Algorithms and Data Structures

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.