Algorithms and Data Structures

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N.Wirth. Algorithms and Data Structures. Oberon version 186 k := (b.m-N+1) DIV 2; (*k = nof items available on page b*) IF k > 0 THEN FOR i := N-2 TO 0 BY -1 DO a.e[i+k] := a.e[i] END; a.e[k-1] := c.e[s]; a.e[k-1].p := a.p0; (*move k-1 items from b to a, one to c*) DEC(b.m, k); FOR i := k-2 TO 0 BY -1 DO a.e[i] := b.e[i+b.m+1] END; c.e[s] := b.e[b.m]; a.p0 := c.e[s].p; c.e[s].p := a; a.m := N-1+k; h := FALSE ELSE (*merge pages a and b, discard a*) c.e[s].p := a.p0; b.e[N] := c.e[s]; FOR i := 0 TO N-2 DO b.e[i+N+1] := a.e[i] END; b.m := 2*N; DEC(c.m); h := c.m < N END END END underflow; PROCEDURE delete (x: INTEGER; a: Page; VAR h: BOOLEAN); (*search and delete key x in B-tree a; if a page underflow arises, balance with adjacent page or merge; h := "page a is undersize"*) VAR i, L, R: INTEGER; q: Page; PROCEDURE del (p: Page; VAR h: BOOLEAN); VAR k: INTEGER; q: Page; (*global a, R*) BEGIN k := p.m-1; q := p.e[k].p; IF q # NIL THEN del(q, h); IF h THEN underflow(p, q, p.m, h) END ELSE p.e[k].p := a.e[R].p; a.e[R] := p.e[k]; DEC(p.m); h := p.m < N END END del; BEGIN IF a # NIL THEN L := 0; R := a.m; (*binary search*) WHILE L < R DO i := (L+R) DIV 2; IF x
N.Wirth. Algorithms and Data Structures. Oberon version 187 END END delete; PROCEDURE ShowTree (VAR W: Texts.Writer; p: Page; level: INTEGER); VAR i: INTEGER; BEGIN IF p # NIL THEN FOR i := 1 TO level DO Texts.Write(W, 9X) END; FOR i := 0 TO p.m-1 DO Texts.WriteInt(W, p.e[i].key, 4) END; Texts.WriteLn(W); IF p.m > 0 THEN ShowTree(W, p.p0, level+1) END; FOR i := 0 TO p.m-1 DO ShowTree(W, p.e[i].p, level+1) END END END ShowTree; Extensive analysis of B-tree performance has been undertaken and is reported in the referenced article (Bayer and McCreight). In particular, it includes a treatment of the question of optimal page size, which strongly depends on the characteristics of the storage and computing system available. Variations of the B-tree scheme are discussed in Knuth, Vol. 3, pp. 476-479. The one notable observation is that page splitting should be delayed in the same way that page merging is delayed, by first attempting to balance neighboring pages. Apart from this, the suggested improvements seem to yield marginal gains. A comprehensive survey of B-trees may be found in [4-8]. 4.7.2 Binary B-Trees The species of B-trees that seems to be least interesting is the first order B-tree (n = 1). But sometimes it is worthwhile to pay attention to the exceptional case. It is plain, however, that first-order B-trees are not useful in representing large, ordered, indexed data sets involving secondary stores; approximately 50% of all pages will contain a single item only. Therefore, we shall forget secondary stores and again consider the problem of search trees involving a one-level store only. A binary B-tree (BB-tree) consists of nodes (pages) with either one or two items. Hence, a page contains either two or three pointers to descendants; this suggested the term 2-3 tree. According to the definition of B-trees, all leaf pages appear at the same level, and all non-leaf pages of BB-trees have either two or three descendants (including the root). Since we now are dealing with primary store only, an optimal economy of storage space is mandatory, and the representation of the items inside a node in the form of an array appears unsuitable. An alternative is the dynamic, linked allocation; that is, inside each node there exists a linked list of items of length 1 or 2. Since each node has at most three descendants and thus needs to harbor only up to three pointers, one is tempted to combine the pointers for descendants and pointers in the item list as shown in Fig. 4.47. The B-tree node thereby loses its actual identity, and the items assume the role of nodes in a regular binary tree. It remains necessary, however, to distinguish between pointers to descendants (vertical) and pointers to siblings on the same page (horizontal). Since only the pointers to the right may be horizontal, a single bit is sufficient to record this distiction. We therefore introduce the Boolean field h with the meaning horizontal. The definition of a tree node based on this representation is given below. It was suggested and investigated by R. Bayer [4-3] in 1971 and represents a search tree organization guaranteeing p = 2*log(N) as maximum path length. TYPE Node = POINTER TO RECORD key: INTEGER; ........... left, right: Node; h: BOOLEAN (*right branch horizontal*)
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Algorithms and Data Structures

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