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### Lecture25

Course: COP 5725, Spring 2012
School: University of Florida
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Word Count: 790

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you What should have learned after this lecture ... how algebraic optimization is done rule 8: permutation of a selection with a join or a cross product, if it only uses attributes of one of the two operand relations. { , } : F(R1 R2) = F(R1) R2 (attr(F) R1) rule 9: permutation of a selection with a join or a cross product { , } : F(R1 R2) = F1(R1) F2(R2), if F = F1 F2, attr(F1) R1, attr(F2) R2...

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you What should have learned after this lecture ... how algebraic optimization is done rule 8: permutation of a selection with a join or a cross product, if it only uses attributes of one of the two operand relations. { , } : F(R1 R2) = F(R1) R2 (attr(F) R1) rule 9: permutation of a selection with a join or a cross product { , } : F(R1 R2) = F1(R1) F2(R2), if F = F1 F2, attr(F1) R1, attr(F2) R2 rule 10: permutation of a projection with a join A(R1 F R2) = A(A1(R1) F A2(R2)) with A1 = {a | a R1 A} {a | a R1 attr(F)} and A2 = {a | a R2 A} {a | a R2 attr(F)} Join attributes must be maintained until the join is executed. rule 11: permutation of a projection with a cross product A(R1 R2) = A1(R1) A2(R2) with A = A1 A2, A1 A2 = , A1 R1, A2 R2 rule 12: permutation of a selection with the set operations union, intersection and difference {, , } : F(R1 R2) = F(R1) F(R2) rule 13: permutation of a projection with the union operator A(R1 R2) = A(R1) A(R2) permutation with the intersection operator and the difference operator inadmissible rule 14: combination of a selection and a cross product to a join, if the selection condition represents a join condition, i.e., if the condition compares attributes of one argument relation with the attributes of the other relation. For an equi join holds e.g.: R1.A1=R2.A2(R1 R2) = R1 R1.A1=R2.A2 R2 rule 15: DeMorgans laws (F1 F2) = F1 F2 (F1 F2) = F1 F2 needed for the formation of cascades of selections, for the shift of negations from outside to inside A simple algorithm for algebraic optimization The following algorithm essentially performs algebraic optimization and additionally summarizes groups of operations that can be combined for a sequential processing of the operand(s). The algorithm only shows a part of the optimization problem. Features are: only sequential processing relations, no consideration of existing access paths no generation of alternative access plans no cost estimation and no choice of an access plan output of the algorithm is a program which consists of the following kinds of steps: application of a selection or a projection application of a selection and a projection application of cartesian product, union, difference to two operands, possibly with previous and following selection and/or projection algorithm AlgebraicOptimization // input: operator tree representation of an expression of the relational algebra // output: program for evaluating this expression step 1: Subdivide a cascade of selections F1F2...Fn(R) single into selections according to rule 4. step 2: Shift each selection according to the rules 1, 2, 3, 6, 8, 9, 12, 15 as far as possible downward in the tree. step 3: Apply the rules 1, 2, 5, 6, 7, 10, 11, 13 to shift projections as far as possible downward in the tree. Eliminate projections to all attributes of its operand. step 4: Apply the rules 4, 5, 6, 7 to transform cascades of selections and projections into a single selection, a single projection, or a selection followed by a projection. step 5: Apply rule 14 to summarize selections and cross products to joins. step 6: Combine the inner nodes of the resulting tree to groups. Each binary, inner node (, , ) is attached with its direct predecessors , , furthermore with each subtree (son) which has the form of a chain of unary operators followed by a leaf. Exception: If the inner node is a cartesian product without subsequent selection (i.e., not a join), then evaluate the chain of unary operators separately. step 7: Create a program which contains a statement for the evaluation of each group. The evaluation order shall be that no group is evaluated before its successors in the tree. example relations books(title, author, pcompany, bookno) publisher(pcompany, paddr, pcity) borrower(name, addr, city, bno) borrowed(bno, bookno, date) We consider the view Xborrowed = A(F(borrowed borrower books)) with A = {title, author, pcompany, bookno, name, addr, city, bno, date} and F = (borrower.bno = borrowed.bno books.bookno = borrowed.bookno) With respect to this view the following query is posed: title(date<1.1.2001(Xborrowed)) We obtain the tree title and after step 2: title date<1.1.2001 title, author, ..., date title, author, ..., date books.bookno = borrowed.bookno borrower.bno = borrowed.bno books.bookno = borrowed.bookno borrower.bno = borrowed.bno borrowed books borrower date<1.1.2001 borrowed borrower books After step 3 when applying rule 5 we obtain: title books.bookno = borrowed.bookno borrower.bno = borrowed.bno books date<1.1.2001 borrower borrowed After step 3 when applying the rules 1, 7, 11 we obtain: (see next page) step 4: not applicable. step 5: not performed/drawn step 6: see next page step 7: Write a program segment for each group. Execute the program segment for the lower group first. title books.bookno = borrowed.bookno borrowed.bookno title, books.bookno borrower.bno = borrowed.bno books borrowed.bookno, borrowed.bno borrower.bno date<1.1.2001 borrower borrowed
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