1. Yue (Jenny) Cui and William Perrizo North Dakota State University
Aggregate Function Computation and Iceberg Querying in Vertical Databases
Yue (Jenny) Cui and William Perrizo
North Dakota State University

2. Outline
Introduction
Review of Aggregate Functions
Review of Iceberg Queries
Algorithms of Aggregate Function Computation Using P-trees
SUM, COUNT, and AVERAGE.
MAX, MIN, MEDIAN, RANK, and TOP-K.
Iceberg Query Operation Using P-trees
An Iceberg Query Example
Performance Analysis
Conclusion

3. Introduction
Commonly used aggregation functions include COUNT, SUM, AVERAGE, MIN, MAX, MEDIAN, RANK, and TOP-K.
Iceberg queries perform aggregate functions across attributes and then eliminate aggregate values that are below some specified threshold.
We use an example to review iceberg queries.
SELECT Location, Product Type, Sum (# Product)
FROM Relation Sales
GROUPBY Location, Product Type
HAVING Sum (# Product) >= T

4. Introduction (Cont.)
We illustrate the procedure of calculating by three steps.
Step one: Generate Location-list.
SELECT Location, Sum (# Product)
FROM Relation Sales
GROUPBY Location
HAVING Sum (# Product) >= T
Step Two: Generate Product Type-list.
SELECT Type, Sum (# Product)
FROM Relation Sales
GROUPBY Product Type
HAVING Sum (# Product) >= T
Step Three: Generate location & Product Type pair groups.
From the Location-list and the Type-list we generated in first two steps, we can eliminate many of the location & Product Type pair groups

5. Algorithms of Aggregate Function Computation Using P-trees
The dataset we used in our example.
We use the data in relation Sales to illustrate algorithms of aggregate function. Id
Mon
Loc
Type
On line
# Product 1
Jan
New York
Notebook Y 10 2
Minneapolis
Desktop N 5 3
Feb
Printer 6 4
Mar 7 11
Chicago 9
Apr
Fax
Table 1. Relation Sales.

6. Algorithms of Aggregate Function Computation Using P-trees (Cont.)
Table 2 shows the binary representation of data in relation Sales. Id
Mon
Loc
Type
On line
# Product
P0,3 P0,2 P0,1 P0,0
P1,4 P1,3 P1,2 P1,1 P1,0
P2,2 P2,1 P2,0
P3,0
P4,3 P4,2 P4,1 P4,0 1
0001
00001
001
1010 2
00101
010
0101 3
0010
100
0110 4
0011
0111 5
1011 6
00110
1001 7
0100
101
Table 2. Binary Form of Sales.

7. Algorithm of Aggregate Function COUNT
COUNT function: It is not necessary to write special function for COUNT because P-tree RootCount function has already provided the mechanism to implement it. Given a P-tree Pi, RootCount(Pi) returns the number of 1s in Pi. Id
Mon
Loc
Type
On line
# Product
P0,3 P0,2 P0,1 P0,0
P1,4 P1,3 P1,2 P1,1 P1,0
P2,2 P2,1 P2,0
P3,0
P4,3 P4,2 P4,1 P4,0 1
0001
00001
001
1010 2
00101
010
0101 3
0010
100
0110 4
0011
0111 5
1011 6
00110
1001 7
0100
101
Table 1. Relation Sales.

8. Algorithm of Aggregate Function SUM
SUM function: Sum function can total a field of numerical values.
Algorithm 4.1 Evaluating sum () with P-tree.
total = 0.00;
For i = 0 to n {
total = total + 2i * RootCount (Pi); }
Return total
Algorithm Sum Aggregate

9. Algorithm of Aggregate Function SUM
P4, P4, P4, P4,0 10 5 6 7 11 9 3 1 1 1 1
For example, if we want to know the total number of products which were sold out in relation Sales, the procedure is showed on left
{3}
{3}
{5}
{5}
23 * * * * = 51

10. Algorithm of Aggregate Function AVERAGE
Average function: Average function will show the average value in a field. It can be calculated from function COUNT and SUM.
Average () = Sum ()/Count ().

11. Algorithm of Aggregate Function MAX
Max function: Max function returns the largest value in a field.
Algorithm 4.2 Evaluating max () with P-tree.
max = 0.00;
c = 0;
Pc is set all 1s
For i = n to 0 {
c = RootCount (Pc AND Pi);
If (c >= 1)
Pc = Pc AND Pi;
max = max + 2i; }
Return max;
Algorithm Max Aggregate.

12. Algorithm of Aggregate Function MAX
Steps IF Pos Bits
P4, P4, P4, P4,0
1. Pc = P4,3
RootCount (Pc) = >= 1 10 5 6 7 11 9 3 1 1 1 1
{1}
2. RootCount (Pc AND P4,2) = 0 <
Pc = Pc AND P’4,2
{0}
3. RootCount (Pc AND P4,1 ) = 2 >= 1
Pc = Pc AND P4,1
{1}
4. RootCount (Pc AND P4,0 ) = 1 >= 1
{1}
23 * * * * =
{1}
{0}
{1}
{1} 11

13. Algorithm of Aggregate Function MIN
Min function: Min function returns the smallest value in a field.
Algorithm Evaluating Min () with P-tree.
min = 0.00;
c = 0;
Pc is set all 1s
For i = n to 0 {
c = RootCount (Pc AND NOT (Pi));
If (c >= 1)
Pc = Pc AND NOT (Pi);
Else
min = min + 2i; }
Return min;
Algorithm Max Aggregate.

14. Algorithm of Aggregate Function MIN
Steps IF Pos Bits
P4, P4, P4, P4,0
1. Pc = P’4,3
RootCount (Pc) = >= 1 10 5 6 7 11 9 3 1 1 1 1
{0}
2. RootCount (Pc AND P’4,2) = 1 >= 1
Pc = Pc AND P’4,2
{0}
3. RootCount (Pc AND P’4,1 ) = 0 < 1
Pc = Pc AND P4,1
{1}
4. RootCount (Pc AND P’4,0 ) = 0 < 1
{1}
23 * * * * =
{0}
{0}
{1}
{1} 3

15. Algorithms of Aggregate Function MEDIAN and RANK
Algorithm 4.4. Evaluating Median () with P-tree
median = 0.00;
pos = N/2; for rank pos = K;
c = 0;
Pc is set all 1s for single attribute
For i = n to 0 {
c = RootCount (Pc AND Pi);
If (c >= pos)
median = median + 2i;
Pc = Pc AND Pi;
Else
pos = pos - c;
Pc = Pc AND NOT (Pi); }
Return median;
Median function returns the median value in a field.
Rank (K) function returns the value that is the kth largest value in a field.
Algorithm Median Aggregate.

16. Algorithm of Aggregate Function MEDIAN
Steps IF Pos Bits
P4, P4, P4, P4,0
1. Pc = P4,3
RootCount (Pc) = < 4
Pc = P’4,3
pos = 4 – 3 = 1 10 5 6 7 11 9 3 1 1 1 1
{0}
2. RootCount (Pc AND P4,2) = 3 >= 1
Pc = Pc AND P4,2
{1}
3. RootCount (Pc AND P4,1 ) = 2 >= 1
Pc = Pc AND P4,1
{1}
4. RootCount (Pc AND P4,0 ) = 1 >= 1
{1}
23 * * * * =
{0}
{1}
{1}
{1} 7

17. P P P P0
Rank = 3
1. Count = 3
3 >= 3 B3 = 1
2. Count = 1
1 < B2 = 0 r = r-c = 3-1 = 2
3. Count = 2
2 >= 2 B1 = 1
4. Count = 1
1 < B0 = 0
Rank = 4
1. Count = 3
3 < B3 = 0 r = r – c = 1
2. Count = 4
4 > B2 = 1
3. Count = 2
2 >= 1 B1 = 1
4. Count = 1
1 >= 1 B0 = 1 7 3 11 1 5 14 10 6 4 P3
R=4 C=3
C<R P3
R=3 C=3
C>=R
root
P3’P2
R=1 C=4
C>=R
P3P2
R=3 C=1
C<R
P3 P2 P1 P0
P3 P2 P1 P0
P P3’
P3’P2P1
R=1 C=2
C>=R
P3P2’P1
R=2 C=2
C>=R
P3P P3P2’ P3’P P3’P2’
P3’P2P1P0
R=1 C=1
C>=R
P3P2’P1P0
R=2 C=1
C<R
P3P2P P3P2P1’ P3P2’P P3P2’P1’ P3’P2P P3’P2P1’ P3’P2’P P3’P2’P1’

18. Algorithm of Aggregate Function TOP-K
Top-k function: In order to get the largest k values in a field, first, we will find rank k value Vk using function Rank (K).
Second, we will find all the tuples whose values are greater than or equal to Vk. Using ENRING technology of P-tree

19. Performance Analysis
Figure 15. Iceberg Query with multi-attributes aggregation Performance Time Comparison

20. Performance Analysis
Our experiments are implemented in the C++ language on a 1GHz Pentium PC machine with 1GB main memory running on Red Hat Linux.
In figure 15, we compare the running time of P-tree method and bitmap method on calculating multi-attribute iceberg query. In this case P-trees are proved to be substantially faster.

21. Conclusion
we believe our study confirms that the P-tree approach is superior to the bitmap approach for aggregation of all types and multi-attribute iceberg queries.
It also proves that the advantages of basic P-tree representations of files are:
First, there is no need for redundant, auxiliary structures.
Second basic P-trees are good at calculating multi-attribute aggregations, numeric value, and fair to all attributes.