Algorithms and Data Structures

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N.Wirth. Algorithms and Data Structures. Oberon version 192 1. h = 0: the subtree p requires no changes of the tree structure. 2. h = 1: node p has obtained a sibling. 3. h = 2: the subtree p has increased in height. PROCEDURE search (VAR p: Node; x: INTEGER; VAR h: INTEGER); VAR q, r: Node; (* ADenS472_BBtrees *) BEGIN (*h=0*) IF p = NIL THEN (*insert new node*) NEW(p); p.key := x; p.L := NIL; p.R := NIL; p.lh := FALSE; p.rh := FALSE; h := 2 ELSIF x < p.key THEN search(p.L, x, h); IF h > 0 THEN (*left branch has grown or received sibling*) q := p.L; IF p.lh THEN h := 2; p.lh := FALSE; IF q.lh THEN (*LL*) p.L := q.R; q.lh := FALSE; q.R := p; p := q ELSE (*q.rh, LR*) r := q.R; q.R := r.L; q.rh := FALSE; r.L := p.L; p.L := r.R; r.R := p; p := r END ELSE DEC(h); IF h = 1 THEN p.lh := TRUE END END END ELSIF x > p.key THEN search(p.R, x, h); IF h > 0 THEN (*right branch has grown or received sibling*) q := p.R; IF p.rh THEN h := 2; p.rh := FALSE; IF q.rh THEN (*RR*) p.R := q.L; q.rh := FALSE; q.L := p; p := q ELSE (*q.lh, RL*) r := q.L; q.L := r.R; q.lh := FALSE; r.R := p.R; p.R := r.L; r.L := p; p := r END ELSE DEC(h); IF h = 1 THEN p.rh := TRUE END END END END END search; Note that the actions to be taken for node rearrangement very strongly resemble those developed in the AVL-balanced tree search algorithm. It is evident that all four cases can be implemented by simple pointer rotations: single rotations in the LL and RR cases, double rotations in the LR and RL cases. In fact, procedure search appears here slightly simpler than in the AVL case. Clearly, the SBB-tree scheme
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
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Algorithms and Data Structures

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