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<h1>Tree Design</h1>
<h2><a name="overview" id="overview">Overview</a></h2>
<p>The tree-based container has the following declaration:</p>
<pre>
<b>template</b><
<b>typename</b> Key,
<b>typename</b> Mapped,
<b>typename</b> Cmp_Fn = std::less<Key>,
<b>typename</b> Tag = <a href="rb_tree_tag.html">rb_tree_tag</a>,
<b>template</b><
<b>typename</b> Const_Node_Iterator,
<b>typename</b> Node_Iterator,
<b>typename</b> Cmp_Fn_,
<b>typename</b> Allocator_>
<b>class</b> Node_Update = <a href=
"null_tree_node_update.html">null_tree_node_update</a>,
<b>typename</b> Allocator = std::allocator<<b>char</b>> >
<b>class</b> <a href=
"tree.html">tree</a>;
</pre>
<p>The parameters have the following meaning:</p>
<ol>
<li><tt>Key</tt> is the key type.</li>
<li><tt>Mapped</tt> is the mapped-policy.</li>
<li><tt>Cmp_Fn</tt> is a key comparison functor</li>
<li><tt>Tag</tt> specifies which underlying data structure
to use.</li>
<li><tt>Node_Update</tt> is a policy for updating node
invariants. This is described in <a href="#invariants">Node
Invariants</a>.</li>
<li><tt>Allocator</tt> is an allocator
type.</li>
</ol>
<p>The <tt>Tag</tt> parameter specifies which underlying
data structure to use. Instantiating it by <a href=
"rb_tree_tag.html"><tt>rb_tree_tag</tt></a>, <a href=
"splay_tree_tag.html"><tt>splay_tree_tag</tt></a>, or
<a href="ov_tree_tag.html"><tt>ov_tree_tag</tt></a>,
specifies an underlying red-black tree, splay tree, or
ordered-vector tree, respectively; any other tag is illegal.
Note that containers based on the former two contain more types
and methods than the latter (<i>e.g.</i>,
<tt>reverse_iterator</tt> and <tt>rbegin</tt>), and different
exception and invalidation guarantees.</p>
<h2><a name="invariants" id="invariants">Node
Invariants</a></h2>
<p>Consider the two trees in Figures <a href=
"#node_invariants">Some node invariants</a> A and B. The first
is a tree of floats; the second is a tree of pairs, each
signifying a geometric line interval. Each element in a tree is refered to as a node of the tree. Of course, each of
these trees can support the usual queries: the first can easily
search for <tt>0.4</tt>; the second can easily search for
<tt>std::make_pair(10, 41)</tt>.</p>
<p>Each of these trees can efficiently support other queries.
The first can efficiently determine that the 2rd key in the
tree is <tt>0.3</tt>; the second can efficiently determine
whether any of its intervals overlaps
<tt>std::make_pair(29,42)</tt> (useful in geometric
applications or distributed file systems with leases, for
example). (See <a href=
"http://gcc.gnu.org/viewcvs/*checkout*/trunk/libstdc%2B%2B-v3/testsuite/ext/pb_ds/example/tree_order_statistics.cc"><tt>tree_order_statistics.cc</tt></a>
and <a href=
"http://gcc.gnu.org/viewcvs/*checkout*/trunk/libstdc%2B%2B-v3/testsuite/ext/pb_ds/example/tree_intervals.cc"><tt>tree_intervals.cc</tt></a>
for examples.) It should be noted that an <tt>std::set</tt> can
only solve these types of problems with linear complexity.</p>
<p>In order to do so, each tree stores some <i>metadata</i> in
each node, and maintains node invariants <a href=
"references.html#clrs2001">clrs2001</a>]. The first stores in
each node the size of the sub-tree rooted at the node; the
second stores at each node the maximal endpoint of the
intervals at the sub-tree rooted at the node.</p>
<h6 class="c1"><a name="node_invariants" id=
"node_invariants"><img src="node_invariants.png" alt=
"no image" /></a></h6>
<h6 class="c1">Some node invariants.</h6>
<p>Supporting such trees is difficult for a number of
reasons:</p>
<ol>
<li>There must be a way to specify what a node's metadata
should be (if any).</li>
<li>Various operations can invalidate node invariants.
<i>E.g.</i>, Figure <a href=
"#node_invariant_invalidations">Invalidation of node
invariants</a> shows how a right rotation, performed on A,
results in B, with nodes <i>x</i> and <i>y</i> having
corrupted invariants (the grayed nodes in C); Figure <a href=
"#node_invariant_invalidations">Invalidation of node
invariants</a> shows how an insert, performed on D, results
in E, with nodes <i>x</i> and <i>y</i> having corrupted
invariants (the grayed nodes in F). It is not feasible to
know outside the tree the effect of an operation on the nodes
of the tree.</li>
<li>The search paths of standard associative containers are
defined by comparisons between keys, and not through
metadata.</li>
<li>It is not feasible to know in advance which methods trees
can support. Besides the usual <tt>find</tt> method, the
first tree can support a <tt>find_by_order</tt> method, while
the second can support an <tt>overlaps</tt> method.</li>
</ol>
<h6 class="c1"><a name="node_invariant_invalidations" id=
"node_invariant_invalidations"><img src=
"node_invariant_invalidations.png" alt="no image" /></a></h6>
<h6 class="c1">Invalidation of node invariants.</h6>
<p>These problems are solved by a combination of two means:
node iterators, and template-template node updater
parameters.</p>
<h3><a name="node_it" id="node_it">Node Iterators</a></h3>
<p>Each tree-based container defines two additional iterator
types, <a href=
"tree_const_node_iterator.html"><tt>const_node_iterator</tt></a>
and <a href=
"tree_node_iterator.html"><tt>node_iterator</tt></a>.
These iterators allow descending from a node to one of its
children. Node iterator allow search paths different than those
determined by the comparison functor. <a href=
"tree.html">tree</a>
supports the methods:</p>
<pre>
<a href="tree_const_node_iterator.html"><tt>const_node_iterator</tt></a>
node_begin() <b>const</b>;
<a href="tree_node_iterator.html"><tt>node_iterator</tt></a>
node_begin();
<a href="tree_const_node_iterator.html"><tt>const_node_iterator</tt></a>
node_end() <b>const</b>;
<a href="tree_node_iterator.html"><tt>node_iterator</tt></a>
node_end();
</pre>
<p>The first pairs return node iterators corresponding to the
root node of the tree; the latter pair returns node iterators
corresponding to a just-after-leaf node.</p>
<h3><a name="node_up" id="node_up">Node Updater
(Template-Template) Parameters</a></h3>
<p>The tree-based containers are parametrized by a
<tt>Node_Update</tt> template-template parameter. A tree-based
container instantiates <tt>Node_Update</tt> to some
<tt>node_update</tt> class, and publicly
subclasses <tt>node_update</tt>. Figure
<a href="#tree_node_update_cd">A tree and its update
policy</a> shows this scheme, as well as some predefined
policies (which are explained below).</p>
<h6 class="c1"><a name="tree_node_update_cd" id=
"tree_node_update_cd"><img src=
"tree_node_update_policy_cd.png" alt="no image" /></a></h6>
<h6 class="c1">A tree and its update policy.</h6>
<p><tt>node_update</tt> (an instantiation of
<tt>Node_Update</tt>) must define <tt>metadata_type</tt> as
the type of metadata it requires. For order statistics,
<i>e.g.</i>, <tt>metadata_type</tt> might be <tt>size_t</tt>.
The tree defines within each node a <tt>metadata_type</tt>
object.</p>
<p><tt>node_update</tt> must also define the following method
for restoring node invariants:</p>
<pre>
void
operator()(<a href=
"tree_node_iterator.html"><tt>node_iterator</tt></a> nd_it, <a href=
"tree_const_node_iterator.html"><tt>const_node_iterator</tt></a> end_nd_it)
</pre>
<p>In this method, <tt>nd_it</tt> is a <a href=
"tree_node_iterator.html"><tt>node_iterator</tt></a>
corresponding to a node whose A) all descendants have valid
invariants, and B) its own invariants might be violated;
<tt>end_nd_it</tt> is a <a href=
"tree_const_node_iterator.html"><tt>const_node_iterator</tt></a>
corresponding to a just-after-leaf node. This method should
correct the node invariants of the node pointed to by
<tt>nd_it</tt>. For example, say node <i>x</i> in Figure
<a href="#restoring_node_invariants">Restoring node
invariants</a>-A has an invalid invariant, but its' children,
<i>y</i> and <i>z</i> have valid invariants. After the
invocation, all three nodes should have valid invariants, as in
Figure <a href="#restoring_node_invariants">Restoring node
invariants</a>-B.</p>
<h6 class="c1"><a name="restoring_node_invariants" id=
"restoring_node_invariants"><img src=
"restoring_node_invariants.png" alt="no image" /></a></h6>
<h6 class="c1">Invalidation of node invariants.</h6>
<p>When a tree operation might invalidate some node invariant,
it invokes this method in its <tt>node_update</tt> base to
restore the invariant. For example, Figure <a href=
"#update_seq_diagram">Insert update sequence diagram</a> shows
an <tt>insert</tt> operation (point A); the tree performs some
operations, and calls the update functor three times (points B,
C, and D). (It is well known that any <tt>insert</tt>,
<tt>erase</tt>, <tt>split</tt> or <tt>join</tt>, can restore
all node invariants by a small number of node invariant updates
[<a href="references.html#clrs2001">clrs2001</a>].)</p>
<h6 class="c1"><a name="update_seq_diagram" id=
"update_seq_diagram"><img src="update_seq_diagram.png" alt=
"no image" /></a></h6>
<h6 class="c1">Insert update sequence diagram.</h6>
<p>To complete the description of the scheme, three questions
need to be answered:</p>
<ol>
<li>How can a tree which supports order statistics define a
method such as <tt>find_by_order</tt>?</li>
<li>How can the node updater base access methods of the
tree?</li>
<li>How can the following cyclic dependency be resolved?
<tt>node_update</tt> is a base class of the tree, yet it
uses node iterators defined in the tree (its child).</li>
</ol>
<p>The first two questions are answered by the fact that
<tt>node_update</tt> (an instantiation of
<tt>Node_Update</tt>) is a <tt><b>public</b></tt> base class
of the tree. Consequently:</p>
<ol>
<li>Any public methods of <tt>node_update</tt> are
automatically methods of the tree [<a href=
"references.html#alexandrescu01modern">alexandrescu01modern</a>].
Thus an order-statistics node updater, <a href=
"tree_order_statistics_node_update.html"><tt>tree_order_statistics_node_update</tt></a>
defines the <tt>find_by_order</tt> method; any tree
instantiated by this policy consequently supports this method
as well.</li>
<li>In C++, if a base class declares a method as
<tt><b>virtual</b></tt>, it is <tt><b>virtual</b></tt> in its
subclasses. If <tt>node_update</tt> needs to access one of
the tree's methods, say the member function <tt>end</tt>, it simply
declares that method as <tt><b>virtual</b></tt>
abstract.</li>
</ol>
<p>The cyclic dependency is solved through template-template
parameters. <tt>Node_Update</tt> is parametrized by the tree's node iterators, its comparison
functor, and its allocator type. Thus,
instantiations of <tt>Node_Update</tt> have all information required.</p>
<p class="c1"><tt>pb_ds</tt> assumes that constructing a metadata object and modifying it
are exception free. Suppose that during some method, say
<tt>insert</tt>, a metadata-related operation
(<i>e.g.</i>, changing the value of a metadata) throws an
exception. Ack! Rolling back the method is unusually complex.</p>
<p>In <a href=
"concepts.html#concepts_null_policies">Interface::Concepts::Null
Policy Classes</a> a distinction was made between <i>redundant
policies</i> and <i>null policies</i>. Node invariants show a
case where null policies are required.</p>
<p>Assume a regular tree is required, one which need not
support order statistics or interval overlap queries.
Seemingly, in this case a redundant policy - a policy which
doesn't affect nodes' contents would suffice. This, would lead
to the following drawbacks:</p>
<ol>
<li>Each node would carry a useless metadata object, wasting
space.</li>
<li>The tree cannot know if its <tt>Node_Update</tt> policy
actually modifies a node's metadata (this is halting
reducible). In Figure <a href=
"#rationale_null_node_update">Useless update path</a> ,
assume the shaded node is inserted. The tree would have to
traverse the useless path shown to the root, applying
redundant updates all the way.</li>
</ol>
<h6 class="c1"><a name="rationale_null_node_update" id=
"rationale_null_node_update"><img src=
"rationale_null_node_update.png" alt="no image" /></a></h6>
<h6 class="c1">Useless update path.</h6>
<p>A null policy class, <a href=
"null_tree_node_update.html"><tt>null_tree_node_update</tt></a>
solves both these problems. The tree detects that node
invariants are irrelevant, and defines all accordingly.</p>
<h2><a name="add_methods" id="add_methods">Additional
Methods</a></h2>
<p>Tree-based containers support split and join methods.
It is possible to split a tree so that it passes
all nodes with keys larger than a given key to a different
tree. These methods have the following advantages over the
alternative of externally inserting to the destination
tree and erasing from the source tree:</p>
<ol>
<li>These methods are efficient - red-black trees are split
and joined in poly-logarithmic complexity; ordered-vector
trees are split and joined at linear complexity. The
alternatives have super-linear complexity.</li>
<li>Aside from orders of growth, these operations perform
few allocations and de-allocations. For red-black trees, allocations are not performed,
and the methods are exception-free. </li>
</ol>
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