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| DATE | 2014-12-12 |
| FROM | Ruben Safir
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| SUBJECT | Subject: [LIU Comp Sci] Fwd: Re: relational algrabra division?
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Subject: Re: relational algrabra division?
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Xref: panix comp.databases.theory:75273
Relational division is one of the eight basic operations in Codd's
relational algebra. The idea is that a divisor table is used to
partition a dividend table and produce a quotient or results table. The
quotient table is made up of those values of one column for which a
second column had all of the values in the divisor. There is a really
good presentation on four ways to do this at:
http://www.cs.arizona.edu/people/mccann/research/divpresentation.pdf
This is easier to explain with an example. We have a table of pilots and
the planes they can fly (dividend); we have a table of planes in the
hangar (divisor); we want the names of the pilots who can fly every
plane (quotient) in the hangar. To get this result, we divide the
PilotSkills table by the planes in the hangar.
CREATE TABLE PilotSkills
(pilot CHAR(15) NOT NULL,
plane CHAR(15) NOT NULL,
PRIMARY KEY (pilot, plane));
PilotSkills
pilot plane
=========================
'Celko' 'Piper Cub'
'Higgins' 'B-52 Bomber'
'Higgins' 'F-14 Fighter'
'Higgins' 'Piper Cub'
'Jones' 'B-52 Bomber'
'Jones' 'F-14 Fighter'
'Smith' 'B-1 Bomber'
'Smith' 'B-52 Bomber'
'Smith' 'F-14 Fighter'
'Wilson' 'B-1 Bomber'
'Wilson' 'B-52 Bomber'
'Wilson' 'F-14 Fighter'
'Wilson' 'F-17 Fighter'
CREATE TABLE Hangar
(plane CHAR(15) NOT NULL PRIMARY KEY);
Hangar
plane
=============
'B-1 Bomber'
'B-52 Bomber'
'F-14 Fighter'
PilotSkills DIVIDED BY Hangar
pilot
=============================
'Smith'
'Wilson'
In this example, Smith and Wilson are the two pilots who can fly
everything in the hangar. Notice that Higgins and Celko know how to fly
a Piper Cub, but we don't have one right now. In Codd's original
definition of relational division, having more rows than are called for
is not a problem.
The important characteristic of a relational division is that the CROSS
JOIN (Cartesian product) of the divisor and the quotient produces a
valid subset of rows from the dividend. This is where the name comes
from, since the CROSS JOIN acts like a multiplication operator.
Division with a Remainder
There are two kinds of relational division. Division with a remainder
allows the dividend table to have more values than the divisor, which
was Codd's original definition. For example, if a pilot can fly more
planes than just those we have in the hangar, this is fine with us. The
query can be written in SQL-89 as
SELECT DISTINCT pilot
FROM PilotSkills AS PS1
WHERE NOT EXISTS
(SELECT *
FROM Hangar
WHERE NOT EXISTS
(SELECT *
FROM PilotSkills AS PS2
WHERE (PS1.pilot = PS2.pilot)
AND (PS2.plane = Hangar.plane)));
The quickest way to explain what is happening in this query is to
imagine an old World War II movie where a cocky pilot has just walked
into the hangar, looked over the fleet, and announced, "There ain't no
plane in this hangar that I can't fly!" We are finding the pilots for
whom there does not exist a plane in the hangar for which they have no
skills. The use of the NOT EXISTS() predicates is for speed. Most SQL
systems will look up a value in an index rather than scan the whole
table. The SELECT * clause lets the query optimizer choose the column to
use when looking for the index.
This query for relational division was made popular by Chris Date in his
textbooks, but it is not the only method nor always the fastest. Another
version of the division can be written so as to avoid three levels of
nesting. While it is not original with me, I have made it popular in my
books.
SELECT PS1.pilot
FROM PilotSkills AS PS1, Hangar AS H1
WHERE PS1.plane = H1.plane
GROUP BY PS1.pilot
HAVING COUNT(PS1.plane) = (SELECT COUNT(plane) FROM Hangar);
There is a serious difference in the two methods. Burn down the hangar,
so that the divisor is empty. Because of the NOT EXISTS() predicates in
Date's query, all pilots are returned from a division by an empty set.
Because of the COUNT() functions in my query, no pilots are returned
from a division by an empty set.
In the sixth edition of his book, INTRODUCTION TO DATABASE SYSTEMS
(Addison-Wesley; 1995 ;ISBN 0-201-82458-2), Chris Date defined another
operator (DIVIDEBY ... PER) which produces the same results as my query,
but with more complexity.
Exact Division
The second kind of relational division is exact relational division. The
dividend table must match exactly to the values of the divisor without
any extra values.
SELECT PS1.pilot
FROM PilotSkills AS PS1
LEFT OUTER JOIN
Hangar AS H1
ON PS1.plane = H1.plane
GROUP BY PS1.pilot
HAVING COUNT(PS1.plane) = (SELECT COUNT(plane) FROM Hangar)
AND COUNT(H1.plane) = (SELECT COUNT(plane) FROM Hangar);
This says that a pilot must have the same number of certificates as
there planes in the hangar and these certificates all match to a plane
in the hangar, not something else. The "something else" is shown by a
created NULL from the LEFT OUTER JOIN.
Please do not make the mistake of trying to reduce the HAVING clause
with a little algebra to:
HAVING COUNT(PS1.plane) = COUNT(H1.plane)
because it does not work; it will tell you that the hangar has (n)
planes in it and the pilot is certified for (n) planes, but not that
those two sets of planes are equal to each other.
Note on Performance
The nested EXISTS() predicates version of relational division was made
popular by Chris Date's textbooks, while the author is associated with
popularizing the COUNT(*) version of relational division. The Winter
1996 edition of DB2 ON-LINE MAGAZINE (http://www.db2mag.com/96011ar:htm)
had an article entitled "Powerful SQL:Beyond the Basics" by Sheryl
Larsen which gave the results of testing both methods. Her conclusion
for DB2 was that the nested EXISTS() version is better when the quotient
has less than 25% of the dividend table's rows and the COUNT(*) version
is better when the quotient is more than 25% of the dividend table.
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