This video gives a live demonstration
of the recursive constructs in sequel
that we introduced in the previous video.
As a reminder, recursion has been
introduced intosequal as part of
the with statement where we can
set up relations that are defined
by queries that themselves refer to
the relation being defined, and finally
we have a query that can
involve the recursively defined relations
as well as other relations or other tables in the database.
The typical expression within a
with statement for a recursively
defined relation would be to
have a base query that doesn't
depend on R and
then a recursive query that does
depend on R. We gave
three examples in the introductory
video and those are the
same examples that we'll be demonstrated shortly.
The first one was to compute
ancestors when we have only a
parent relation and the family
tree could be arbitrarily deep.
Our second example was a
case where we have an arbitrarily
deep company hierarchy and we
want to compute the total salary
cost of a project starting
at that project's manager and summing
the salary of the entire sub-tree.
and our third example was about airplane flights.
Where we want to find the cheapest
way to fly from point A
to point B. And we're willing
to change planes as many kinds,
as we might need to in order to bring down the cost.
We saw that all of these
examples, involved, basically a
notion of transit of closure computed
as a recursively defined relation.
The last portion of our demo, after
we see these three queries solved using recursion, will introduce one more twist,
which is what happens when we introduce cycles.
So in the Airline example.
We'll set up a case where
you can fly from one city to another one and back.
Which is of course true in
reality, and we'll see what
happens when we try to answer our query in that setting.
I've started by creating a table called Parent of.
With parent child relationships.
So we have, Alice Carol, Bob Carol, Carol Dave, and so on.
You might actually want to write
this down on a piece of paper
to see what the actual tree
looks like, but the query
we want to run is to
find all of Mary's ancestors
so we're going to, of course,
have Eve as a parent, and Dave as a parent.
And then Dave's parent is Carol, and Carol's parent is Bob, and so on.
We'll get most of the data in our database in our query.
So here is the query,
our first example of our recursive query.
Let me say right off, that
even more than anything else,
we've done in these videos, I am
going to encourage you to download
the script and take a close
look at the query, and preferable actually
run the queries and play with them on the postscript system.
And we are using for
this demo postscript a sequel
lite and my sequel do not
support forth the Withery cursive statement at this point in time.
So, anyway here's our query
and it is the form
that we described in the introduction,
it's a width-statement with
recursive that's going to
set up a recursive relation called
ancestor, so this is what we were calling "r" earlier.
This is our ancestor with a
schema AD for ancestor and descendant.
Our final query, once
ancestor is all set up,
is very simple, it just
says, take the 'A'
attribute from ancestor, where a
descendant is Mary so that will give us Mary's defendant.
Of course what's interesting is what's
right here,inside these parans
because this our recursive query.
And it does take the form
we described of having a base
query, that is the first
line, and then the recursive
query with a union between them.
So we're going to start by
saying that whenever we have
a parent child relationship that's also an ancestor relationship.
So we're going to take from our parent of table.
The parent and child, and
we have to rename them as
A and D, and that says
that parent children are an ancestor.
What else in an ancestor?
Well if we have a
tuple an ancestor, an ancestor
and a descendant, and that
descendant is the parent of a
another person, then the A
and the ancestor, together with
the child from the parent
of, is also an ancestor relationship.
So this is a kind of doing the join.
Not just "kind of", it's actually
joining our ancestor as its
being created and extending that
relationship by joining with another instance of parent.
So you can kind of think of
going down the ancestor tree
adding relationships as we go down.
Again, I really can't encourage
you enough to download this query
and play with it yourself
to fully understand what's going on.
Let's go ahead and run it.
It's going to be rather anticlimatic.
When we run it we do
discover that these five people
are Mary's ancestors, and if
you've drawn the little tree of
the data you can verify that, that's the correct answer.
Let's play around a little bit.
Let me try a few other people's ancestors.
Let's try Frank.
We don't see Frank here because
Frank actually happens to be
a child of Mary's, so we
should get even more ancestors when
we run this one, especially
Mary should be included, and in
fact, she is, there she is, and these are Frank's ancestors.
Let's try George, I think
George was somewhere in the
middle of the tree there.
Yes, George has three ancestors,
and finally let's try Bob.
Bob is at the root so
we should get an empty result
and we do, because again Bob
has no ancestors in our database.
Now lets take a look at our second example.
That was the one where we had
a hierarchy of management chain
in a company, and then we
were interested in computing the total
salary cost of a project,
so I've set up our three tables.
The first one is the Employee
table, it just gives the IDs
of the employees and the salary of each employee.
The second table is the Manager relationship.
So, again, you might
want to draw the little tree
here although it's pretty simple this time.
123 is at the root
of our little management structure,
as 234 as a subordinate.
234 has two subordinates, 345 and 456 and 345 is another one.
So it's only a
three level tree.
Of course if we knew it was only three levels we wouldn't need recursion at all.
But we're going to write a query that will work for arbitrary numbers of levels.
So that's our management structure.
And finally, our third table,
the Project table says that
employee 123 is the
manager of project X, so
what we want to do is
find the manager of project
Y in the hierarchy,and then take
that manager's salary along with
the salary's of all the
manager's subordinates recursively down
to the management structure, and
of course that's everybody in in our little database.
Again, I can't encourage you
to download the script and run it for yourself.
So here's our query to find
the total salary of project
x and I'm actually going to
give a couple of different ways
of running this for you.
The way we've done it the first
time is to effectively
expand the management structure
into a relation called superior.
So that's really pretty much doing
the ancestor computation, which by the way is a transitive closure.
I should have mentioned that earlier for those of you familiar with transitive closures.
It's basically that operation.
So we're going to compute
these superiors so that
we'll have every manager
and employee relationship with a
manager is arbitrarily above the employee.
And then once we have that
superior relationship computed, then
we write a actually a fairly
complicated query, so this
is the final query of our with statement.
And this one says we've got this recursive relation superior.
We're going to the salaries
from the employee relation where the
ID is either the manager
of the project X, so that's
the first half here, or an
employee, that's managed by the
manager, of project X. Okay.
Now this down here, I just
want to emphasize this is not
recursive, it just so happens
to have that same structure of
union, but there is nothing recursive happening down here.
So this is just a regular sequel query.
Once we have the superior
relation, that's the transitive
closure of the manager relation.
So let's take a look at superior.
Superior, here this is
recursive with the union, says
that if we have a
manager and that's the MID and EID.
Then if somebody is managing someone else then they are their superior.
Notice by the way, I didn't
specify a schema here, so
the schema is implicitly going to be M-I-D-E-I-D.
So we're going to put manager relationships in.
And then, if we have
a superior relationship, so if
we have an MID managing an
EID in the S relationship
then we can add one more
level, because we join
with the managers saying that if
S is a superior
of Y and why is
the manager of Z been X's
superior of Z. This
parallels exactly what we
did with the ancestor computation in the previous example.
Again, it's going to be rather
anti-climactic to run the query, but let's do it.
And, we find out that four
hundred is the total salary
cost of Project X when
we count the manager of project
X together with all of
the people underneath that manager in the hierarchical structure.
So when we think of recursion we
often think of transitive closure
or expanding hierarchies as we've done with our examples so far.
But if we step back for a
second we can see
that there is a quite a bit simpler
way to express the query
that finds the salary burden
of project X. Now
not only is this actually nicer
to look at, it's probably much
more efficient, depending on how
smart the query processor is.
In our previous example, if
the query processor executes the
query in a straight forward way,
it would compute this superior relationship
for the absolute entire company hierarchy
before it figured out
which of those people were involved
in project X. Now, a
really good query processor might actually
figure out to fold in
a project X but, not necessarily.
Here's an example and here's a new formulation of the query.
We're actually going to tie
X specifically to our recursion.
What we're going to compute in
our recursive [xx] statement here.
So this is the temporary relation
we're computing is a relation
containing just a list of
the IDs of the employees
who are involved in project X.
Once we have all the employees
involved in project X, the query down here is trivial.
We just find those employees who
are among the X employees and we sum up their salaries.
So, let's take a look at
the recursive definition here and
again, it's taking the usual
form of a base query,
union and recursive query... and here's what we do.
Well, obviously the manager of
project X is one
of the IDs involved in project
X. So here we find in
the project, the project name text
and we take the manager of that
project and we put that person's ID into Xemps.
That's the first ID that's going to go in there.
That's going to seed the recursion.
That's again the base query.
Then we add in our
recursive step any employee
who is managed by anybody
who's in the X employees.
So, we'll take our manager relationship,
our X employees relationship and
if the employee's manager is an
X, then that employee is
also involved in X. So
we seed the recursion with
the manager of project X
and then we just recursively go
down the tree adding all
of the employees that are underneath one by one.
We don't have to know the depth
of the tree because the recursion will
continue until nobody else is added.
I guess I should have mentioned that earlier in my earlier examples.
Again the recursion sort of
adds a data over and
over again until there's nothing
new to add, and that's when it terminates.
So let's go ahead and run the query.
Anti-climatic again, but we
get the same answer
400 as the salary cost of Project
X. Now we use
the same form of query to
find the total salary cost of
two projects Y and Z.
And that will also demonstrate having two
relations that are defined in the width recursive command.
So I have added project Y
and Z to our project
table, and they're both
managed by employees who
are already in the database, so they're a little lower down the hierarchy.
We should expect those projects have lower total cost.
That's for project X, whose manager was at the root of our hierarchy.
So here's our query, it's a
big one; we're going to
define YM and ZM
exactly as we defined X amps in the previous example.
So Y amps is a table
of a recursively defined relation,
temporary, that's gonna contain
a list of IDs of
the people, the employees that are
involved in Project Y. So
we are going to put the manager of
Project Y as our base
query, and then we're
going to add to it in
the recursion all of the
employees who are managed by
someone who's in the YMs.
And ZM's exactly the same.
We start the manager of project
Z, and then we add
to it, all of the people
are managed transitively down the
tree by someone who's in the ZM's relation.
And then our final query down
here for the statement is a union of two queries.
The first one gets the total
salary for Y and
it labels it as Y total.
So it takes all the Ids that
are in the Y table and from
the employee table get their salaries and sums them up.
And similarly the Z total.
So now we'll run this query,
it will be slightly less is anti-climactic.
We do have now two tuples in our result.
We see that the total salary
for Y is 300 in
the total salaries for Z is 70.
And if you check, cross-check
this result against the data
you'll see that these are
indeed the total salaries
when we take the managers
we specified for projects Y
and Z. And finally our last and most fun example.
The one to find how to
fly from point A to
point be when all we're
concerned about is cost, and
we don't care how many times we have to change planes.
So, here's the little flights table
I've set up, and I
used A and B so
we can literally fly from point
A to point B. All of
our intermediate destinations are actually
real airport codes, and I've
put in some airlines, although they're
not actually going to be used in our query.
And then I've put in the cost of the flights.
You might want to draw
yourself a little graph so
you can see what's going on,
and we can fly from
A to Chicago for 200,
from Chicago to B
for another 100, or we can
go from A to Phoenix and then Phoenix to Las Vegas.
to Las Vegas to oh-oh, I don't remember what this is.
CMH, Detroit, Cincinnati, somewhere in the midwest.
And from there to point
B, or we can take
a non-stop from A to
B on good old Jet Blue for 195.
So clearly we're never
going to be going through Chicago for
a total of 300 with that Jet Blue flight.
but I've set up the
data, as you're probably not
surprised, so that this
long route through Phoenix
and Las Vegas and somewhere in
the midwest is, in fact, gonna be our cheapest way to go.
So now let's take
a look at the recursive query that's
going to find us our
root from point A to
point B, or at least
find us the cheapest way to
get from point A to point
B. So the first
query I'm going to show
actually gives us all the different
costs of getting from A
to B, just so we can see
those enumerated for us, and
then we'll modify the query to give us the cheapest cost.
So here's the recursive query and
we're going to use
a recursively defined relation called root.
And root says that we
can get from an origin to
a destination for a particular total cost.
OK?
So, we again in
our recursion have the base query and the recursive query.
This is exactly what you'd imagine.
We can certainly get from
point X to point Y
for a total cost, if we
can take a direct flight from
point X to point Y for a given cost.
So that's our base query,
we start out with all
of the direct flights in our route relation.
And then we start adding routes
by doing the JOIN of a route with a additional flight.
So basically what this
join here says, "If I
can get from the origin
in a route to the destination and
then that destination is the
origin of another flight, then
I can add that flight.
I can start with my original origin,
final destinationand the cost
of that is going to be the total that I already had.
Plus the cost of the new flight and that's my new total.
So again, this is another
transitive closer like recursion.
It's very similar to the ancestor recursion.
Very similar to expanding the company hierarchy.
The only real difference here
is that we're also accumulating these costs as we do the recursion.
So once we've done this recursion
then we have a
complete specification of all
the roots within our
flights database, all of the
way's we can get from
A to B and the total cost.
Now one thing I should say
is this is not actually giving
us the ways of getting from one place to another.
If we wanted to accumulate
the actual route that we
take so the flights and
the costs and the airlines and
so on, we have to
kind of use a structured structure
inside our data base to accumulate those.
There are ways of doing that
but I'm not going to demonstrate that here.
I'm just going to demonstrate the is
accused of recursion and computing the total cost.
OK, so let's go ahead
and run this query so we've
computed all of the
routes, and then we're just
gonna start by finding the
routes from A to B
and what the total cost of those are.
So we'll run the query and we'll
find out that there are three
ways of getting from A
to B. The first one
happens to be that direct Jet Blue flight for 195.
The second was the
flight through Chicago for a total cost of 300.
You can go back and look at the data and verify these.
And the third one was that
complicated routing where we
stopped several times, but we
save a lot of money, well
twenty dollars over the direct
flight, by going
through those cities because the total sub-cost is 175.
I'll leave it up to
you whether it's worth twenty dollars
to stop several times versus the direct flight.
So now since my actual specification
of what I wanted to know was
the cheapest way to go then
I just say min total instead
of in my final
query and I run
that and my answer is
that 175 is the cheapest way to get
from A to B. Now here
is an alternative formulation of the
same query That essentially parallels
the alternative that we looked
at with our project cross where
we built in project X
into our recursion that simplified
the recursion in that case.
In this case it's not simpler, but it could potentially be more efficient.
We're going to build in the
fact that we're starting from origin
A so instead of
finding all roots from any
point to any other point
in our recursion and then finding
the roots from A to
B, let's create a
relation recursively that says,
"Starting from point A,
I can get to a particular
destination for a particular total cost.
So this is going to build
up roots starting from A
the base query is going
to, of course, start by looking
at direct flights where the
origin is A and is
going to put the destination and the
cost into our relation called
from A. So that starts
with that first gives us
direct start from A where
we can get on the direct, where
we can get to, and how
much it will cost us, and
then our recursion is going
to add flights to that one.
Again, it really parallels what
we did with the Project X.
Our recursion is going to
say, ok we know we
can get a, from particular,
we can get to a particular
place from point A for a certain cost and that's our destination.
If we add 1 more flight so
the origin of that flight
is the destination of where
we can get then that will
also be a destination that
we can get to from point
A and we'll just add the
cost of the additional flight on.
One more time a
strong suggestion that you download and try these things for yourself.
Once we found all the
places we can get from A
then we'll use that to
figure out the cheapest way to
get to Point B. But let's
just start by running the
With statement where all we
do is see the places we can
get to from A and the total cost of getting there.
So here we go.
And we can get to Chicago,
Phoenix or we can get
to B a couple of different
ways, three different ways
actually as we already know.
We can also get to
Las Vegas and this mysterious
CMH I wish I remembered what it were.
So now if we're interested in
finding the cheapest way to get
from A to B then We'll
add where the destination equals
B on here and we'll
add the minimum total cost
and hopefully that will
be our good old 175 and indeed it is.
By the way we can do the same basic idea but backwards.
Instead of finding all the
places that we can get from
city A how about if
we find all the places from
which we can get to city
B. So here's the query that does that.
To B is going to
be our recursively define relation that's
going to give us the origin,
the place from which we
can get to B and the total
cost of getting to B from that place.
So again, the structure is exactly parallel.
We start out with our base
query saying if we have
a direct flight to B
then we can get from the
origin of that direct flight
at the cost of the flight to
B and we then
recursively add flights on
that you can think of if your
going from left to right
adding flights from the left.
So if we know we
can get from a place to B
and then we can go from...take
a direct flight from somewhere
else to that place And we can get from that somewhere else to be.
Anyway so we do that
again by joining so we're
going to take our origin
from which we can get to B
we're going to find flight that take
us to that origin, we're
going to add the cost of that
flight and that gives us
a new way to get
to B. And then let
me start by just writing the
query that all of
the places from which we can get to B and the cost of getting there.
We'll run the query.
And we can see that we
can get to B from point
A in 3 different ways and
from our other cities in our database as well.
Similarly, to what we did
previously, if we're particularly
interested in getting from A
to B, whoops, let's make that
our origin then we add
where origin equals a and
if we want the minimum it would
be our minimum total again paralleling
exactly what we did before.
We run it and good old 175 comes out.
Now we're going to have
some real fun because I
added another flight to our
database, and this flight
takes us from Columbus, I
now know its Columbus, to Phoenix
and creates a loop in our flights.
So that means that we
can fly from A to
B next to Las Vegas, to
Columbus, back to Phoenix
and then to Las Vegas and Columbus again.
So, we're going to have
arbitrarily, actually unbounded, actually
infinite, length routes that we can take now.
Now, obviously those routes aren't
going to ever be the cheapest
way because as we take
those roots it's going to
get more and more expensive,
none of them are negative costs paying us to take flights.
But if we just do
our naive recursion where we
generate all of our roots
before we take a
look at our final query then
we're going to be generating an infinite number of roots.
So here's our original enquiry, the
first one we wrote, where we
were just finding all of the
costs of getting from A
to B by computing all of
the roots in the entire database
and then looking at those from
A to B. Now with
our additional flight that creates
a loop we run this command
and nothing happens.
Actually if we wait long enough we're going to get an error.
Well, okay we waited for a while.
appears that the user
interface we're using isn't going to show us the error.
But if you try running this in
post risk command line interface,
I assure you if you
wait long enough, eventually it
will tell you that the recursion effectively overflowed.
So it is trying to compute
this unbounded number of routes
in the recursive part of the
with statement and never even
gets to the query that we want to execute.
OK, here's my first attempt at fixing the problem.
We know that we're never going
to want to take a arbitrarily long route.
We're never going to want to go
around a cycle lots of
times as our cheapest way
to get from point A to point
B so what I've
done here, and I'm not going
to go into this in great
detail, but I have added
a condition in the recursion
that says "I'm only going
to add a new route, a
new route into my recursively
defined route table when the
total cost of that route, and
that's defined as the cost
plus total here when we
added, is less then
all of the ways we can
already get from that place,
to that origin to that destination.
So in other words, I'm only
going to add cheaper routes
than the ones that are already there.
And by the way, if there
are no routes already from
the origin to the destination then this
will be satisfied and we will
add that first route, then after that only adding cheaper ones.
So let's try running this query and see what happens.
Well, we got an error.
Now this is not a runtime execution error.
This is actually an error
that says we're not allowed
to refer to our recursively
defined relation in a sub
query within our recursion.
The SQL standard actually might
allow this particular use, but
I don't know that any implementation actually handles it.
It can be fairly difficult
to handle a case where
you have the recursively defined
relation in a subquery as well as in the outer query here.
So that's obviously not going to solve our problem.
Now there's actually a feature of
Basic SQL that can help us here with our problem.
There's something called Limit.
We actually didn't discuss this
in the SQL videos, but that
says just give us this number of results.
So let's say that we're
going to have our recursion here
for the roots, but down here
we're going to say I only need
up to 20 results for how
I get from point A to
point B. And the posary
system actually makes use of
the limit command in the final
query to restrict the recursion.
It's a nice feature, and it
was added specifically for this
problem of possibly infinite
recursions where we actually don't
want it to be infinite because we only need a finite number of answers.
Okay, so, let's go with that
here and we'll see that Ah, great.
Everything worked well.
We got our roots from A
to B. And I do have
20 roots, I mean, so they're getting very expensive.
Down here I'm going to go
around and around the mid west
while lots and lots of
times but that did
the limit the recursion it did
stop unlike our query
where we didn't have the limit and it just went on indefinitely.
So that looks pretty good,
with one unfortunate problem,
which is if we still want
the minimum, we're going to
again get a infinite execution.
So the old result
is still sitting here but now
the system is chunking on because
the limit here is applied to
this min to the
number of tupimit the recursion,
we're always going to get only one tuple in our result.
So even if we said limit one
here, we'd still get the
infinite behavior, so we haven't
quite solved our problem.
Okay, so, here's what we're going to do.
Aesthetically, maybe it's not
the absolutely best solution but
I'm going to argue that it's a pretty reasonable one.
We tried limiting our
recursion to only add
new routes that were cheaper
than existing routes to and from the same place.
We weren't allowed to do that
syntactically the recursive with
statement didn't allow the sub
query with the recursively defined relation in it.
So we're going to do a
different change here where
we're not going to add
new roots to our flight
when the length of the
root, in other words the number
of flights contributing to that
root, is greater than
or equal to ten.
So how do we do that?
We're going to add to our
recursively defined relation route
the origin, destination and total cost of that.
And then we are going to add the length.
And so that's going to put
in each root tupple, how
many flights were involved in the root.
So let's see how we do that with our recursion.
We still have the base case
here and the recursively defined union.
In our base case, we're going
to be adding to our route
the non-stop flights, so
we'll have exactly what thought
we had before, and then we'll
have the constant one to say
that this non-stop flight is just one flight.
Then when we do our recursion,
we're joining our route relation
that we're building up by extending
it with an additional flight exactly
as before, but there is two changes here.
One of them is that we're
going to compute our new length
by adding one to
the existing length of the
root for our new root because we're adding one flight.
And then we're only going to
add tupples to the root
relation when the length of
the route that we're adding is
less than ten.
So now let's see what happens.
I'm going to start again by
looking at all of the
costs of getting from point
A to point B and then we'll take to look at finding the least.
So, we'll go ahead and execute
the query and we see
that we have, one,
two, three, four, five ways of
getting from A to B,
where the length of the number
of flights involved is less than or equal to 10.
We see our friends here.
This was the nonstop flight.
This was the one through Boston.
Here's our favorite one and
there's a few more so these
are going to go through that
cycle a couple of times.
But once we get to
the length of ten, we're not going
to add any more, so we've
got termination and if we
want to change it to
find the minimum cost flight,
then it's just the min total
as before, and we'll find good old 175.
Now what 's unaesthetic
about this, is that we're
actually limiting the amount of recursion.
So, the whole point of writing
recursive queries is when we
don't know the minimum
number of computations that we need to do to get our answer.
So maybe it would so
happen to turn out that
more than ten flights were required
to get the cheapest and, if
that was the case, then we wouldn't get our right answer.
Of course, we could change it to
a hundred and we'd still
get the one seventy five.
And, you know, honestly we could
change it to 10,000 and you
can't see it happening here
but it is actually recomputing that
175 even when I put in 10,000.
I can even do a
100,000 and still going to work for me.
So, if we
presume that nobody wants to
take more than a 100,000
flights in order to get
from Point A to Point B
in the cheapest fashion then, this
would be a reasonable way to
bound the recursion and get the answer that we want.
Even in the presence of cycles in our relation.