized graphs.
+</p>
+</section>
+
+<section id='sotd'>
+<p>This document is an experimental work in progress.</p>
+<!-- <p>
+This document has been reviewed by W3C Members, by software
+developers, and by other W3C groups and interested parties, and is
+endorsed by the Director as a W3C Recommendation. It is a stable
+document and may be used as reference material or cited from another
+document. W3C's role in making the Recommendation is to draw attention
+to the specification and to promote its widespread deployment. This
+enhances the functionality and interoperability of the Web.
+</p> -->
+</section>
+
+<section>
+<h1>Introduction</h1>
+
+<p>When data scientists discuss graph normalization, they
+do so in the context of achieving a particular set of goals. Since a directed
+graph can express the same information in a variety of different ways, it
+becomes necessary to be able to transform a graph of information into a
+standard form for the purposes of calculating differences between two graphs,
+generating a cryptographically-strong hash identifier for a graph, or digitally
+signing a graph.</p>
+
+<p>Many RDF graphs, containing nodes with unique identifiers, can be normalized
+quite easily. However, when a node does not have a unique identifier,
+graph normalization becomes exponentially more difficult. This problem is
+called the <tdef>graph isomorphism</tdef> problem, and it is believed to be a
+NP-hard problem in the worst cases. This means that the problem is very
+difficult to solve in a reasonable timeframe for certain inputs. Luckily, these
+types of inputs are extremely rare in the real world. Graph isomorphisms are
+most commonly used as a denial of service attack and thus any software system
+attempting to solve the graph normalization problem should be able to detect
+graph isomorphisms correctly.</p>
+
+<p>This document outlines an algorithm for generating a normalized RDF
+graph given an input RDF graph. The algorithm is called the
+<strong>Universal RDF Graph Normalization Algorithm 2011</strong> or
+<strong>URGNA2011</strong>.</p>
+
+<section>
+<h2>How to Read this Document</h2>
+
+<p>
+This document is a detailed specification for an RDF graph normalization
+algorithm. The document is primarily intended for the following audiences:
+</p>
+
+<ul>
+ <li>Software developers that want to implement an RDF graph normalization
+algorithm.</li>
+ <li>Masochists.</li>
+</ul>
+
+<p>
+To understand the basics in this specification you must be familiar with basic
+RDF concepts [[!RDF-CONCEPTS]]. A working knowledge of
+<a href="http://en.wikipedia.org/wiki/Graph_theory">graph theory</a> is
+also recommended.</p>
+</section>
+
+<section>
+<h2>Contributing</h2>
+
+<p>There are a number of ways that one may participate in the development of
+this specification:</p>
+
+<ul>
+<li>Technical discussion typically occurs on the public mailing list:
+<a href="http://lists.w3.org/Archives/Public/public-linked-json/">public-linked-json@w3.org</a>
+</li>
+
+<li><a href="http://json-ld.org/minutes/">Public teleconferences</a> are held
+on Tuesdays at 1500UTC on the second and fourth week of each month.
+</li>
+
+<li>Specification bugs and issues should be reported in the
+<a href="https://github.com/json-ld/json-ld.org/issues">issue tracker</a>.</li>
+
+<li><a href="https://github.com/json-ld/json-ld.org/tree/master/spec">Source code</a> for the
+specification can be found on Github.</li>
+
+<li>The <a href="http://webchat.freenode.net/?channels=#json-ld">#json-ld</a>
+IRC channel is available for real-time discussion on irc.freenode.net.</li>
+</ul>
+
+</section>
+
+</section>
+
+<section>
+<h1>Normalization</h1>
+
+<p class="issue">This algorithm is a work in progress, do not implement it.</p>
+
+<p>Normalization is the process of taking <tref>input graph</tref> and
+performing a transformation on that input that results in all
+aspects of the graph being arranged in a deterministic way in the
+<tref>output graph</tref>. That is, for two <tref>input graph</tref>s
+containing the same
+information, regardless of their arrangement, the <tref>output graph</tref>s
+will be identical. The problem is a fairly difficult technical
+problem to solve because it requires a directed graph to be ordered into a
+set of nodes and edges in a deterministic way. This is easy to do when all of
+the nodes have unique names, but very difficult to do when some of the nodes
+are not labeled and thus names must be generated for those nodes in a
+deterministic fashion.
+</p>
+
+<p>In time, there may be more than one normalization algorithm that will need
+to be identified. For identification purposes, this algorithm is named the
+"Universal RDF Graph Normalization Algorithm 2011"
+(<abbr title="Universal RDF Graph Normalization Algorithm 2011">URGNA2011</abbr>).
+</p>
+
+<section>
+<h3>Normalization Algorithm Terms</h3>
+ <dl>
+ <dt><tdef>input graph</tdef></dt>
+ <dd>
+ The data structure that is provided as input to the algorithm.
+ </dd>
+ <dt><tdef>output graph</tdef></dt>
+ <dd>
+ The data structure that is produced as output by the algorithm.
+ </dd>
+ <dt><tdef>label</tdef></dt>
+ <dd>
+ The subject IRI associated with a graph node.
+ </dd>
+ <dt><tdef>list of expanded nodes</tdef></dt>
+ <dd>
+ A list of all nodes in the <tref>input graph</tref>.
+ </dd>
+ <dt><tdef>alpha</tdef> and <tdef>beta</tdef> values</dt>
+ <dd>
+ The words <tref>alpha</tref> and <tref>beta</tref> refer to the first and
+ second nodes or values being examined in an algorithm. The names are
+ merely used to refer to each input value to a comparison algorithm.
+ </dd>
+ <dt><tdef>renaming counter</tdef></dt>
+ <dd>
+ A counter that is used during the
+ <a href="#node-relabeling-algorithm">Node Relabeling Algorithm</a>. The
+ counter typically starts at one (1) and counts up for every node that is
+ relabeled. There will be two such renaming counters in an implementation
+ of the normalization algorithm. The first is the
+ <tref>labeling counter</tref> and the second is the
+ <tref>deterministic labeling counter</tref>.
+ </dd>
+ <dt><tdef>serialization label</tdef></dt>
+ <dd>
+ An identifier that is created to aid in the normalization process in the
+ <a href="#deep-comparison-algorithm">Deep Comparison Algorithm</a>. The
+ value typically takes the form of <code>s<NUMBER></code> or
+ <code>c<NUMBER></code>.
+ </dd>
+</dl>
+</section>
+
+<section>
+<h3>Normalization State</h3>
+
+<p>When performing the steps required by the normalization algorithm,
+it is helpful to track the many pieces of information in a
+data structure called the <tdef>normalization state</tdef>. Many of these
+pieces simply provide indexes into the graph. The information
+contained in the <tref>normalization state</tref> is described below.</p>
+
+<dl>
+ <dt><tdef>node state</tdef></dt>
+ <dd>
+ Each node in the graph will be assigned a <tref>node state</tref>. This
+ state contains the information necessary to deterministically
+ <tref>label</tref> all nodes in the graph. A <tref>node state</tref>
+ includes:
+ <dl>
+ <dt><tdef>node reference</tdef></dt>
+ <dd>
+ A <tref>node reference</tref> is a reference to a node in the graph.
+ For a given <tref>node state</tref>, its <tref>node reference</tref>
+ refers to the node that the state is for. When a
+ <tref>node state</tref> is created, its <tref>node reference</tref>
+ should be to the node it is created for.
+ </dd>
+ <dt><tdef>outgoing list</tdef></dt>
+ <dd>
+ Lists the <tref>label</tref>s for all nodes that are properties of
+ the <tref>node reference</tref>. This list should be initialized
+ by iterating over every object associated with a property in the
+ <tref>node reference</tref> adding its label if it is another node.
+ </dd>
+ <dt><tdef>incoming list</tdef></dt>
+ <dd>
+ Lists the <tref>label</tref>s for all nodes in the graph for which
+ the <tref>node reference</tref> is a property. This list is
+ initialized to an empty list.
+ </dd>
+ <dt><tdef>outgoing serialization map</tdef></dt>
+ <dd>
+ Maps node <tref>label</tref>s to <tref>serialization label</tref>s.
+ This map is initialized to an empty map. When this map is populated,
+ it will be filled with keys that are the <tref>label</tref>s of every node in the
+ graph with a label that begins with <code>_:</code> and that has a
+ path, via properties, that starts with the
+ <tref>node reference</tref>.
+ </dd>
+ <dt><tdef>outgoing serialization</tdef></dt>
+ <dd>
+ A string that can be lexicographically compared to the
+ <tref>outgoing serialization</tref>s of other
+ <tref>node state</tref>s. It is a representation of the
+ <tref>outgoing serialization map</tref> and other related
+ information. This string is initialized to an empty string.
+ </dd>
+ <dt><tdef>incoming serialization map</tdef></dt>
+ <dd>
+ Maps node <tref>label</tref>s to <tref>serialization label</tref>s.
+ This map is initialized to an empty map. When this map is populated,
+ it will be filled with keys that are the <tref>label</tref>s of every
+ node in the graph with a <tref>label</tref> that begins with
+ <code>_:</code> and that has a path, via properties, that ends with
+ the <tref>node reference</tref>.
+ </dd>
+ <dt><tdef>incoming serialization</tdef></dt>
+ <dd>
+ A string that can be lexicographically compared to the
+ <tref>outgoing serialization</tref>s of other
+ <tref>node state</tref>s. It is a representation of the
+ <tref>incoming serialization map</tref> and other related
+ information. This string is initialized to an empty string.
+ </dd>
+ </dl>
+ </dd>
+ <dt><tdef>node state map</tdef></dt>
+ <dd>
+ A mapping from a node's <tref>label</tref> to a <tref>node state</tref>.
+ It is initialized to an empty map.
+ </dd>
+ <dt><tdef>labeling prefix</tdef></dt>
+ <dd>
+ The labeling prefix is a string that is used as the beginning of a node
+ <tref>label</tref>. It should be initialized to a random base string that
+ starts with the characters <code>_:</code>, is not used by any other
+ node's <tref>label</tref> in the <tref>input graph</tref>, and does not
+ start with the characters <code>_:c14n</code>. The prefix has two uses.
+ First it is used to temporarily name nodes during the normalization
+ algorithm in a way that doesn't collide with the names that already
+ exist as well as the names that will be generated by the normalization
+ algorithm. Second, it will eventually be set to <code>_:c14n</code> to
+ generate the final, deterministic labels for nodes in the graph. This
+ prefix will be concatenated with the <tref>labeling counter</tref> to
+ produce a node <tref>label</tref>. For example, <code>_:j8r3k</code> is
+ a proper initial value for the <tref>labeling prefix</tref>.
+ </dd>
+ <dt><tdef>labeling counter</tdef></dt>
+ <dd>
+ A counter that is used to label nodes. It is appended to the
+ <tref>labeling prefix</tref> to create a node <tref>label</tref>. It is
+ initialized to <code>1</code>.
+ </dd>
+ <dt><tdef>map of flattened nodes</tdef></dt>
+ <dd>
+ A map containing a representation of all nodes in the graph where the
+ key is a node <tref>label</tref> and the value is a single
+ <tref>JSON object</tref> that has no nested sub-objects
+ and has had all properties for the same node merged into a single
+ <tref>JSON object</tref>.
+ </dd>
+</dl>
+
+</section>
+
+<section>
+<h3>Normalization Algorithm</h3>
+
+<p>The normalization algorithm expands the <tref>input graph</tref>,
+flattens the data structure, and creates an initial set of names for all
+nodes in the graph. The flattened data structure is then processed by a
+node labeling algorithm in order to get a fully expanded and named list of
+nodes which is then sorted. The result is a deterministically named and
+ordered list of graph nodes.
+</p>
+
+<ol class="algorithm">
+<li>Expand the <tref>input graph</tref> according to the steps in
+the <a href="#expansion-algorithm">Expansion Algorithm</a> and store the
+result as the <strong>expanded input</strong>.</li>
+<li>Create a <tref>normalization state</tref>.</li>
+<li>Initialize the <tref>map of flattened nodes</tref> by recursively
+processing every <tdef>expanded node</tdef> in the
+<strong>expanded input</strong> in depth-first order:
+ <ol class="algorithm">
+ <li>If the <tref>expanded node</tref> is an unlabeled node, add a
+ new key-value pair to the <tref>expanded node</tref>
+ where the key is <code>@subject</code> and the value is the
+ concatenation of the <tref>labeling prefix</tref>
+ and the string value of the <tref>labeling counter</tref>.
+ Increment the <tref>labeling counter</tref>.</li>
+ <li>Add the <tref>expanded node</tref> to the
+ <tref>map of flattened nodes</tref>:
+ <ol class="algorithm">
+ <li>If the <tref>expanded node</tref>'s <tref>label</tref> is already
+ in the
+ <tref>map of flattened nodes</tref> merge all properties from the
+ entry in the <tref>map of flattened nodes</tref> into the
+ <tref>expanded node</tref>.</li>
+ <li>Go through every property associated with an array in the
+ <tref>expanded node</tref> and remove any duplicate IRI entries from
+ the array. If the resulting array only has one IRI entry, change it
+ from an array to an object.</li>
+ <li>Set the entry for the <tref>expanded node</tref>'s <tref>label</tref>
+ in the <tref>map of flattened nodes</tref> to the
+ <tref>expanded node</tref>.
+ </li></ol></li>
+ <li>After exiting the recursive step, replace the reference to the
+ <tref>expanded node</tref> with an object containing a single
+ key-value pair where the key is <code>@iri</code> and the value is
+ the value of the <code>@subject</code> key in the node.</li>
+ </ol></li>
+<li>For every entry in the <tref>map of flattened nodes</tref>, insert a
+ key-value pair into the <tref>node state map</tref> where the key is the
+ key from the <tref>map of flattened nodes</tref> and the value is a
+ <tref>node state</tref> where its <tref>node reference</tref> refers to
+ the value from the <tref>map of flattened nodes</tref>.
+<li>Populate the <tref>incoming list</tref> for each <tref>node state</tref>
+ by iterating over every node in the graph and adding its <tref>label</tref>
+ to the <tref>incoming list</tref> associated with each node found in its
+ properties.</li>
+<li>For every entry in the <tref>node state map</tref> that has a
+<tref>label</tref> that begins with <code>_:c14n</code>, relabel the node
+using the <a href="#node-relabeling-algorithm">Node Relabeling Algorithm</a>.
+<li>Label all of the nodes that contain a <code>@subject</code> key associated
+with a value starting with <code>_:</code> according to the steps in the
+<a href="#deterministic-labeling-algorithm">Deterministic Labeling Algorithm</a>.
+</li>
+</ol>
+</section>
+
+<section>
+<h4>Node Relabeling Algorithm</h4>
+
+<p>This algorithm renames a node by generating a unique
+<tdef>new label</tdef> and updating all references to that <tref>label</tref>
+in the <tref>node state map</tref>. The <tdef>old label</tdef> and the
+<tref>normalization state</tref> must be given as an input to the
+algorithm. The <tref>old label</tref> is the current <tref>label</tref> of
+the node that is to be relabeled.
+
+<p>The node relabeling algorithm is as follows:</p>
+
+<ol class="algorithm">
+ <li>If the <tref>labeling prefix</tref> is <code>_:c14n</code> and the
+ <tref>old label</tref> begins with <code>_:c14n</code> then return as
+ the node has already been renamed.
+ </li>
+ <li>Generate the <tdef>new label</tdef> by concatenating the
+ <tref>labeling prefix</tref> with the string value of the
+ <tref>labeling counter</tref>. Increment the <tref>labeling counter</tref>.
+ </li>
+ <li>For the <tref>node state</tref> associated with the
+ <tref>old label</tref>, update every node in the <tref>incoming list</tref>
+ by changing all the properties that reference the <tref>old label</tref> to
+ the <tref>new label</tref>.
+ </li>
+ <li>Change the <tref>old label</tref> key in the <tref>node state map</tref>
+ to the <tref>new label</tref> and set the associated
+ <tref>node reference</tref>'s <tref>label</tref> to the
+ <tref>new label</tref>.
+ </li>
+</ol>
+</section>
+
+<section>
+<h4>Deterministic Labeling Algorithm</h4>
+
+<p>The deterministic labeling algorithm takes the
+<tref>normalization state</tref>
+and produces a <tdef>list of finished nodes</tdef> that is sorted and
+contains deterministically named and expanded nodes from the graph.
+
+<ol class="algorithm">
+ <li>Set the <tref>labeling prefix</tref> to <code>_:c14n</code>, the
+ <tref>labeling counter</tref> to <code>1</code>,
+ the <tdef>list of finished nodes</tdef> to an empty array, and create
+ an empty array, the <tdef>list of unfinished nodes</tdef>.</li>
+ <li>For each <tref>node reference</tref> in the <tref>node state map</tref>:
+ <ol class="algorithm">
+ <li>If the node's <tref>label</tref> does not start with <code>_:</code>
+ then put the <tref>node reference</tref> in the
+ <tref>list of finished nodes</tref>.
+ </li>
+ <li>If the node's <tref>label</tref> does start with <code>_:</code>
+ then put the <tref>node reference</tref> in the
+ <tref>list of unfinished nodes</tref>.
+ </li>
+ </ol>
+ </li>
+ <li>Append to the <tref>list of finished nodes</tref> by processing
+ the remainder of the <tref>list of unfinished nodes</tref> until it is
+ empty:
+ <ol class="algorithm">
+ <li>Sort the <tref>list of unfinished nodes</tref> in descending order
+ according to the
+ <a href="#deep-comparison-algorithm">Deep Comparison Algorithm</a> to
+ determine the sort order.</li>
+ <li>Create a <tdef>list of labels</tdef> and initialize it to an
+ empty array.</li>
+ <li>For the first node from the <tref>list of unfinished nodes</tref>:
+ <ol class="algorithm">
+ <li>Add its <tref>label</tref> to the <tref>list of labels</tref>.
+ </li>
+ <li>For each key-value pair from its associated
+ <tref>outgoing serialization map</tref>, add the key to a list and
+ then sort the list according to the lexicographical order of the
+ keys' associated values. Append the list to the
+ <tref>list of nodes to label</tref>.
+ </li>
+ <li>For each key-value pair from its associated
+ <tref>incoming serialization map</tref>, add the key to a list and
+ then sort the list according to the lexicographical order of the
+ keys' associated values. Append the list to the
+ <tref>list of nodes to label</tref>.
+ </li></ol></li>
+ <li>For each <tref>label</tref> in the <tref>list of labels</tref>,
+ relabel the associated node according to the
+ <a href="#node-relabeling-algorithm">Node Relabeling Algorithm</a>. If
+ any <tref>outgoing serialization map</tref> contains a key that
+ matches the <tref>label</tref>, clear the map and set the associated
+ <tref>outgoing serialization</tref> to an empty string. If any
+ <tref>incoming serialization map</tref> contains a key that
+ matches the <tref>label</tref>, clear the map and set the associated
+ <tref>incoming serialization</tref> to an empty string.
+ </li>
+ <li>
+ Remove each node with a <tref>label</tref> that starts with
+ <code>_:c14n</code> from the <tref>list of unfinished nodes</tref> and
+ add it to the <tref>list of finished nodes</tref>.
+ </li>
+ </ol>
+ </li>
+ <li>Sort the <tref>list of finished nodes</tref> in descending order
+ according to the
+ <a href="#deep-comparison-algorithm">Deep Comparison Algorithm</a> to
+ determine the sort order.</li>
+</ol>
+</section>
+
+<section>
+<h4>Shallow Comparison Algorithm</h4>
+
+<p>
+The shallow comparison algorithm takes two unlabeled nodes,
+<tref>alpha</tref> and <tref>beta</tref>, as input and
+determines which one should come first in a sorted list. The following
+algorithm determines the steps that are executed in order to determine the
+node that should come first in a list:
+</p>
+
+<ol class="algorithm">
+ <li>Compare the total number of node properties. The node with fewer
+ properties is first.</li>
+ <li>Lexicographically sort the property IRIs for each node and compare
+ the sorted lists. If an IRI is found to be lexicographically smaller, the
+ node containing that IRI is first.</li>
+ <li>Compare the values of each property against one another:
+ <ol class="algorithm">
+ <li>The node associated with fewer property values is first.
+ </li>
+ <li>Create an <tdef>alpha list</tdef> by adding all values associated
+ with the <tref>alpha</tref> property that are not unlabeled nodes.
+ </li>
+ <li>Create a <tdef>beta list</tdef> by adding all values associated
+ with the <tref>beta</tref> property that is not an unlabeled node.
+ </li>
+ <li>Compare the length of <tref>alpha list</tref> and
+ <tref>beta list</tref>. The node associated with the list containing
+ the fewer number of items is first.</li>
+ <li>Sort <tref>alpha list</tref> and <tref>beta list</tref> according to
+ the
+ <a href="#object-comparison-algorithm">Object Comparison Algorithm</a>.
+ For each offset into the <tref>alpha list</tref>, compare the item
+ at the offset against the item at the same offset in the
+ <tref>beta list</tref> according to the
+ <a href="#object-comparison-algorithm">Object Comparison Algorithm</a>.
+ The node associated with the lesser item is first.
+ </ol></li>
+ <li>Process the <tref>incoming list</tref>s associated with each node to
+ determine order:
+ <ol class="algorithm">
+ <li>The node with the shortest <tref>incoming list</tref> is first.</li>
+ <li>Sort the <tref>incoming list</tref>s according to incoming property
+ and then incoming <tref>label</tref>.
+ <li>The node associated with the fewest number of incoming nodes is
+ first.</li>
+ <li>For each offset into the <tref>incoming list</tref>s,
+ compare the associated properties and <tref>label</tref>s:
+ <ol class="algorithm">
+ <li>The node associated with a <tref>label</tref> that does not begin with
+ <code>_:</code> is first.
+ </li>
+ <li>If the nodes' <tref>label</tref>s do not begin with
+ <code>_:</code>, then the node associated with the
+ lexicographically lesser <tref>label</tref> is first.</li>
+ </li>
+ <li>The node associated with the lexicographically lesser associated
+ property is first.
+ </li>
+ <li>The node with the <tref>label</tref> that does not begin with
+ <code>_:c14n</code> is first.
+ </li>
+ <li>The node with the lexicographically lesser <tref>label</tref>
+ is first.
+ </li>
+ </ol>
+ </ol></li>
+ <li>Otherwise, the nodes are equivalent.</li>
+</section>
+
+<section>
+<h4>Object Comparison Algorithm</h4>
+
+<p>
+The object comparison algorithm is designed to compare two graph node
+property values, <tref>alpha</tref> and <tref>beta</tref>, against the other.
+The algorithm is useful when sorting two lists of graph node properties.
+</p>
+
+<ol class="algorithm">
+ <li>If one of the values is a <tref>string</tref> and the other is not, the value that is
+ a string is first.
+ </li>
+ <li>If both values are <tref>string</tref>s, the lexicographically lesser string is
+ first.
+ </li>
+ <li>If one of the values is a literal and the other is not, the value that is
+ a literal is first.
+ </li>
+ <li>If both values are literals:
+ <ol class="algorithm">
+ <li>The lexicographically lesser string associated with
+ <code>@literal</code> is first.
+ </li>
+ <li>The lexicographically lesser string associated with
+ <code>@datatype</code> is first.
+ </li>
+ <li>The lexicographically lesser string associated with
+ <code>@language</code> is first.
+ </li>
+ </ol>
+ </li>
+ <li>If both values are expanded IRIs, the
+ lexicographically lesser string associated with <code>@iri</code>
+ is first.</li>
+ <li>Otherwise, the two values are equivalent.</li>
+</ol>
+
+</section>
+
+<section>
+<h4>Deep Comparison Algorithm</h4>
+
+<p>
+The deep comparison algorithm is used to compare the difference between two
+nodes, <tref>alpha</tref> and <tref>beta</tref>.
+A deep comparison takes the incoming and outgoing node edges in
+a graph into account if the number of properties and value of those properties
+are identical. The algorithm is helpful when sorting a list of nodes and will
+return whichever node should be placed first in a list if the two nodes are
+not truly equivalent.
+</p>
+
+<p>When performing the steps required by the deep comparison algorithm, it
+is helpful to track state information about mappings. The information
+contained in a <tref>mapping state</tref> is described below.</p>
+
+<dl class="algorithm">
+ <dt><tdef>mapping state</tdef></dt>
+ <dd>
+ <dl>
+ <dt><tdef>mapping counter</tdef></dt>
+ <dd>
+ Keeps track of the number of nodes that have been mapped to
+ <tref>serialization labels</tref>. It is initialized to
+ <code>1</code>.
+ </dd>
+ <dt><tdef>processed labels map</tdef></dt>
+ <dd>
+ Keeps track of the <tref>label</tref>s of nodes that have already
+ been assigned <tref>serialization label</tref>s. It is initialized
+ to an empty map.
+ </dd>
+ <dt><tdef>serialized labels map</tdef></dt>
+ <dd>
+ Maps a node <tref>label</tref> to its associated
+ <tref>serialization label</tref>. It is initialized to an empty map.
+ </dd>
+ <dt><tdef>adjacent info map</tdef></dt>
+ <dd>
+ Maps a <tref>serialization label</tref> to the node
+ <tref>label</tref> associated with it, the list of sorted
+ <tref>serialization label</tref>s for adjacent nodes, and the map of
+ adjacent node <tref>serialiation label</tref>s to their associated
+ node <tref>label</tref>s. It is initialized to an empty map.
+ </dd>
+ <dt><tdef>key stack</tdef></dt>
+ <dd>
+ A stack where each element contains an array of adjacent
+ <tref>serialization label</tref>s and an index into that array. It
+ is initialized to a stack containing a single element where its
+ array contains a single string element <code>s1</code> and its
+ index is set to <code>0</code>.
+ </dd>
+ <dt><tdef>serialized keys</tdef></dt>
+ <dd>
+ Keeps track of which <tref>serialization label</tref>s have already
+ been written at least once to the <tref>serialization string</tref>.
+ It is initialized to an empty map.
+ </dd>
+ <dt><tdef>serialization string</tdef></dt>
+ <dd>
+ A string that is incrementally updated as a serialization is built.
+ It is initialized to an empty string.
+ </dd>
+ </dl>
+ </dd>
+</dl>
+
+<p>The deep comparison algorithm is as follows:</p>
+
+<ol class="algorithm">
+ <li>Perform a comparison between <tref>alpha</tref> and <tref>beta</tref>
+ according to the
+ <a href="#shallow-comparison-algorithm">Shallow Comparison Algorithm</a>.
+ If the result does not show that the two nodes are equivalent, return
+ the result.
+ </li>
+ <li>Compare incoming and outgoing edges for each node, updating their
+ associated <tref>node state</tref> as each node is processed:
+ <ol class="algorithm">
+ <li>If the <tref>outgoing serialization map</tref> for <tref>alpha</tref>
+ is empty, generate the serialization according to the
+ <a href="#node-serialization-algorithm">Node Serialization Algorithm</a>.
+ Provide <tref>alpha</tref>'s <tref>node state</tref>, a new
+ <tref>mapping state</tref>,
+ <code>outgoing direction</code> to the algorithm as inputs.
+ <li>If the <tref>outgoing serialization map</tref> for <tref>beta</tref>
+ is empty, generate the serialization according to the
+ <a href="#node-serialization-algorithm">Node Serialization Algorithm</a>.
+ Provide <tref>beta</tref>'s <tref>node state</tref>, a new
+ <tref>mapping state</tref>, and
+ <code>outgoing direction</code> to the algorithm as inputs.
+ <li>If <tref>alpha</tref>'s <tref>outgoing serialization</tref> is
+ lexicographically less than <tref>beta</tref>'s, then
+ <tref>alpha</tref> is first. If it is greater, then <tref>beta</tref>
+ is first.</li>
+ <li>If the <tref>incoming serialization map</tref> for <tref>alpha</tref>
+ is empty, generate the serialization according to the
+ <a href="#node-serialization-algorithm">Node Serialization Algorithm</a>.
+ Provide <tref>alpha</tref>'s <tref>node state</tref>, a new
+ <tref>mapping state</tref> with its <tref>serialized labels map</tref>
+ set to a copy of <tref>alpha</tref>'s
+ <tref>outgoing serialization map</tref>, and
+ <code>incoming direction</code> to the algorithm as inputs.
+ <li>If the <tref>incoming serialization map</tref> for <tref>beta</tref>
+ is empty, generate the serialization according to the
+ <a href="#node-serialization-algorithm">Node Serialization Algorithm</a>.
+ Provide <tref>beta</tref>'s <tref>node state</tref>, a new
+ <tref>mapping state</tref> with its <tref>serialized labels map</tref>
+ set to a copy of <tref>beta</tref>'s
+ <tref>outgoing serialization map</tref>, and
+ <code>incoming direction</code> to the algorithm as inputs.
+ <li>If <tref>alpha</tref>'s <tref>incoming serialization</tref> is
+ lexicographically less than <tref>beta</tref>'s, then
+ <tref>alpha</tref> is first. If it is greater, then <tref>beta</tref>
+ is first.</li>
+ </ol></li>
+</ol>
+</section>
+
+<section>
+<h4>Node Serialization Algorithm</h4>
+
+<p>
+The node serialization algorithm takes a <tref>node state</tref>, a
+<tref>mapping state</tref>, and a <tdef>direction</tdef> (either
+<code>outgoing direction</code> or <code>incoming direction</code>) as
+inputs and generates a deterministic serialization for the
+<tref>node reference</tref>.
+</p>
+
+<ol class="algorithm">
+<li>If the <tref>label</tref> exists in the
+ <tref>processed labels map</tref>, terminate the algorithm as the
+ <tref>serialization label</tref> has already been created.
+</li>
+<li>Set the value associated with the <tref>label</tref> in the
+ <tref>processed labels map</tref> to <code>true</code>.
+</li>
+<li>Generate the next <tdef>serialization label</tdef> for the
+ <tref>label</tref> according to the
+ <a href="#serialization-label-generation-algorithm">Serialization Label Generation Algorithm</a>.
+</li>
+<li>Create an empty map called the <tdef>adjacent serialized labels map</tdef>
+that will store mappings from <tref>serialized label</tref>s to adjacent
+node <tref>label</tref>s.</li>
+<li>Create an empty array called the
+<tdef>adjacent unserialized labels list</tdef> that will store
+<tref>label</tref>s of adjacent nodes that haven't been assigned
+<tref>serialization label</tref>s yet.
+</li>
+<li>For every <tref>label</tref> in a list, where the list the <tref>outgoing list</tref> if
+the <tref>direction</tref> is <code>outgoing direction</code> and the
+<tref>incoming list</tref> otherwise, if the <tref>label</tref> starts with
+<code>_:</code>, it is the <tdef>target node label</tdef>:
+ <ol class="algorithm">
+ <li>Look up the <tref>target node label</tref> in the
+ <tref>processed labels map</tref> and if a mapping exists,
+ update the <tref>adjacent serialized labels map</tref> where the key is
+ the value in the <tref>serialization map</tref> and the value is the
+ <tref>target node label</tref>.</li>
+ <li>Otherwise, add the <tref>target node label</tref> to the
+ <tref>adjacent unserialized labels list</tref>.
+ </ol>
+</li>
+<li>Set the <tdef>maximum serialization combinations</tdef> to
+ <code>1</code> or the length of the
+ <tref>adjacent unserialized labels list</tref>, whichever is greater.</li>
+<li>While the <tref>maximum serialization combinations</tref> is greater than
+ <code>0</code>, perform the
+ <a href="#combinatorial-serialization-algorithm">Combinatorial Serialization Algorithm</a>
+ passing the <tref>node state</tref>, the <tref>mapping state</tref> for the
+ first iteration and a copy of it for each subsequent iteration, the
+ generated <tref>serialization label</tref>, the <tref>direction</tref>,
+ the <tref>adjacent serialized labels map</tref>, and the
+ <tref>adjacent unserialized labels list</tref>.
+ Decrement the <tref>maximum serialization combinations</tref> by
+ <code>1</code> for each iteration.
+</ol>
+
+</section>
+
+<section>
+<h4>Serialization Label Generation Algorithm</h4>
+
+<p>
+The algorithm generates a <tref>serialization label</tref> given a
+<tref>label</tref> and a <tref>mapping state</tref> and returns the
+<tref>serialization label</tref>.
+</p>
+
+ <ol class="algorithm">
+ <li>If the <tref>label</tref> is already in the
+ <tref>serialization labels map</tref>, return its associated value.
+ </li>
+ <li>If the <tref>label</tref> starts with the string <code>_:c14n</code>,
+ the <tref>serialization label</tref> is the letter <code>c</code>
+ followed by the number that follows <code>_:c14n</code> in the
+ <tref>label</tref>.
+ </li>
+ <li>Otherwise, the <tref>serialization label</tref> is the
+ letter <code>s</code> followed by the string value of
+ <tref>mapping count</tref>. Increment the <tref>mapping count</tref> by
+ <code>1</code>.
+ </li>
+ <li>Create a new key-value pair in the <tref>serialization labels map</tref>
+ where the key is the <tref>label</tref> and the value is the
+ generated <tref>serialization label</tref>.
+ </li>
+ </ol>
+</section>
+
+<section>
+<h4>Combinatorial Serialization Algorithm</h4>
+
+<p>
+The combinatorial serialization algorithm takes a <tref>node state</tref>, a
+<tref>mapping state</tref>, a <tref>serialization label</tref>, a
+<tref>direction</tref>, a <tref>adjacent serialized labels map</tref>,
+and a <tref>adjacent unserialized labels list</tref> as inputs and generates
+the lexicographically least serialization of nodes relating to the
+<tref>node reference</tref>.
+</p>
+
+<ol class="algorithm">
+ <li>If the <tref>adjacent unserialized labels list</tref> is not empty:
+ <ol class="algorithm">
+ <li>Copy the <tref>adjacent serialized labels map</tref> to the
+ <tdef>adjacent serialized labels map copy</tdef>.</li>
+ <li>Remove the first <tref>unserialized label</tref> from the
+ <tref>adjacent unserialized labels list</tref> and create a new
+ <tdef>new serialization label</tdef> according to the
+ <a href="#serialization-label-generation-algorithm">Serialization Label Generation Algorithm</a>.
+ <li>Create a new key-value mapping in the
+ <tref>adjacent serialized labels map copy</tref>
+ where the key is the <tref>new serialization label</tref> and the value
+ is the <tref>unserialized label</tref>.
+ <li>Set the <tdef>maximum serialization rotations</tdef> to
+ <code>1</code> or the length of the
+ <tref>adjacent unserialized labels list</tref>, whichever is greater.
+ </li>
+ <li>While the <tref>maximum serialization rotations</tref> is greater than
+ <code>0</code>:
+ <ol class="algorithm">
+ <li>Recursively perform the
+ <a href="#combinatorial-serialization-algorithm">Combinatorial Serialization Algorithm</a>
+ passing the <tref>mapping state</tref> for the first iteration of the
+ loop, and a copy of it for each subsequent iteration.
+ </li>
+ <li>Rotate the elements in the
+ <tref>adjacent unserialized labels list</tref> by shifting each of
+ them once to the right, moving the element at the end of the list
+ to the beginning of the list.
+ </li>
+ <li>Decrement the <tref>maximum serialization rotations</tref> by
+ <code>1</code> for each iteration.
+ </li>
+ </ol>
+ </li>
+ </ol>
+ </li>
+ <li>If the <tref>adjacent unserialized labels list</tref> is empty:
+ <ol class="algorithm">
+ <li>Create a <tdef>list of keys</tdef> from the keys in the
+ <tref>adjacent serialized labels map</tref> and sort it
+ lexicographically.
+ </li>
+ <li>Add a key-value pair to the <tref>adjacent info map</tref> where
+ the key is the <tref>serialization label</tref> and the value is
+ an object containing the <tref>node reference</tref>'s label, the
+ <tref>list of keys</tref> and the
+ <tref>adjacent serialized labels map</tref>.
+ </li>
+ <li>Update the <tref>serialization string</tref> according to the
+ <a href="#mapping-serialization-algorithm">Mapping Serialization Algorithm</a>.
+ </li>
+ <li>If the <tref>direction</tref> is <code>outgoing direction</code>
+ then <tdef>directed serialization</tdef> refers to the
+ <tref>outgoing serialization</tref> and the
+ <tdef>directed serialization map</tdef> refers to the
+ <tref>outgoing serialization map</tref>, otherwise it refers to the
+ <tref>incoming serialization</tref> and the
+ <tref>directed serialization map</tref> refers to the
+ <tref>incoming serialization map</tref>. Compare the
+ <tref>serialization string</tref> to the
+ <tref>directed serialization</tref> according to the
+ <a href="#mapping-serialization-algorithm">Serialization Comparison Algorithm</a>.
+ If the <tref>serialization string</tref> is less than or equal to
+ the <tref>directed serialization</tref>:
+ <ol class="algorithm">
+ <li>For each value in the <tref>list of keys</tref>, run the
+ <a href="#node-serialization-algorithm">Node Serialization Algorithm</a>.
+ </li>
+ <li>Update the <tref>serialization string</tref> according to the
+ <a href="#mapping-serialization-algorithm">Mapping Serialization Algorithm</a>.
+ </li>
+ <li>Compare the <tref>serialization string</tref> to the
+ <tref>directed serialization</tref> again and if it is less than
+ or equal and the length of the <tref>serialization string</tref> is
+ greater than or equal to the length of the
+ <tref>directed serialization</tref>, then set the
+ <tref>directed serialization</tref> to the
+ <tref>serialization string</tref> and set the
+ <tref>directed serialization map</tref> to the
+ <tref>serialized labels map</tref>.
+ </li>
+ </ol>
+ </li>
+ </ol>
+ </li>
+</ol>
+
+</section>
+
+<section>
+<h4>Serialization Comparison Algorithm</h4>
+
+<p>
+The serialization comparison algorithm takes two serializations,
+<tref>alpha</tref> and <tref>beta</tref> and returns either which of the two
+is less than the other or that they are equal.
+</p>
+
+<ol class="algorithm">
+ <li>Whichever serialization is an empty string is greater. If they are
+ both empty strings, they are equal.</li>
+ <li>Return the result of a lexicographical comparison of <tref>alpha</tref>
+ and <tref>beta</tref> up to the number of characters in the shortest of
+ the two serializations.
+ </li>
+</ol>
+</section>
+
+<section>
+<h4>Mapping Serialization Algorithm</h4>
+
+<p>
+The mapping serialization algorithm incrementally updates the
+<tref>serialization string</tref> in a <tref>mapping state</tref>.
+</p>
+
+<ol class="algorithm">
+ <li>If the <tref>key stack</tref> is not empty:
+ <ol class="algorithm">
+ <li>Pop the <tdef>serialization key info</tdef> off of the
+ <tref>key stack</tref>.
+ </li>
+ <li>For each <tdef>serialization key</tdef> in the
+ <tref>serialization key info</tref> array, starting at
+ the <tdef>serialization key index</tdef> from the
+ <tref>serialization key info</tref>:
+ <ol class="algorithm">
+ <li>If the <tref>serialization key</tref> is not in the
+ <tref>adjacent info map</tref>, push the
+ <tref>serialization key info</tref> onto the
+ <tref>key stack</tref> and exit from this loop.
+ </li>
+ <li>If the <tref>serialization key</tref> is a key in
+ <tref>serialized keys</tref>, a cycle has been detected. Append
+ the concatenation of the <code>_</code> character and the
+ <tref>serialization key</tref> to the
+ <tref>serialization string</tref>.
+ <li>Otherwise, serialize all outgoing and incoming edges in the
+ related node by performing the following steps:
+ <ol class="algorithm">
+ <li>Mark the <tref>serialization key</tref> as having
+ been processed by adding a new key-value pair to
+ <tref>serialized keys</tref> where the key
+ is the <tref>serialization key</tref> and the value is
+ <code>true</code>.
+ </li>
+ <li>Set the <tdef>serialization fragment</tdef> to the value of
+ the <tref>serialization key</tref>.</li>
+ <li>Set the <tref>adjacent info</tref> to the value of the
+ <tref>serialization key</tref> in the
+ <tref>adjacent info map</tref>.
+ </li>
+ <li>Set the <tref>adjacent node label</tref> to the node
+ <tref>label</tref> from the <tref>adjacent info</tref>.
+ </li>
+ <li>If a mapping for the <tref>adjacent node label</tref>
+ exists in the <tref>map of all labels</tref>:
+ <ol class="algorithm">
+ <li>Append the result of the
+ <a href="">Label Serialization Algorithm</a> to the
+ <tref>serialization fragment</tref>.
+ </li>
+ </ol>
+ </li>
+ <li>Append all of the keys in the <tref>adjacent info</tref>
+ to the <tref>serialization fragment</tref>.
+ </li>
+ <li>Append the <tref>serialization fragment</tref> to the
+ <tref>serialization string</tref>.
+ </li>
+ <li>Push a new key info object containing the keys from the
+ <tref>adjacent info</tref> and an index of <code>0</code>
+ onto the <tref>key stack</tref>.
+ </li>
+ <li>Recursively update the <tref>serialization string</tref>
+ according to the
+ <a href="#mapping-serialization-algorithm">Mapping Serialization Algorithm</a>.
+ </li>
+ </ol>
+ </li>
+ </ol>
+ </li>
+ </ol>
+ </li>
+</ol>
+
+</section>
+
+<section>
+<h4>Label Serialization Algorithm</h4>
+
+<p>
+The label serialization algorithm serializes information about a node that
+has been assigned a particular <tref>serialization label</tref>.
+</p>
+
+<ol class="algorithm">
+ <li>Initialize the <tref>label serialization</tref> to an empty string.</li>
+ <li>Append the <code>[</code> character to the
+ <tref>label serialization</tref>.</li>
+ <li>Append all properties to the <tref>label serialization</tref> by
+ processing each key-value pair in the <tref>node reference</tref>,
+ excluding the
+ <code>@subject</code> property. The keys should be processed in
+ lexicographical order and their associated values should be processed
+ in the order produced by the
+ <a href="#object-comparison-algorithm">Object Comparison Algorithm</a>:
+ <ol class="algorithm">
+ <li>Build a string using the pattern <code><</code><strong>KEY</strong><code>></code>
+ where <strong>KEY</strong> is the current key. Append string to the
+ <tref>label serialization</tref>.</li>
+ <li>The value may be a single object or an array of objects.
+ Process all of the objects that are associated with the key, building
+ an <tdef>object string</tdef> for each item:
+ <ol class="algorithm">
+ <li>If the object contains an <code>@iri</code> key with a
+ value that starts
+ with <code>_:</code>, set the <tref>object string</tref> to
+ the value <code>_:</code>. If the value does not
+ start with <code>_:</code>, build the <tref>object string</tref>
+ using the pattern
+ <code><</code><strong>IRI</strong><code>></code>
+ where <strong>IRI</strong> is the value associated with the
+ <code>@iri</code> key.</li>
+ <li>If the object contains a <code>@literal</code> key and a
+ <code>@datatype</code> key, build the <tref>object string</tref>
+ using the pattern
+ <code>"</code><strong>LITERAL</strong><code>"^^<</code><strong>DATATYPE</strong><code>></code>
+ where <strong>LITERAL</strong> is the value associated with the
+ <code>@literal</code> key and <strong>DATATYPE</strong> is the
+ value associated with the <code>@datatype</code> key.</li>
+ <li>If the object contains a <code>@literal</code> key and a
+ <code>@language</code> key, build the <tref>object string</tref>
+ using the pattern
+ <code>"</code><strong>LITERAL</strong><code>"@</code><strong>LANGUAGE</strong>
+ where <strong>LITERAL</strong> is the value associated with the
+ <code>@literal</code> key and <strong>LANGUAGE</strong> is the
+ value associated with the <code>@language</code> key.</li>
+ <li>Otherwise, the value is a string. Build the
+ <tref>object string</tref> using the pattern
+ <code>"</code><strong>LITERAL</strong><code>"</code>
+ where <strong>LITERAL</strong> is the value associated with the
+ current key.</li>
+ <li>If this is the second iteration of the loop,
+ append a <code>|</code> separator character to the
+ <tref>label serialization</tref>.</li>
+ <li>Append the <tref>object string</tref> to the
+ <tref>label serialization</tref>.</li>
+ </ol>
+ </ol>
+ </li>
+ <li>Append the <code>]</code> character to the
+ <tref>label serialization</tref>.</li>
+ <li>Append the <code>[</code> character to the
+ <tref>label serialization</tref>.</li>
+ <li>Append all incoming references for the current
+ <tref>label</tref> to the <tref>label serialization</tref> by
+ processing all of the items associated with the <tref>incoming list</tref>:
+ <ol class="algorithm">
+ <li>Build a <tdef>reference string</tdef>
+ using the pattern <code><</code><strong>PROPERTY</strong><code>></code><code><</code><strong>REFERER</strong><code>></code>
+ where <strong>PROPERTY</strong> is the property associated with the
+ incoming reference and <strong>REFERER</strong> is either the subject of
+ the node referring to the <tref>label</tref> in the incoming reference
+ or <code>_:</code> if <strong>REFERER</strong> begins with
+ <code>_:</code>.
+ <li>If this is the second iteration of the loop,
+ append a <code>|</code> separator character to the
+ <tref>label serialization</tref>.</li>
+ <li>Append the <tref>reference string</tref> to the
+ <tref>label serialization</tref>.</li>
+ </ol>
+ <li>Append the <code>]</code> character to the
+ <tref>label serialization</tref>.</li>
+ <li>Append all <tref>adjacent node labels</tref> to the
+ <tref>label serialization</tref> by concatenating the string value
+ for all of them, one after the other, to the
+ <tref>label serialization</tref>.</li>
+ <li>Push the <tref>adjacent node labels</tref> onto the
+ <tref>key stack</tref> and append the result of the
+ <a href="#mapping-serialization-algorithm">Mapping Serialization Algorithm</a>
+ to the <tref>label serialization</tref>.
+</ol>
+
+</section>
+
+</section>
+
+<section class="appendix">
+<h1>Acknowledgements</h1>
+
+<p>The editors would like to thank Jeremy Carroll for his work on the
+graph normalization problem. Gavin Carothers for providing valuable feedback
+and testing input for the algorithm defined in this specification. Sir
+Tim Berners Lee for his thoughts on graph normalization over the years.
+Jesús Arias Fisteus for his work on a similar algorithm. Finally, a huge thank
+you goes out to Dave Longley who designed and implemented the algorithms
+used in this specification, which turned out to be a monumentally difficult
+design challenge.
+</p>
+
+</section>
+
+</body>
+</html>