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CHAPTER 14 TREE-STRUCTURED DATA
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LOOP : WHILE TRUE BEGIN ; INSERT OUTLIST PREORDER ( RIGHT FROM TLR ) ; IF NOT ( IS_EMPTY ( LR WHERE LEFT = ( RIGHT FROM TLR ) ) THEN TLR := TUPLE FROM ( LR WHERE LEFT = ( RIGHT FROM TLR ) ) ; ELSE LEAVE LOOP : END IF ; END ; END WHILE ; END ; END IF ; END ; END IF ; RETURN OUTLIST ; END ; END OPERATOR ; Note: Even if you agree with me that this code looks a bit complicated, that s not really the point. Rather, the point is that (as already mentioned) the simple expression PREORDER(1) can now be used to produce the preorder, and of course that expression can be freely incorporated into other expressions in particular, into ad hoc queries as desired. Also, it s not impossible that the code that implements PREORDER might be provided by the system instead of by some application programmer (meaning PREORDER could become a built-in operator, just like, e.g., JOIN).
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Consider Figure 14-3, which shows in tree form how a certain part namely, part P1 is constructed from certain other parts. To be specific, it shows that part P1 contains parts P2, P3, and P4 as immediate components; part P2 contains parts P5 and P6 as immediate components; and so on. Note: If you d prefer a more concrete example, take part P1 to be a bicycle wheel; then parts P2, P3, and P4 might be a hub, a spoke, and a rim, respectively (and so on).4 Incidentally, note that the tree of Figure 14-3, unlike those we were considering in the previous two sections, is unordered; for example, interchanging the two subtrees rooted at the node for part P2 has no effect (no effect on the interpretation of the tree, that is).
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4. In practice, of course, we d also want to know how many of each immediate component we need for a given part; for example, a bicycle wheel has one hub but many spokes. I omit such considerations here for space reasons.
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CHAPTER 14 TREE-STRUCTURED DATA
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CHAPTER 14 TREE-STRUCTURED DATA
Now I need to confess that I ve been practicing a slight deception on you ... The fact is, part-structure trees like the one we ve been talking about are not, in general, trees at all! That s because the same part might appear at more than one point in the overall structure. Figure 14-5 is a revised version of Figure 14-3 in which part P6 has been replaced by part P4 (and part P8 has therefore been replaced by part P9 as well). While it s true that the revised structure still looks like a tree, it does so only because I ve drawn it with repeated nodes. The fact is, the node for part P4 is now a child with two distinct parents the nodes for part P2 and part P1 and the structure thus violates the definitions I gave for a tree structure earlier in this chapter. Thus, a more accurate rendition of the structure in question is the graph shown in Figure 14-6. A relational analog of that graph is shown in Figure 14-7. Note: The relation in that figure is an example of what s sometimes called a digraph relation, because it can be represented as a graph of nodes and directed edges (as Figure 14-6 indicates).
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CHAPTER 14 TREE-STRUCTURED DATA
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CHAPTER 14 TREE-STRUCTURED DATA
Let r be a binary relation with attributes X and Y, both of the same type T. Then the transitive closure of r is a relation r+ defined as follows: The tuple <x,y> appears in r+ if and only if (a) it appears in r or (b) there exists a value z of type T such that the tuple <x,z> appears in r and the tuple <z,y> appears in r+. In other words, the tuple <x,y> appears in r+ if and only if there s a path in the graph from node x to node y, loosely speaking. Transitive closure is important for the obvious reason that it enables us to obtain a solution to the bill-of-materials problem very directly. Thus, what we need is a way of computing the transitive closure of a relation like MM. The following not very sophisticated Tutorial D code will suffice: OPERATOR TRANCLO ( XY RETURNS RETURN ( WITH ( XY UNION ( ( XY JOIN ( XY RENAME RENAME IF TC = XY THEN TC ELSE TRANCLO END IF ) ; END OPERATOR ; RELATION { X P#, Y P# } ) RELATION { X P#, Y P# } ;
( Y AS Z, X AS Y ) ) ) { X, Z } ( Z AS Y ) ) ) AS TC :
( TC )
If relation MM is as shown in Figure 14-7, the expression TRANCLO ( MM RENAME ( MAJOR_P# AS X , MINOR_P# AS Y ) ) RENAME ( X AS MAJOR_P# , Y AS MINOR_P# ) will now return the result shown in Figure 14-8. Explanation: The following nonrecursive pseudocode might help you understand how TRANCLO works: TC := XY ; do until TC reaches a "fixpoint" ; TC := TC UNION ( XY <*> TC ) ; end ; Loosely speaking, this procedure works by repeatedly forming an intermediate result consisting of the union of (a) the previous intermediate result and (b) the join of XY and that previous intermediate result, until that intermediate result reaches a fixpoint i.e., a point at which it ceases to grow. The expression XY <*> TC is shorthand for join XY and TC over XY.Y and TC.X and project the result over XY.X and TC.Y ; for brevity, I m ignoring here the attribute renaming operations that Tutorial D would require in order to make this operation work. Exercise: Try working through this algorithm (both nonrecursive pseudocode and recursive Tutorial D versions) using the sample data of Figure 14-7. Compare your results with Figure 14-8. Can you see any coding improvements that could be made in the interests of efficiency
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