Excerpt from Presentation by S. Geffner Dynamic Data Cubes ...OLAP (Online Analytical Processing):...

1USC - CSCI585 - Spring 2008 - Farnoush Banaei-Kashani4/9/2008

OLAP (Online Analytical Processing):Dynamic Data Cubes

Excerpt from Presentation by S. Geffner


Content

� Introduction to Multidimensional Databases (from A.R. 20 and 21)

� Focus Application: OLAP� Prefix-Sum Data Cube (from A.R. 16)� Dynamic Data Cube (from A.R. 17)� Iterative Data Cube (from A.R. 18)� Wavelet-based approaches

• Compact Data Cube (from A.R. 19)• ProPolyne (from A.R. 22 and 23)


The Dynamic Data Cube

S. Geffner, D. Agrawal, and A. EL Abbadi

Dynamic Data Cube


� Problem Description� Dynamic Data Cube� Improving Update� Conclusion

Outline


Problem Description: with original array A

191322427

3

2

1

8

5

2

3

7

39172546

58163325

74331244

25351233

43332422

17862371

64221530

6543210Index

Size= N2


� Complexity:� Arbitrary range queries : O(nd)� Update: O(1)

where:d: # of dimensionn: size in each dimension

Problem Description: with original array A


2302051681271036955277

206

172

142

117

88

62

26

7

182154115936149256

15112695755040215

12310380611535194

998667513529153

786753402924122

575093292118101

231713119830

6543210Index

191322427

3

2

1

8

5

2

3

7

39172546

58163325

74331244

25351233

43332422

17862371

64221530

6543210Index

Precomputearray P

Original array A

Problem Description: with prefix-sum array P


Query with Prefix-sum?

2302051681271036955277

206

172

142

117

88

62

26

7

182154115936149256

15112695755040215

12310380611535194

998667513529153

786753402924122

575093292118101

231713119830

6543210Index

Size= N2


Update with Prefix-sum?

2302051681271036955277

206

172

142

117

88

62

26

7

182154115936149256

15112695755040215

12310380611535194

998667513529153

7867534029*24122

575093292118101

231713119830

6543210Index

Size= N2


� Complexity:� Arbitrary range queries : O(1)� Update in worst case: O(nd)

where:d: # of dimensionn: size in each dimension

Problem Description: with prefix-sum array P


� Same as for Prefix-sum, DDC can also be applied to obtain count (special case of SUM) and average (SUM/COUNT).

� Generalization: applicable to any binary operator “+”which has an inverse operator “–” such that a+b-b=a

� We focus on SUM

Solution: Dynamic Data Cube


� Overlay Box

� Step1: A set of disjoint rectangles of equal size that completely partition cells of array A

� Step2: Each box stores exactly (kd-(k-1)d) values, where

k: The length of the overlay box in each dimensiond: # of dimension

Basic Data Structure: Overlay Box


191322427

3

2

1

8

5

2

3

7

39172546

58163325

74331244

25351233

43332422

17862371

64221530

6543210Index

Overlay Box: Step 1Original array A


n=8, k=n/2=4, # of values in the overlay box=(kd-(k-1)d)

6154308523426127

47

31

15

66

48

33

15

7

426

245

104

483516513529153

402

291

110

6543210Index

k

kk

k

Overlay Box: Step 2


Dynamic Data Cube

� A tree structure which recursively partitions original array A into overlay boxes

� Each overlay box will contain information regarding relative sums of the corresponding regions of A

� Organize overlay boxes into a tree to recursively partition array A into non-overlapping regions

� Each node forms children by dividing its range in each dimension in half


How to Construct DDC?

7

7

6

5

4

3

2

1

0

6543210Index


7

7

6

5

4

3

2

1

0

6543210Index

How to Construct DDC? Level 2

n=8, k=n/2, # of values in the overlay box=(kd-(k-1)d)


6154308523426127

47

31

15

66

48

33

15

7

426

245

104

483516513529153

402

291

110

6543210Index

n=8, k=n/2, # of values in the overlay box=(kd-(k-1)d)



513529153

402

291

110

3210Index

k=n/4=2113115

56

1131810

38k=n/2=4

# of values in the overlay box=(kd-(k-1)d) =3



1131153

562

11318101

380

3210Index

k=n/8=1

k=n/4=2

37

53

37

53

51

32

62

21



The DDC Hierarchy

Level 2

(Root node)

113115

56

1131810

38

1612144

610

1512164

87

134156

99

134116

46

196146

96

1272110

96

Level 1

6154308523426127

47

31

15

66

48

33

15

7

426

245

104

483516513529153

402

291

110

6543210

37

53

42

54

32

24

23

42

21

36

78

42

62

21

51

32

19

33

22

72

63

31

81

43

25

17

82

54

53

33

13

91

Level 0

(Leaf node)


6154308523426127

47

31

15

66

48

33

15

7

426

*245

104

483516513529153

402

291

110

6543210Index

How to Query DDC?


6154308523426127

47

31

15

66

48

33

15

7

426

*245

104

483516513529153

402

291

110

6543210Index

How to Query DDC? (cont’d)


6154308523426127

47

31

15

66

48

33

15

7

426

*245

104

483516513529153

402

291

110

6543210

Level 2

(Root node)

113115

56

1131810

38

1612144

610

15*12164

87

134156

99

134116

46

196146

96

1272110

96

Level 1

51+48+24+16+12=151

How to Query DDC? (cont’d)


Level 2

(Root node)

113115

56

1131810

38

1612144

610

1512164

87

134156

99

134116

46

196146

96

1272110

96

Level 1

6154308523426127

47

31

15

66

48

33

15

7

426

245

104

483516513529153

402

291

110

6543210

37

53

42

54

32

24

23

42

21

36

78

42

62

21

51

32

19

33

22

72

63

31

81

43

25

17

82

54

53

33

13

91

Level 0

(Leaf node)

6

How to Update DDC?


Level 2

(Root node)

113115

56

1131810

38

1612144

610

1512*164

87

134156

99

134116

46

196146

96

1272110

96

Level 1

6154308523426127

47

31

15

66

48

33

15

7

426

*245

104

483516513529153

402

291

110

6543210

37

53

42

54

32

24

23

42

21

36

78

42

62

21

51

32

19

33

22

72

63

31

81

43

2*5

17

82

54

53

33

13

91

Level 0

(Leaf node)

6

How to Update DDC? (cont’d)


Level 2

(Root node)

113115

56

1131810

38

1612144

610

1613164

87

134156

99

134116

46

196146

96

1272110

96

Level 1

6255308523426127

48

32

15

66

48

33

15

7

426

245

104

483516513529153

402

291

110

6543210

37

53

42

54

32

24

23

42

21

36

78

42

62

21

51

32

19

33

22

72

63

31

81

43

26

17

82

54

53

33

13

91

Level 0

(Leaf node)

How to Update DDC? (cont’d)


� Query cost for arbitrary range queries : O(log n)

� Update cost in worst case should be evaluated considering:1. O(log n) hierarchy traversal cost

• Whichever cell you update, only one overlay box is updated at each tree level

2. The cost of updating the values in the relevant overlay box at each level

The 2nd cost might be high depending on the placement of the updated cell. Hence, the total cost may become as bad as O(n)

Query and Update Complexity with DDC


� The high update cost of the overlay boxes is a consequence of dependencies between successive row sum values

Improving Update: Problem

X8X7X6X5X4X3X2X1


Storing row sum values in an array

Store row sum values separately in

Cumulative B Tree (BC Tree)

Improving Update: Solution


� Each leaf of the BC tree corresponds to one row-sum cell

� Interior nodes of the BC tree maintain subtree sums (STS)

� For each node entry, the STS stores the sum of the subtree from the left branch associated with leaf value

How to Construct BC Tree?


665345332314

Key 4STS:33

Key 3STS:9

Key 2STS:14

Key 6STS:8

Key 5STS:12

Leaf 1Value 14

Leaf 6Value 13

Leaf 5Value 8

Leaf 4Value 12

Leaf 3Value 10

Leaf 2Value 9

Overlay box

How to Construct BC Tree?

23-14 33-23


Query with BC Tree?

665345332314

Key 3STS:9

Key 2STS:14

Leaf 1Value 14

Leaf 6Value 13

Leaf 4Value 12

Leaf 3Value 10

Leaf 2Value 9

� Traverse the tree using the cell’s index as the key� If descending to a node’s right branch, we sum each

preceding STS in the node with the key less than or equal to the query key

654321

Key 4STS:33

5

5

Key 6STS:8

Key 5STS:12

Leaf 5Value 8

Key 6 is not added because it does not precede key 5


Update with BC Tree?

665345332314

Key 3STS:9

Key 2STS:14

Leaf 1Value 14

Leaf 6Value 13

Leaf 4Value 12

Leaf 3Value 10

Leaf 2Value 9

� Reflect the change using a bottom-up method

654321

Leaf 5Value 8

Key 6STS:8

Key 5STS:12

Key 4STS:33 38

15

38 50 58 71

Key 3 is not updatedbecause leaf 3 is not in its left subtree


� A two dimensional overlay box has two groups of row sum values, each of which is one dimensional

� In general, an overlay box of d dimensions has dgroups of row sum values and each group is (d-1) dimensional

� Hence, both query and update complexity is O(logdn)

Query and Update Complexity with Improved DDC


Conclusion

O(logdn)O(logdn)DDC

O(nd)O(1)Prefix sum

O(1)O(nd)Naïve approach

UpdateQueryMethod

Excerpt from Presentation by S. Geffner Dynamic Data Cubes ...OLAP (Online Analytical Processing):...

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