Algorithms and Data Structures

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N.Wirth. Algorithms and Data Structures. Oberon version 180 increasing order from left to right if the B-tree is squeezed into a single level by inserting the descendants in between the keys of their ancestor page. This arrangement represents a natural extension of binary search trees, and it determines the method of searching an item with given key. Consider a page of the form shown in Fig. 4.43 and a given search argument x. Assuming that the page has been moved into the primary store, we may use conventional search methods among the keys k 0 ... k m-1 . If m is sufficiently large, one may use binary search; if it is rather small, an ordinary sequential search will do. (Note that the time required for a search in main store is probably negligible compared to the time it takes to move the page from secondary into primary store.) If the search is unsuccessful, we are in one of the following situations: 1. k i < x < k i+1 for 0 < i < m-1 The search continues on page p i^ 2. k m-1 < x The search continues on page p m-1^. 3. x < k 0 The search continues on page p -1^. p -1 k 0 p 0 k 1 p 1 . . . p m-2 k m-1 p m-1 Fig. 4.43. B-tree page with m keys If in some case the designated pointer is NIL, i.e., if there is no descendant page, then there is no item with key x in the whole tree, and the search is terminated. Surprisingly, insertion in a B-tree is comparatively simple too. If an item is to be inserted in a page with m < 2n items, the insertion process remains constrained to that page. It is only insertion into an already full page that has consequences upon the tree structure and may cause the allocation of new pages. To understand what happens in this case, refer to Fig. 4.44, which illustrates the insertion of key 22 in a B-tree of order 2. It proceeds in the following steps: 1. Key 22 is found to be missing; insertion in page C is impossible because C is already full. 2. Page C is split into two pages (i.e., a new page D is allocated). 3. The 2n+1 keys are equally distributed onto C and D, and the middle key is moved up one level into the ancestor page A. A 20 A 20 30 7 10 15 18 26 30 35 40 B C 7 10 15 18 22 26 35 40 B C D Fig. 4.44. Insertion of key 22 in B-tree This very elegant scheme preserves all the characteristic properties of B-trees. In particular, the split pages contain exactly n items. Of course, the insertion of an item in the ancestor page may again cause that page to overflow, thereby causing the splitting to propagate. In the extreme case it may propagate up to the root. This is, in fact, the only way that the B-tree may increase its height. The B-tree has thus a strange manner of growing: it grows from its leaves upward to the root. We shall now develop a detailed program from these sketchy descriptions. It is already apparent that a recursive formulation will be most convenient because of the property of the splitting process to propagate
N.Wirth. Algorithms and Data Structures. Oberon version 181 back along the search path. The general structure of the program will therefore be similar to balanced tree insertion, although the details are different. First of all, a definition of the page structure has to be formulated. We choose to represent the items in the form of an array. TYPE Page = POINTER TO PageDescriptor; Item = RECORD key: INTEGER; p: Page; count: INTEGER (*data*) END; PageDescriptor = RECORD m: INTEGER; (* 0 .. 2n *) p0: Page; e: ARRAY 2*n OF Item END Again, the item component count stands for all kinds of other information that may be associated with each item, but it plays no role in the actual search process. Note that each page offers space for 2n items. The field m indicates the actual number of items on the page. As m ≥ n (except for the root page), a storage utilization of a least 50% is guaranteed. The algorithm of B-tree search and insertion is formulated below as a procedure called search. Its main structure is straightforward and similar to that for the balanced binary tree search, with the exception that the branching decision is not a binary choice. Instead, the "within-page search" is represented as a binary search on the array e of elements. The insertion algorithm is formulated as a separate procedure merely for clarity. It is activated after search has indicated that an item is to be passed up on the tree (in the direction toward the root). This fact is indicated by the Boolean result parameter h; it assumes a similar role as in the algorithm for balanced tree insertion, where h indicates that the subtree had grown. If h is true, the second result parameter, u, represents the item being passed up. Note that insertions start in hypothetical pages, namely, the "special nodes" of Fig. 4.19; the new item is immediately handed up via the parameter u to the leaf page for actual insertion. The scheme is sketched here: PROCEDURE search (x: INTEGER; a: Page; VAR h: BOOLEAN; VAR u: Item); BEGIN IF a = NIL THEN (*x not in tree, insert*) assign x to item u, set h to TRUE, indicating that an item u is passed up in the tree ELSE binary search for x in array a.e; IF found THEN process data ELSE search(x, descendant, h, u); IF h THEN (*an item was passed up*) IF no. of items on page a^ < 2n THEN insert u on page a^ and set h to FALSE ELSE split page and pass middle item up END END END END END search If the parameter h is true after the call of search in the main program, a split of the root page is requested. Since the root page plays an exceptional role, this process has to be programmed separately. It consists merely of the allocation of a new root page and the insertion of the single item given by the paramerter u.
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Algorithms and Data Structures

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