Answer:

1. select * from R where not a=k

   Since the set of values for each attribute is uniformly distributed, and since there are 10 possible values for the a attribute, we would expect that 1/10 of the tuples would satisfy a=k. This means that 90% of the tuples would not satisfy the condition. Since there are 1000 tuples, we would expect 900 of them to be in the result of the query.

   An alternative formulation of the above

   • Prob(a=k) = 0.1
   • Prob(a!=k) = 1 - Prob(a=k) = 0.9
   • #result tuples = r × Prob(a!=k) = 1000 × 0.9 = 900

2. select * from R where a=k and b=j

   If attributes are independent, then we'd expect that the chance of attribute R.a having a specific value was 1/10 and the chance of attribute R.b then have a particular value was 1/100 of that (i.e. the likelihoods multiply).

   Using the probability-based formulation:

   • Prob(a=k) = 0.1,   Prob(b=j) = 0.01
   • Prob(A=k and B=j) = Prob(A=k) × Prob(B=j) = 0.001
   • #result tuples = r × Prob(A=k and B=j) = 1000 × 0.001 = 1

3. select * from R where a in (k,l,m,n)

   If we have alternatives for possible values for attribute R.a then we simply sum up the chances for any one of the values to occur.

   Using the probability-based formulation:

   • Prob(a=x) = 0.1, where x is any value in domain(a)
   • Prob(a∈{x,y,...}) = Prob(a=x) + Prob(a=y) + ...
   • Prob(a in (k,l,m,n)) = 4 × Prob(a=k) (assuming uniform distribution)
   • #result tuples = r × Prob(a in (k,l,m,n)) = 1000 × 0.4 = 400

create table Item (
    iname       text,
    category    text,
    primary key (name)
);
create table Store (
    sname       text,
    city        text,
    street      text,
    primary key (city,street)
);
create table Transaction (
    item        text references Item(iname),
    store       text references Store(sname),
    tdate       date,
    primary key (item,store,tdate)
);

select category,city
from   Item, Store, Transaction
where  iname=item and store=sname and
       tdate='20-12-2004' and city='Sydney';

JoinResult   = Item Join[iname=item] Transaction Join[store=sname] Store
SelectResult = Sel[tdate='20-12-2004' and city='Sydney'](JoinResult)
FinalResult  = Proj[category,city](SelectResult)

Answer:

The three most promising relation algebra expressions:

Exp1:
    Temp1   = Sel[tdate='20-12-2004'](Transaction)
    Temp2   = Sel[city='Sydney'](Store)
    Temp3   = Temp1 Join[store=sname] Temp2
    Temp4   = Item Join[iname=item] Temp3
    Result  = Proj[category,city](Temp4)

Exp2:
    Temp1   = Sel[city='Sydney'](Store)
    Temp2   = Transaction Join[store=sname] Temp1
    Temp3   = Sel[tdate='20-12-2004'](Temp2)
    Temp4   = Item Join[iname=item] Temp3
    Result  = Proj[category,city](Temp4)

Exp2:
    Temp1   = Sel[tdate='20-12-2004'](Transaction)
    Temp2   = Temp1 Join[store=sname] Store
    Temp3   = Sel[city='Sydney'](Temp2)
    Temp4   = Item Join[iname=item] Temp3
    Result  = Proj[category,city](Temp4)

The most efficient relational algebra expression is a variation on Exp2:

Best:
    Temp1   = Sel[city='Sydney'](Store)
    Temp2   = Temp1 Join[sname=store] Transaction
    Temp3   = Sel[tdate='20-12-2004'](Temp2)
    Temp4   = Temp3 Join[item=iname] Item
    Result  = Proj[category,city](Temp4)

Costs for the various operations (assuming pipelining, so no writes):

1. Cost of Temp1: 2 + 2 = 4
   Assume 10 tuples/city (uniform distribution). Since there are 5 tuples/page and the file is sorted on city, these 10 Sydney tuples should appear in 2 pages (worst-case 3 pages). Cost of traversing B+tree to find the first matching tuple is 2 (since tree has depth 2), plus cost of reading 2 data pages, gives a total of 4 page reads.

2. Cost of Temp2+Temp3: 10 * 3 = 30
   Use indexed nested loop join and combine it with the selection on tdate as follows: the index on Transaction is on the pair of attributes (store,tdate); the join condition involves just store; however, the selection also gives us the tdate value; so, for each store, we can form an index key (store,tdate) and do an index lookup. Since there are 10 tuples in the result of Temp1, we will perform 10 index lookups on Transaction. Each index lookup will yield 10 results (we are given that 10 items are sold in each store on each day). Since the Transaction data file is sorted (clustered) by store (then tdate), and there are 25 tuples per page, all 10 tuples will likely be on a single page (worst case 2 pages). The B+tree index is depth 2, so there are 2 index page reads and 1 data page read for each index lookup (i.e. 3 page reads). Since we perform 10 index lookups, this gives a total of 30 reads, which produces 100 tuples.

3. Cost of Temp4: 100 * 1 = 100
   The join between Temp3 and Item can also use an index nested loop join, based on the fact that Item is hashed on the iname value, and each tuple from Temp3 gives us an item value to use as a hash key. For each of the 100 tuples from Temp3, we do a hash access to the page containing the matching tuple (there will be only 1 matching tuple, since iname is a key). This requires exactly 100 page reads.

Total cost is the sum of these: 4 + 30 + 100 = 134 page reads.

Note that we ignore the cost of the projection in the above calculation. We can do this because we generally ignore the cost of writing results (especially in the context of comparing relational algebra expressions for a given query, since all expressions produce the same result). Also, we don't need to worry about filtering duplicates in this case, since the query didn't ask for it (i.e. not select distinct).