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IV Data Storage and Querying
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146 Show that the following equivalences hold Explain how you can apply then to improve the ef ciency of certain queries: a E1 1 (E2 E3 ) = (E1 1 E2 E1 1 E3 ) b ( A GF (E)) = A GF ( (E)), where uses only attributes from A c (E1 1 E2 ) = (E1 ) 1 E2 where uses only attributes from E1
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147 Show how to derive the following equivalences by a sequence of transformations using the equivalence rules in Section 1431 a 1 2 3 (E) = 1 ( 2 ( 3 (E))) b 1 2 (E1 1 3 E2 ) = 1 (E1 1 3 ( 2 (E2 ))), where 2 involves only attributes from E2 148 For each of the following pairs of expressions, give instances of relations that show the expressions are not equivalent a A (R S) and A (R) A (S) b B<4 ( A Gmax (B) (R)) and A Gmax (B) ( B<4 (R)) c In the preceding expressions, if both occurrences of max were replaced by min would the expressions be equivalent d (R 1 S) 1 T and R 1 (S 1 T ) In other words, the natural left outer join is not associative (Hint: Assume that the schemas of the three relations are R(a, b1), S(a, b2), and T (a, b3), respectively) e (E1 1 E2 ) and E1 1 (E2 ), where uses only attributes from E2 149 SQL allows relations with duplicates ( 4) a De ne versions of the basic relational-algebra operations , , , 1, , , and that work on relations with duplicates, in a way consistent with SQL b Check which of the equivalence rules 1 through 7b hold for the multiset version of the relational-algebra de ned in part a
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1410 Show that, with n relations, there are (2(n 1))!/(n 1)! different join orders Hint: A complete binary tree is one where every internal node has exactly two children Use the fact that the number of different complete binary trees 1 with n leaf nodes is n 2(n 1) (n 1) If you wish, you can derive the formula for the number of complete binary trees with n nodes from the formula for the number of binary trees with n 1 nodes The number of binary trees with n nodes is n+1 2n ; this number is n known as the Catalan number, and its derivation can be found in any standard textbook on data structures or algorithms 1411 Show that the lowest-cost join order can be computed in time O(3n ) Assume that you can store and look up information about a set of relations (such as the optimal join order for the set, and the cost of that join order) in constant time (If you nd this exercise dif cult, at least show the looser time bound of O(22n ))
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IV Data Storage and Querying
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The McGraw Hill Companies, 2001
Bibliographical Notes
1412 Show that, if only left-deep join trees are considered, as in the System R optimizer, the time taken to nd the most ef cient join order is around n2n Assume that there is only one interesting sort order 1413 A set of equivalence rules is said to be complete if, whenever two expressions are equivalent, one can be derived from the other by a sequence of uses of the equivalence rules Is the set of equivalence rules that we considered in Section 1431 complete Hint: Consider the equivalence 3=5 (r) = { } 1414 Decorrelation: a Write a nested query on the relation account to nd for each branch with name starting with B , all accounts with the maximum balance at the branch b Rewrite the preceding query, without using a nested subquery; in other words, decorrelate the query c Give a procedure (similar that that described in Section 1445) for decorrelating such queries 1415 Describe how to incrementally maintain the results of the following operations, on both insertions and deletions a Union and set difference b Left outer join 1416 Give an example of an expression de ning a materialized view and two situations (sets of statistics for the input relations and the differentials) such that incremental view maintenance is better than recomputation in one situation, and recomputation is better in the other situation
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