1. 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

2. 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 S, the procedure is showed on left
{3}
{3}
{5}
{5}
23 * * * * = 51

3. 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 ().

4. 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.

5. 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

6. 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.

7. Algorithm of Aggregate Function MIN
Steps IF Pos Bits
P4, P4, P4, P4,0
1. Pc = P’4,3
RootCount (Pc) = 4 > = 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

8. 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/Rank: 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.

9. 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

10. 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

11. Iceberg Query Operation Using P-rees
We demonstrate the computation procedure of iceberg querying with the following example:
SELECT Loc, Type, Sum (# Product)
FROM Relation S
GROUPBY Loc, Type
HAVING Sum (# Product) >= 15

12. Iceberg Query Operation Using P-trees (Step One)
Step one: We build value P-trees for the 4 values, {Loc| New York, Minneapolis, Chicago}, of attribute Loc.
PMN 1
PNY
PCH
Figure 4. Value P-trees of Attribute Loc

13. Iceberg Query Operation Using P-trees (Step One)
Figure 5 illustrates the calculation procedure of value P-tree PNY. Because the binary value of New York is 00001, we will get formula 1.
PNY = P’1,4 AND P’1,3 AND P’1,2 AND P’1,1 AND P1, (1)
LOC
P1,4 P1,3 P1,2 P P P’1,4 P’1,3 P’1,2 P’ P PNY 1
Figure 5. Procedure of Calculating PNY

14. Iceberg Query Operation Using P-trees (Step One)
After getting all the value P-trees for each location, we calculate the total number of products sold in each place. We still use the value, New York, as our example.
Sum(# product | New York) =
23 * RootCount (P4,3 AND PNY) +
22 * RootCount (P4,2 AND PNY) +
21 * RootCount (P4,1 AND PNY) +
20 * RootCount (P4,0 AND PNY)
= 8 * * * * 1 = (2)

15. Iceberg Query Operation Using P-trees (Step One)
Table 3 shows the total number of products sold out in each of the three of the locations. Because our threshold T is 15, we eliminate the city Chicago.
Loc Values
Sum (# Product)
Threshold
New York 23 Y
Minneapolis 18
Chicago 9 N
Table 3. the Summary Table of Attribute Loc.

16. Iceberg Query Operation Using P-trees (Step Two)
Step two: Similarly we build value P-trees for every value of attribute Type. Attribute Type has four values {Type | Notebook, desktop, Printer, Fax}. Figure 6 shows the value P-tree of the four values of attribute Type. 1
PNotebook PDesktop PPrinter PFAX
Figure 6. Value P-trees of Attribute Type.

17. Iceberg Query Operation Using P-trees (Step Two)
Similarly we get the summary table for each value of attribute Type.
According to the threshold, T equals 15, only value P-tree of notebook will be used in the future.
Type Values
Sum (# Product)
Threshold
Notebook 28 Y
Desktop 14 N
FAX 3
Printer 6
Table 4. Summary Table of Attribute Type.

18. Iceberg Query Operation Using P-trees (Step Three)
Step three: We only generate candidate Loc and Type pairs for local store and Product type, which can pass the threshold T. By Performing And operation on PNY with PNotebook, we obtain value P-tree
PNY AND Notebook
PNY PNotebook PNY AND Notebook 1 1 1 =
AND
Figure 7. Procedure of Calculating PNY AND Notebook

19. Iceberg Query Operation Using P-trees (Step Three)
We calculate the total number of notebooks sold out in New York by formula 3.
Sum(# Product | New York) =
23 * RootCount (P4,3 AND PNY AND Notebook) +
22 * RootCount (P4,2 AND PNY AND Notebook) +
21 * RootCount (P4,1 AND PNY AND Notebook) +
20 * RootCount (P4,0 AND PNY AND Notebook)
= 8 * * * 2 + 1* 1 = (3)

20. Iceberg Query Operation Using P-trees (Step Three)
By performing And operations on PMN with
P Notebook, we obtain value P-tree PMN AND Notebook 1
PMN PNotebook PMN AND Notebook
AND =
Figure 8. Procedure of Calculating PMN AND Notebook

21. Iceberg Query Operation Using P-trees (Step Three)
We calculate the total number of notebook sold out in Minneapolis by formula 4.
Sum (# product | Minneapolis) =
23 * RootCount (P4,3 AND PMN AND Notbook) +
22 * RootCount (P4,2 AND PMN AND Notbook) +
21 * RootCount (P4,1 AND PMN AND Notbook) +
20 * RootCount (P4,0 AND PMN AND Notbook)
= 8 * * * * 1 = (4)

22. Iceberg Query Operation Using P-trees (Step Three)
Finally, we obtain the summary table 5. According to the threshold T=15, we can see that only group pair “New York And Notebook” pass our threshold T. From value P-tree PNY AND Notebook, we can see that tuple 1 and 4 are in the results of our iceberg query example.
PNY AND Notebook
Type Values
Sum (# Product)
Threshold
New York And Notebook 17 Y
Minneapolis And Notebook 11 N 1
Table 5. Summary Table of Our Example.