1.( /5 marks) Suppose that a B+-tree index with the search key (dept_name, building) is available on
relation department. What would be the best way to handle the following selection?
?(building = “Watson”) ? (budget < 55000) ? (dept_name = “Music”)(department)
a. 2 marks for clearly showing an equivalent relation algebra expression using the
appropriate, well identified rule supplied in the textbook (e.g., the derivation with rule #
applied).
b. 1.5 marks for clearly explaining how the B+ tree index would be used to evaluate the
conditions involving attributes that are specified as part of the index
c. 1.5 marks for clearly explaining how the remaining conditions that involve attributes that
are not specified as part of the index would be evaluated
2. ( /5 marks) Show how to derive the following relational algebra equivalences by a sequence of
transformations using the relational algebra equivalence rules in Section 13.2.1. Hint: make sure
you clearly define all rules you use; as well identify which rule is being applied and apply only one
rule for each line in your solution – show your work.
a. ??1??2??3 (E) = ??1 (??2 (??3 (E))) (2 marks)
b. ??1??2 (E1 ??3 E2) = ??1 (E1 ??3 (??2 (E2))), where ?2 involves only attributes from E2 (3
marks)
3. ( /5 marks) Consider the two expressions ?? (E1 E2) and ?? (E1 ? E2).
a. Show using a concrete example that the two expressions are not equivalent in general
(3 marks). Hint: first identify what the difference is between natural join and left outer join
(what is extra in the left outer join), and then make sure you give the concrete relation instances
for E1 and E2 that result in a different expression result.
b. Give a simple condition on the predicate ?, which if satisfied will ensure that the two
expressions are equivalent. Hint: Think about what extra tuples you would have in the left

1
Some of the problems are adapted from “Database System Concepts”, 6th Edition, 2011, Avis Silberschatz, Henry
F. Korth, S. Sudarshan.
outer join, how they joined tuple is constructed, and a condition that will exclude these extra
tuples.
4. ( /5 marks) Explain how to use a histogram to estimate the size of a selection of the form ?A=v(r).
Hint: Divide the distribution into n buckets {rj | j ? [1,n]} and consider the two cases: Part 1) the
selection value v is greater than or equal to the largest value of a given bucket ri (2 marks) and Part 2) the
selection value v is “part way” between the max and min values of bucket ri+1. (3 marks).
5. ( /5 marks) Give an example of a relational algebra expression defining a materialized view and
two situations (for the differentials) such that incremental view maintenance is better than
recomputation (recomputing the materialized view from scratch) in one situation (2.5 marks),
and recomputation is better in the other situation (2.5 marks). Hint: think about different values for
x between 0 and 100, where you insert x% tuples are deleted.

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