<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> <head> <meta name="generator" content= "HTML Tidy for Linux/x86 (vers 12 April 2005), see www.w3.org" /> <title>Tree-Based Containers</title> <meta http-equiv="Content-Type" content= "text/html; charset=us-ascii" /> </head> <body> <div id="page"> <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> </div> </body> </html>