Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 183
number of items m on the reduced page, because, if m < n, the primary characteristic of B-trees would be
violated. Some additional action has to be taken; this underflow condition is indicated by the Boolean
variable parameter h.
The only recourse is to borrow or annect an item from one of the neighboring pages, say from Q. Since
this involves fetching page Q into main store — a relatively costly operation — one is tempted to make the
best of this undesirable situation and to annect more than a single item at once. The usual strategy is to
distribute the items on pages P and Q evenly on both pages. This is called page balancing.
Of course, it may happen that there is no item left to be annected since Q has already reached its minimal
size n. In this case the total number of items on pages P and Q is 2n-1; we may merge the two pages into
one, adding the middle item from the ancestor page of P and Q, and then entirely dispose of page Q. This is
exactly the inverse process of page splitting. The process may be visualized by considering the deletion of
key 22 in Fig. 4.45. Once again, the removal of the middle key in the ancestor page may cause its size to
drop below the permissible limit n, thereby requiring that further special action (either balancing or merging)
be undertaken at the next level. In the extreme case page merging may propagate all the way up to the
root. If the root is reduced to size 0, it is itself deleted, thereby causing a reduction in the height of the B-
tree. This is, in fact, the only way that a B-tree may shrink in height. Figure 4.46 shows the gradual decay
of the B-tree of Fig. 4.45 upon the sequential deletion of the keys
25 45 24; 38 32; 8 27 46 13 42; 5 22 18 26; 7 35 15;
The semicolons again mark the places where the snapshots are taken, namely where pages are being
eliminated. The similarity of its structure to that of balanced tree deletion is particularly noteworthy.
a)
25
10 20
30 40
5 7 8 13 15 18 22 24 26 27 32 35 38 42 45 46
b)
10 22 30 40
5 7 8 13 15 18 20 26 27 32 35 38 42 46
c)
10 22 30
5 7 8 13 15 18 20 26 27 35 40 42 46
d)
10 22
5 7 15 18 20 26 30 35 40
e)
15
7 10 20 30 35 40
f)
10 20 30 40
Fig. 4.46. Decay of B-tree of order 2
TYPE Page = POINTER TO PageRec; (* ADenS471_Btrees *)
Entry = RECORD