Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 193
emerges as an alternative to the AVL-balancing criterion. A performance comparison is therefore both
possible and desirable.
We refrain from involved mathematical analysis and concentrate on some basic differences. It can be
proven that the AVL-balanced trees are a subset of the SBB-trees. Hence, the class of the latter is larger.
It follows that their path length is on the average larger than in the AVL case. Note in this connection the
worst-case tree (4) in Fig. 4.50. On the other hand, node rearrangement is called for less frequently. The
balanced tree is therefore preferred in those applications in which key retrievals are much more frequent
than insertions (or deletions); if this quotient is moderate, the SBB-tree scheme may be preferred. It is very
difficult to say where the borderline lies. It strongly depends not only on the quotient between the
frequencies of retrieval and structural change, but also on the characteristics of an implementation. This is
particularly the case if the node records have a densely packed representation, and if therefore access to
fields involves part-word selection.
The SBB-tree has later found a rebirth under the name of red-black tree. The difference is that whereas
in the case of the symmetric, binary B-tree every node contains two h-fields indicating whether the
emanating pointers are horizontal, every node of the red-black tree contains a single h-field, indicating
whether the incoming pointer is horizontal. The name stems from the idea to color nodes with incoming
down-pointer black, and those with incoming horizontal pointer red. No two red nodes can immediately
follow each other on any path. Therefore, like in the cases of the BB- and SBB-trees, every search path is
at most twice as long as the height of the tree. There exists a canonical mapping from binary B-trees to
red-black trees.
4.8 Priority Search Trees
Trees, and in particular binary trees, constitute very effective organisations for data that can be ordered
on a linear scale. The preceding chapters have exposed the most frequently used ingenious schemes for
efficient searching and maintenance (insertion, deletion). Trees, however, do not seem to be helpful in
problems where the data are located not in a one-dimensional, but in a multi-dimensional space. In fact,
efficient searching in multi-dimensional spaces is still one of the more elusive problems in computer science,
the case of two dimensions being of particular importance to many practical applications.
Upon closer inspection of the subject, trees might still be applied usefully at least in the two-dimensional
case. After all, we draw trees on paper in a two-dimensional space. Let us therefore briefly review the
characteristics of the two major kinds of trees so far encountered.
1. A search tree is governed by the invariants
p.left ≠ NIL implies p.left.x < p.x,
p.right ≠ NIL implies p.x < p.right.x,
holding for all nodes p with key x. It is apparent that only the horizontal position of nodes is at all
constrained by the invariant, and that the vertical positions of nodes can be arbitrarily chosen such that
access times in searching, (i.e. path lengths) are minimized.
2. A heap, also called priority tree, is governed by the invariants
p.left ≠ NIL implies p.y ≤ p.left.y,
p.right ≠ NIL implies p.y ≤ p.right.y,
holding for all nodes p with key y. Here evidently only the vertical positions are constrained by the
invariants.
It seems straightforward to combine these two conditions in a definition of a tree organization in a twodimensional
space, with each node having two keys x and y which can be regarded as coordinates of the
node. Such a tree represents a point set in a plane, i.e. in a two-dimensional Cartesian space; it is therefore