Algorithms and Data Structures

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N.Wirth. Algorithms and Data Structures. Oberon version 188 END a b a b c Fig. 4.47. Representation of BB-tree nodes Considering the problem of key insertion, one must distinguish four possible situations that arise from growth of the left or right subtrees. The four cases are illustrated in Fig. 4.48. 1. A A B a B a b c b c B 2. A B A B C A C a b C a b c d a b c d c d 3. B A B A B A c a b c a b c a b B 4. B C A B C A C A c d a b c d a b c d a b
N.Wirth. Algorithms and Data Structures. Oberon version 189 Fig. 4.48. Node insertion in BB-tree Remember that B-trees have the characteristic of growing from the bottom toward the root and that the property of all leafs being at the same level must be maintained. The simplest case (1) is when the right subtree of a node A grows and when A is the only key on its (hypothetical) page. Then, the descendant B merely becomes the sibling of A, i.e., the vertical pointer becomes a horizontal pointer. This simple raising of the right arm is not possible if A already has a sibling. Then we would obtain a page with 3 nodes, and we have to split it (case 2). Its middle node B is passed up to the next higher level. Now assume that the left subtree of a node B has grown in height. If B is again alone on a page (case 3), i.e., its right pointer refers to a descendant, then the left subtree (A) is allowed to become B's sibling. (A simple rotation of pointers is necessary since the left pointer cannot be horizontal). If, however, B already has a sibling, the raising of A yields a page with three members, requiring a split. This split is realized in a very straightforward manner: C becomes a descendant of B, which is raised to the next higher level (case 4). It should be noted that upon searching a key, it makes no effective difference whether we proceed along a horizontal or a vertical pointer. It therefore appears artificial to worry about a left pointer in case 3 becoming horizontal, although its page still contains not more than two members. Indeed, the insertion algorithm reveals a strange asymmetry in handling the growth of left and right subtrees, and it lets the BBtree organization appear rather artificial. There is no proof of strangeness of this organization; yet a healthy intuition tells us that something is fishy, and that we should remove this asymmetry. It leads to the notion of the symmetric binary B-tree (SBB-tree) which was also investigated by Bayer [4-4] in 1972. On the average it leads to slightly more efficient search trees, but the algorithms for insertion and deletion are also slightly more complex. Furthermore, each node now requires two bits (Boolean variable lh and rh) to indicate the nature of its two pointers. Since we will restrict our detail considerations to the problem of insertion, we have once again to distinguish among four cases of grown subtrees. They are illustrated in Fig. 4.49, which makes the gained symmetry evident. Note that whenever a subtree of node A without siblings grows, the root of the subtree becomes the sibling of A. This case need not be considered any further. The four cases considered in Fig. 4.49 all reflect the occurrence of a page overflow and the subsequent page split. They are labelled according to the directions of the horizontal pointers linking the three siblings in the middle figures. The initial situation is shown in the left column; the middle column illustrates the fact that the lower node has been raised as its subtree has grown; the figures in the right column show the result of node rearrangement.
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Algorithms and Data Structures

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