PrefixCube: Prefix-sharing
Condensed Data Cube
Jianlin Feng
Qiong Fang
Hulin Ding
Huazhong Univ. of Sci. & Tech.
fengjl@mail.hust.edu.cn
Nov 12, 2004
Outline
 Introduction
 Related
Work
 ODM: Ordered Datacube Model
 BST-Condensed Cube
 Prefix-sharing Condensed Cube
 Comparisons
 Conclusions
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Introduction

Data Cube (ICDE’96)
– N-dimensional cube(A1, A2, …, AN)
– 2N cuboids, i.e. GROUP-BYs

The Huge Size Problem
– When R is sparse, the size of a cuboid is
possibly close to the size of R.
– The I/O cost even for storing the cube
result tuples becomes dominative.
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Related Work
Condensed Cube (ICDE’02)
 Dwarf (SIGMOD’02)
 Quotient Cube (VLDB’02)
 QC-Tree (SIGMOD’03)
 Basic idea: remove redundancies
existing among cube tuples.

– prefix redundancy
– suffix redundancy
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Prefix redundancy

Given an example cube(A, B, C)
– Each value of dimension A occurs in 4
cuboids: cuboid(A), (AB), (AC) and (ABC)
– Possibly many times in each cuboid
except cuboid(A)

Inter-cuboid and Intra-cuboid prefix
redundancy
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Suffix Redundancy


Occurs when cube tuples belonging to
different cuboids are actually aggregated
from the same group of base relation tuples.
An extreme case
– Let the source relation R have only one single
tuple r(a1, a2, …, an, m);
– 2n cube tuples can be condensed into one physical
tuple: (a1, a2, …, an, V), where V = aggr(r);
– together with some information indicating that it
is a representative tuple.
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Thinking…

Condensed cube
– It condenses those cube tuples, aggregated
from one single base tuple, into a physical
tuple in order to reduce cube’s size.

Dwarf
– Besides suffix coalescing, i.e. multi-basetuple condensing, it also realized full prefixsharing so as to achieve high cube size
reducing effectiveness.
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Motivation


HOW to further reduce condensed cube’s
size while taking into account query
characteristics we intend to answer range query?
Augmenting BST-condensing with removing
of intra-cuboid prefix redundancy!
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Ordered Datacube Model
Value ALL(or *) is encoded as 0.
 A dimension D and its cardinality C

– each dimension value is one-to-one mapped to
an integer value between 1 and C inclusively.
N dimensions form a N-dimensional space.
 The origin O(0, 0, …, 0) represents the
grand total.

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Ordered Datacube Model

Under ODM, a range query against a
data cube can actually be reduced to
a sub-query against only one
particular cuboid in the cube or a
union of such sub-queries.
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BST-Condensed Cube

Base Single Tuple (BST)
t1
t2
t3
A
8
1
1
B
1
8
2
C
1
1
3
M
100
50
60
– t1 is a BST on SD {A} and {B}
– t2 is a BST on SD {B}

A unique minimal BST-Condensed Cube
can be got when fully taking advantage of
each BST with all of its SDs - MinCube.
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BU-BST Condensed Cube



BottomUpBST algorithms (ICDE’02)
Each BST corresponds to only one SD.
It’s easier to compute and to restore normal cube tuple
from condensed cube compared with MinCube.
Note: BST Condensing is a special kind of
Prefix-sharing !
A
B
C
M
8
8
8
8
*
1
*
1
*
*
1
1
10
10
10
10
A group of cube tuples
with sharing prefix are
represented by a BST!
ct7
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A
B
C
M
SD
8
1
1
10
{A}
Jianlin Feng
A BU-BST Condensed
Cube Example
Note:
Intra-cuboid prefix
redundancy: ct3 and ct4
Inter-cuboid prefix
redundancy: ct2, ct3 and ct5
t1
t2
t3
A
8
1
1
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B
1
8
2
C
1
1
3
M
100
50
60
ct1
ct2
ct3
ct4
ct5
ct6
ct7
ct8
ct9
ct10
ct11
ct12
13
A
*
1
1
1
1
1
8
*
*
*
*
*
B
*
*
2
8
*
*
1
1
2
8
*
*
C
*
*
3
1
1
3
1
1
3
1
1
3
M
210
110
60
50
50
60
100
100
60
50
150
60
SID
CID
ALL
A
AB
AB
AC
AC
A
B
B
B
C
C
Jianlin Feng
Prefix-sharing Condensed
Cube - PrefixCube
Prefix-sharing
BST Condensing
+
Intra-cuboid prefix-sharing
PrefixCube
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A PrefixCube Example
N-Roots
V-Roots
CID = ALL
210
CID = A
CID = AC
CID = A
1 110
1
1 150
1 50
SID = A
SID = AB
3 60
SID = B
8
1
1
1
2
1 100
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3 60
15
8
2
8
1 50
1 50
3 60
3 60
1 100
Jianlin Feng
Corresponding Dwarf
8
1
1
(node2)
(node1)
A Dimension
8
2
1
1 50 3 60 110
8
2
(node3)
B Dimension
1 150
3 60 210
3 60 60
1 50 50
1 100 100
(node4)
C Dimension
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PrefixCube vs. Dwarf
PrefixCube
Dwarf
Prefix-sharing
Intra-cuboid
Inter- and
Intra-cuboid
Suffix
Coalescing
BST
Condensing
Multi-tuple
Condensing
Compression
Ratio
Lower
Higher
Saving extra
value ALL?
No
Yes
Tuple clustered
by cuboid?
Yes
No
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PrefixCube does
not aim at blindly
achieving effective
compression ratio,
but it is intended
to make a good
compromise among
cube size reducing
ratio, restoring and
updating costs, and
query
characteristics!
Jianlin Feng
Effectiveness of Size Reduction

Datasets
100%
100%
80%
80%
Size Ratio
Size Ratio
– synthetic datasets with uniform distribution
– # of tuples: 1,000,000
60%
40%
BU-BST
PrefixCube
20%
60%
40%
BU-BST
PrefixCube
20%
0%
0%
2
3
4
5
6
7
8
9
2
Number of Dimensions
4
5
6
7
8
9
Number of Dimensions
(a) Cardinality = 100
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(b) Cardinality = 1000
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Effectiveness of Size Reduction

PrefixBUC
– Full Cube (computed by BUC)
– Prefix-sharing
100%
Size Ratio
80%
60%
40%
C=100
C=1000
20%
0%
2
3
4
5
6
7
8
9
Number of Dimensions
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Impact of Data Density

Datasets
–
–
–
–
Uniform distribution
# of dimensions: 6
Cardinality of dimensions: 100
# of tuples: range from 1,000 to 1,000,000
100%
Size Ratio
80%
60%
40%
20%
BU-BST
P refixCube
P refixBUC
0%
1.E+03
1.E+04
1.E+05
1.E+06
Number of Tuples
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Impact of Data Skewness

Datasets
– Zipf distribution
– # of tuples: 1,000,000
– Cardinality of dimensions: range from 1,000 to 500 with 100
interval
– Zipf factor: range from 0 to 0.8 with 0.2 interval
100%
Size Ratio
80%
60%
40%
BU-BST
P refixCube
P refixBUC
20%
0%
0
0.2
0.4
0.6
0.8
Zipf Factors
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Real-world Dataset

Datasets
– Weather Datasets
– # of tuples: 1,015,367
700
100%
BUC
BU-BST
P refixCube
600
Time(sec.)
Size Ratio
80%
60%
40%
BU-BST
P refixCube
P refixBUC
20%
500
400
300
200
100
0
0%
2
3
4
5
6
7
8
2
9
4
5
6
7
8
9
Number of Dimensions
Number of Dimensions
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Conclusion

A new cube structure PrefixCube was
proposed by augmenting BU-BST
condensing with intra-cuboid prefixsharing.
– It can greatly reduce data cube’s size
compared with BU-BST condensed cube.
– It can also reduce the impact of data skew
on BU-BST condensing.
– It can make a quite stable size reduction
on both dense and sparse datasets.
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The End
Thank u!
Any question?
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