Algorithms and Data Structures

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N.Wirth. Algorithms and Data Structures. Oberon version 178 to the youngest (or eldest) offspring assigned to the parent. A possible type definition for this case is the following, and a possible data structure is shown in Fig. 4.40. TYPE Person =POINTER TO RECORD name: alfa; sibling, offspring: Person END JOHN PETER ALBERT MARY ROBERT CAROL CHRIS TINA PAUL GEORGE PAMELA Fig. 4.40. Multiway tree represented as binary tree We now realize that by tilting this picture by 45 degrees it will look like a perfect binary tree. But this view is misleading because functionally the two references have entirely different meanings. One usually dosen't treat a sibling as an offspring and get away unpunished, and hence one should not do so even in constructing data definitions. This example could also be easily extended into an even more complicated data structure by introducing more components in each person's record, thus being able to represent further family relationships. A likely candidate that cannot generally be derived from the sibling and offspring references is that of husband and wife, or even the inverse relationship of father and mother. Such a structure quickly grows into a complex relational data bank, and it may be possible to map serveral trees into it. The algorithms operating on such structures are intimately tied to their data definitions, and it does not make sense to specify any general rules or widely applicable techniques. However, there is a very practical area of application of multiway trees which is of general interest. This is the construction and maintenance of large-scale search trees in which insertions and deletions are necessary, but in which the primary store of a computer is not large enough or is too costly to be used for long-time storage. Assume, then, that the nodes of a tree are to be stored on a secondary storage medium such as a disk store. Dynamic data structures introduced in this chapter are particularly suitable for incorporation of secondary storage media. The principal innovation is merely that pointers are represented by disk store addresses instead of main store addresses. Using a binary tree for a data set of, say, a million items, requires on the average approximately log 10 6 (i.e. about 20) search steps. Since each step now involves a disk access (with inherent latency time), a storage organization using fewer accesses will be highly desirable. The multiway tree is a perfect solution to this problem. If an item located on a secondary store is accessed, an entire group of items may also be accessed without much additional cost. This suggests that a tree be subdivided into subtrees, and that the subtrees are represented as units that are accessed all together. We shall call these subtrees pages. Figure 4.41 shows a binary tree subdivided into pages, each page consisting of 7 nodes.
N.Wirth. Algorithms and Data Structures. Oberon version 179 Fig. 4.41. Binary tree subdivided into pages The saving in the number of disk accesses — each page access now involves a disk access — can be considerable. Assume that we choose to place 100 nodes on a page (this is a reasonable figure); then the million item search tree will on the average require only log 100 (10 6 ) (i.e. about 3) page accesses instead of 20. But, of course, if the tree is left to grow at random, then the worst case may still be as large as 10 4 . It is plain that a scheme for controlled growth is almost mandatory in the case of multiway trees. 4.7.1 Multiway B-Trees If one is looking for a controlled growth criterion, the one requiring a perfect balance is quickly eliminated because it involves too much balancing overhead. The rules must clearly be somewhat relaxed. A very sensible criterion was postulated by R. Bayer and E.M. McCreight [4.2] in 1970: every page (except one) contains between n and 2n nodes for a given constant n. Hence, in a tree with N items and a maximum page size of 2n nodes per page, the worst case requires log n N page accesses; and page accesses clearly dominate the entire search effort. Moreover, the important factor of store utilization is at least 50% since pages are always at least half full. With all these advantages, the scheme involves comparatively simple algorithms for search, insertion, and deletion. We will subsequently study them in detail. The underlying data structures are called B-trees, and have the following characteristics; n is said to be the order of the B-tree. 1. Every page contains at most 2n items (keys). 2. Every page, except the root page, contains at least n items. 3. Every page is either a leaf page, i.e. has no descendants, or it has m+1 descendants, where m is its number of keys on this page. 4. All leaf pages appear at the same level. 25 10 20 30 40 2 5 7 8 13 14 15 18 22 24 26 27 28 32 35 38 41 42 45 46 Fig. 4.42. B-tree of order 2 Figure 4.42 shows a B-tree of order 2 with 3 levels. All pages contain 2, 3, or 4 items; the exception is the root which is allowed to contain a single item only. All leaf pages appear at level 3. The keys appear in
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Algorithms and Data Structures

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