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jlechner |
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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<html>
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<head>
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<title>Node Invariants</title>
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<meta name="GENERATOR" content="Microsoft Visual Studio .NET 7.1">
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<meta name="vs_targetSchema" content="http://schemas.microsoft.com/intellisense/ie5">
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</head>
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<body bgcolor = "white">
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<h1>Node Invariants</h1>
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<p>
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Figure
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<a href = "#node_invariants">Some node invariants</a>
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shows some node invariants. A shows
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a tree whose each node contains, asides from an <tt>double</tt> key, the number
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of nodes at the subtree rooted at the node; B shows a tree whose each node
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contains, asides from a line-interval key, the maximal endpoint of the interval
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of any node in the subtree rooted at the node.
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The first tree allows querying efficiently what is the order statistic
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of any element; the second tree allows querying efficiently if any, or which,
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intervals overlap a given interval.
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</p>
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<h6 align = "center">
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<a name = "node_invariants">
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<img src = "node_invariants.jpg" width = "50%" alt = "no image">
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</a>
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</h6>
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<h6 align = "center">
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Some node invariants.
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</h6>
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<p>
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Maintaining such trees is difficult, for two reasons:
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</p>
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<ol>
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<li> Various operations can invalidate node invariants.
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<i>E.g.</i>, Figure
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<a href = "#node_invariant_invalidations">Invalidation of node invariants</a>
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shows how a right rotation, performed on A, results in B, with nodes <i>x</i>
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and <i>y</i> having corrupted invariants (the greyed nodes in C);
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Figure
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<a href = "#node_invariant_invalidations">Invalidation of node invariants</a>
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shows how an insert, performed on D, results in E, with nodes <i>x</i>
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and <i>y</i> having corrupted invariants (the greyed nodes in F).
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It is not feasible to know outside the tree the effect of an operation on the
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nodes of the tree.
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</li>
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<li>
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Even if node invariants are maintained, it is not possible to know
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in advance which search paths are required (<i>e.g.</i>, searching for all
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line intervals overlapping some interval might require several search paths).
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</li>
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</ol>
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<h6 align = "center">
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<a name = "node_invariant_invalidations">
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<img src = "node_invariant_invalidations.jpg" width = "80%" alt = "no image">
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</a>
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</h6>
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<h6 align = "center">
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Invalidation of node invariants.
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</h6>
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<p>
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These problems are solved by a combination of two means:
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</p>
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<ol>
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<li>
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The tree-based containers are parameterized by a <tt>Node_Updator</tt>
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parameter. When a tree operation might invalidate some node invariant,
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a <tt>Node_Updator</tt> object is invoked to restore the invariant. This object is
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always invoked with three nodes: some node, say <i>x</i> in
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Figure
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<a href = "#restoring_node_invariants">Invalidation of node invariants</a>-A
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has an invalid invariant, but its children, <i>y</i> and <i>z</i> hav valid invariants.
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After the invocation, all three nodes have valid invariants, as
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in
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Figure
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<a href = "#restoring_node_invariants">Invalidation of node invariants</a>-B.
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It is well known that any <tt>insert</tt>, <tt>erase</tt>,
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<tt>split</tt> or <tt>join</tt>, can restore
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all node invariants by a small number of node invariant updates
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[<a href = "references.html#clrs2001">clrs2001</a>].
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For example, Figure
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<a href = "#update_seq_diagram">
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Insert update sequence diagram
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</a>
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shows an <tt>insert</tt> operation (point A); the tree performs some operations, and
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calls the update functor three times (points B, C, and D).
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</li>
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<li>
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The tree based containers all define internally <tt>node_iterator</tt>
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and <tt>const_node_iterator</tt>, iterators which can be used to traverse
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from a node to any of its children or parent.
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</li>
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</ol>
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<h6 align = "center">
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<a name = "restoring_node_invariants">
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<img src = "restoring_node_invariants.jpg" width = "80%" alt = "no image">
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</a>
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</h6>
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<h6 align = "center">
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Invalidation of node invariants.
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</h6>
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<h6 align = "center">
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<a name = "update_seq_diagram">
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<img src = "update_seq_diagram.jpg" width = "50%" alt = "no image">
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</a>
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</h6>
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<h6 align = "center">
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Insert update sequence diagram.
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</h6>
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<p>
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In
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<a href = "concepts.html#concepts_null_policies">Null Policy Classes</a>
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a distinction was made between <i>redundant policies</i>
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and <i>null policies</i>.
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</p>
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<p>
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Seemingly, in this case a redundant policy - a policy which doesn't
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affect nodes' contents would suffice in this case. This, however, would
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lead to performance loss.
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Figure
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<a href = "#rationale_null_node_updator">
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Rationale for null node-invariant functors
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</a>
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shows a typical case where invariants are restored (in this case, to the
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shaded node). In most cases, tree operations such as rotations affect only
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the lower levels of the tree. A null policy allows to know that there
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is no need to traverse the tree to the root.
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</p>
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<h6 align = "center">
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<a name = "rationale_null_node_updator">
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<img src = "rationale_null_node_updator.jpg" width = "50%" alt = "no image">
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</a>
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</h6>
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<h6 align = "center">
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Rationale for null node-invariant functors.
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</h6>
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</body>
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</html>
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