1. Algorithm for the 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
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

2. Algorithm of Aggregates AVERAGE, MAX
Average function: Average function will show the average value in a field. It can be calculated from function COUNT and SUM. Average () = Sum ()/Count ().
Algorithm 4.2 Evaluating max () with P-tree.
max = 0.00; c = 0; /*Pc is set all 1s*/
For i=n to 0 { c=Count (Pc AND Pi); If (c >= 1) Pc = Pc AND Pi; max = max + 2i }
Return 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

3. Algorithm of Aggregate Function MIN
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;
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

4. Algorithms of Aggregate Function MEDIAN, RANK, TOP-K
Rank (K) function returns the value that is the kth largest value in a field.
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&Pi); If (c >= pos) median=median+2i; Pc=PcΠ
Else pos=pos-c; Pc=Pc &NOT(Pi);} Return 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
Top-k: (largest k values): Find rank k value Vk. Find all tuples  Vk using EINRING.

5. Iceberg Query Operation Using P-trees
PMN 1
PNY
PCH
SELECT Loc, Type, Sum (# Product) FROM Relation S GROUPBY Loc, Type HAVING Sum (# Product) >= 15
Step one: Build value P-trees for the 4 values,
{Loc| New York, Minneapolis, Chicago}, of attribute Loc.
Calculation of value P-tree PNY. Because binary value of New York is 00001, we will get (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

6. 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 = 23 (2)
Loc Values
Sum (# Product)
Threshold
New York 23 Y
Minneapolis 18
Chicago 9 N
Table shows total number of products sold out in each location. Because threshold is 15, eliminate Chicago. 1
PNotebook PDesktop PPrinter PFAX
Step 2: Similarly, build P-trees for every value of attribute Type. Attribute Type has 4 values {Type | Notebook, desktop, Printer, Fax}. Figure shows P-tree of 4 values of attrib Type.
Type Values
Sum (# Product)
Threshold
Notebook 28 Y
Desktop 14 N
FAX 3
Printer 6
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.

7. Calculate total notebooks sold in New York by
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)
Iceberg Step three: We only generate candidate Loc and Type pairs for local store and Product type, which can pass T. By Performing And op on PNY with PNotebook, obtain P-tree PNY AND Notebook 1
PNY PNotebook PNY AND Notebook
AND =
By performing And operations on PMN with P Notebook, obtain value P-tree PMN AND Notebook 1
PMN PNotebook PMN AND Notebook
AND =

8. 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)
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