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 P4,3 P4,2 P4,1 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 * + 22 * + 21 * + 20 * = 51

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 P4,3 P4,2 P4,1 P4,0 1. Pc = P4,3 RootCount (Pc) = 3 >= 1 10 5 6 7 11 9 3 1 1 1 1 {1} 2. RootCount (Pc AND P4,2) = 0 < 1 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 * + 22 * + 21 * + 20 * = {1} {0} {1} {1} 11

Algorithm of Aggregate Function MIN Algorithm 4.3. 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 P4,3 P4,2 P4,1 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 * + 22 * + 21 * + 20 * = {0} {0} {1} {1} 3

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 P4,3 P4,2 P4,1 P4,0 1. Pc = P4,3 RootCount (Pc) = 3 < 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 * + 22 * + 21 * + 20 * = {0} {1} {1} {1} 7 Top-k: (largest k values): Find rank k value Vk. Find all tuples  Vk using EINRING.

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 P1,0 (1) LOC 0 0 0 0 1 P1,4 P1,3 P1,2 P1.1 P1.0 P’1,4 P’1,3 P’1,2 P’1.1 P1.0 PNY 1

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 + 4 * 2 + 2 * 3 + 1 * 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.

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 * 1 + 4 * 1 + 2 * 2 + 1* 1 = 17 (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 =

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 * 0 + 2 * 1 + 1 * 1 = 11 (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