Originally posted
here.
In our
previous installment, we saw how to convert Adjacency Lists into Nested Sets using a CTE.
In this episode, we will convert the Adjacency List into a
Nested Intervals
encoding. Specifically, this encoding will make use of the Nested
Intervals with Continued Fractions technique that Tropashko presented
in a
later paper.
The key to this technique lies in using a slightly different form of
materialized path than was used in the last post. Rather than
materializing the EmployeeIds into a path, the path will be created as
an
enumerated
representation, based on sibling ordering. For example, the first
two levels of the AdventureWorks HumanResources.Employee table's tree
look like:
EmployeeId 109 (CEO)
EmployeeId 6 (Marketing Manager)
EmployeeId 12 (VP Engineering)
EmployeeId 42 (IS Manager)
EmployeeId 140 (CFO)
EmployeeId 148 (VP Production)
EmployeeId 273 (VP Sales)
Using the materialized path representation from the previous post, the paths to the second-level employees would be:
6: 109.6
12: 109.12
42: 109.42
140: 109.140
148: 109.148
273: 109.273
However, for this post, paths will instead be materialized based on
sibling ordering. We don't know anything about how siblings
should be ordered in the AdventureWorks employee hierarchy (that's
probably a business rule question, if it matters at all). So
ordering will be done by EmployeeId:
6: 1.1
12: 1.2
42: 1.3
140: 1.4
148: 1.5
273: 1.6
The significance of this encoding is that for each path a rational
number can be generated, using a Euclidian algorithm (described very
well on
this web site).
The algorithm works by iterating over each element on the path,
building a rational number as it goes. The beauty of this, as
shown by Tropashko in his paper, is that by using these paths we can
determine an interval, beween which all children of a given path will
fall.
In order to accomplish getting the path, the ROW_NUMBER function
will be used, but slightly differently than last time. Instead of
creating a second CTE to reference the first, thereby getting the row
number for each element of the path, the ROW_NUMBER function will be
embedded within the recursive CTE itself, as in the following example:
WITH EmployeeRows AS
(
SELECT
EmployeeId,
ManagerId,
ROW_NUMBER() OVER (ORDER BY EmployeeId) AS theRow
FROM HumanResources.Employee
WHERE ManagerId IS NULL
UNION ALL
SELECT
e.EmployeeId,
e.ManagerId,
ROW_NUMBER() OVER (ORDER BY e.EmployeeId) AS theRow
FROM EmployeeRows x
JOIN HumanResources.Employee e ON e.ManagerId = x.EmployeeId
)
SELECT *
FROM EmployeeRows
ORDER BY
ManagerId,
EmployeeId
Interestingly, this example as-is will return
exactly the same results as the following, non-recursive CTE example:
SELECT
EmployeeId,
ManagerId,
ROW_NUMBER() OVER (PARTITION BY ManagerId ORDER BY EmployeeId) AS theRow
FROM HumanResources.Employee
ORDER BY
ManagerId,
EmployeeId
So, why do we care? Take a close look at the two examples.
In the CTE example, the ROW_NUMBER function does not use PARTITION
BY. Yet, results are
implicitly partitioned. This
gives an interesting view into the inner-workings of CTEs. As it
turns out, the recursive part of the CTE is called once per row
returned by the anchor or previous recursion. This is not how I
originally expected CTEs to behave (I thought the recursive part would
be called once per rowset returned by the anchor or previous
recursion), but it does help us with this particular task!
The current row number, at any given point in the recursion, represents
the enumeration for that node. But because we're using a
recursive CTE, we also have access to the parent's enumaration.
For instance, to build an enumerated path, the following T-SQL would be
used:
WITH EmployeeRows AS
(
SELECT
EmployeeId,
ManagerId,
CONVERT(VARCHAR(MAX), ROW_NUMBER() OVER (ORDER BY EmployeeId)) AS thePath
FROM HumanResources.Employee
WHERE ManagerId IS NULL
UNION ALL
SELECT
e.EmployeeId,
e.ManagerId,
x.thePath + '.' +
CONVERT(VARCHAR(MAX), ROW_NUMBER() OVER (ORDER BY e.EmployeeId)) AS
thePath
FROM EmployeeRows x
JOIN HumanResources.Employee e ON e.ManagerId = x.EmployeeId
)
SELECT *
FROM EmployeeRows
ORDER BY
thePath
Note that this sample can be made a bit more readable (and more
functional for later) by embeding the ROW_NUMBER in a derived table:
WITH EmployeeRows AS
(
SELECT
y.EmployeeId,
y.ManagerId,
CONVERT(VARCHAR(MAX), y.theRow) AS thePath
FROM
(
SELECT
EmployeeId,
ManagerId,
ROW_NUMBER() OVER (ORDER BY EmployeeId) AS theRow
FROM HumanResources.Employee
WHERE ManagerId IS NULL
) y
UNION ALL
SELECT
y.EmployeeId,
y.ManagerId,
y.thePath + '.' + CONVERT(VARCHAR(MAX), y.theRow) AS thePath
FROM
(
SELECT
e.EmployeeId,
e.ManagerId,
x.thePath,
ROW_NUMBER() OVER (ORDER BY e.EmployeeId) AS theRow
FROM EmployeeRows x
JOIN HumanResources.Employee e ON e.ManagerId = x.EmployeeId
) y
)
SELECT *
FROM EmployeeRows
ORDER BY
thePath
From here, it's a simple step to implement the Euclidian algorithm. The algorithm is quite simple:
- Set parentNumerator <- 1
- Set parentDenominator <- 0
- Set theElement <- first enumeration in the path
- Set currentNumerator <- theElement
- Set currentDenominator <- 1
- Set theElement <- next enumeration in the path
- Set previousParentNumerator <- parentNumerator
- Set previousParentDenominator <- parentDenominator
- Set parentNumerator <- currentNumerator
- Set parentDenominator <- currentDenominator
- Set currentNumerator <- (parentNumerator * theElement) + previousParentNumerator
- Set currentDenominator <- (parentDenominator * theElement) + previousParentDenominator
- If the current element is not the final node in the path, goto 6.
This seems a bit hairy, but I think that looking at the algorithm
and spending a few minutes with our old friends pencil and paper will
make it quite clear. Also, look at the web site linked
above. Here is how I've implemented the algorithm using a CTE:
WITH EmployeeRows AS
(
SELECT
EmployeeId,
CONVERT(VARCHAR(MAX), theRow) AS thePath,
CONVERT(BIGINT, 1) AS prevNumer,
CONVERT(BIGINT, 0) AS prevDenom,
CONVERT(BIGINT, theRow) AS currNumer,
CONVERT(BIGINT, 1) AS currDenom
FROM
(
SELECT
EmployeeId,
ROW_NUMBER() OVER (ORDER BY EmployeeId) AS theRow
FROM HumanResources.Employee
WHERE ManagerId IS NULL
) y
UNION ALL
SELECT
y.EmployeeId,
y.thePath + '.' + CONVERT(VARCHAR(MAX), y.theRow) AS thePath,
prevNumer = y.currNumer,
prevDenom = y.currDenom,
(y.currNumer * y.theRow) + y.prevNumer AS currNumer,
(y.currDenom * y.theRow) + y.prevDenom AS currDenom
FROM
(
SELECT
e.EmployeeID,
x.thePath,
x.currNumer,
x.currDenom,
x.prevNumer,
x.prevDenom,
ROW_NUMBER() OVER (ORDER BY e.EmployeeID) AS therow
FROM EmployeeRows x
JOIN HumanResources.Employee e ON e.ManagerId = x.EmployeeId
) y
)
Note that in this case I didn't require the use
of the temporary variables; the previous anchor/recursive parts act as
temporary storage enough.
Readers will also hopefully notice that I haven't yet included a SELECT
to get the data from the CTE! This is because I'd like to explain
briefly what it will do. In his paper, Tropashko explains that for
each node, the intervals for the children of that node will fall into
an interval between the encoding of that node (currNumer / currDenom)
and the numerator of the previous node plus the numerator for the
current node, divided by the denominator of the previous node plus the
denominator for the current node ((currNumer + prevNumer) / (currDenom
+ prevDenom)). Quite wordy here. Refer to the paper for a better
explanation and proof.
Anyway, the completed query follows:
WITH EmployeeRows AS
(
SELECT
EmployeeId,
CONVERT(VARCHAR(MAX), theRow) AS thePath,
CONVERT(BIGINT, 1) AS prevNumer,
CONVERT(BIGINT, 0) AS prevDenom,
CONVERT(BIGINT, theRow) AS currNumer,
CONVERT(BIGINT, 1) AS currDenom
FROM
(
SELECT
EmployeeId,
ROW_NUMBER() OVER (ORDER BY EmployeeId) AS theRow
FROM HumanResources.Employee
WHERE ManagerId IS NULL
) y
UNION ALL
SELECT
y.EmployeeId,
y.thePath + '.' + CONVERT(VARCHAR(MAX), y.theRow) AS thePath,
prevNumer = y.currNumer,
prevDenom = y.currDenom,
(y.currNumer * y.theRow) + y.prevNumer AS currNumer,
(y.currDenom * y.theRow) + y.prevDenom AS currDenom
FROM
(
SELECT
e.EmployeeID,
x.thePath,
x.currNumer,
x.currDenom,
x.prevNumer,
x.prevDenom,
ROW_NUMBER() OVER (ORDER BY e.EmployeeID) AS therow
FROM EmployeeRows x
JOIN HumanResources.Employee e ON e.ManagerId = x.EmployeeId
) y
)
SELECT
EmployeeId,
thePath,
currNumer AS startNumer,
currDenom AS startDenom,
currNumer + prevNumer AS endNumer,
currDenom + prevDenom AS endDenom
FROM EmployeeRows
For each node (EmployeeId), you now have an interval (start and end)
within which all children intervals will fall. Note that computation
must be done by the interval, not by the current node's encoding. The
reason becomes apparent when looking at the encodings for 1.1.1 and
1.2. They are the same; however, their intervals do not overlap. As a
matter of fact, encodings will be the same for every next sibling/first
child pair. But the intervals remain nested, and if proper queries are
written there will be no confusion.
So that's a first step towards using the Nested Intervals Model in SQL
Server 2005. Stay tuned for more... And as always, feel free to post
questions or comments. I know some of this material can be confusing
(at least, it was to me before I wrote this post!)