An Improved Prefix Labeling Scheme: A Binary String Approach for Dynamic Ordered XML

An Improved Prefix Labeling Scheme: A Binary String Approach for Dynamic

Ordered XML

Changqing Li Tok Wang Ling

Department of Computer Science

School of Computing

National University of Singapore

2

Overview

Related work and motivation Labeling with our ImprovedBinary scheme Order-sensitive Update and Query Experimental results Conclustion

3

Related work – Containment scheme

Containment scheme [1] Node u is an ancestor of v iff label(u.start) <

label(v.start) < label(v.end) < label(u.end) Or interval of v is contained in interval of u

[1] Rakesh Agrawal, Alexander Borgida, H. V. Jagadish: Efficient Management of Transitive Relationships in Large Data and Knowledge Bases. SIGMOD Conference 1989: 253-262

1,20

2,3 4,5 6,11 12,13 14,19

7,8 9,10 15,16 17,18

4

2,3

1,20

2,3 4,5 6,11 12,13 14,19

7,8 9,10 15,16 17,18


Update is bad Need to re-label all the ancestor nodes and all the nodes after the

inserted node in document order

1,20

2,3 4,5 6,11 12,13 14,19

7,8 9,10 15,16 17,18

Insert a node1,22

4,5 6,7 8,13 14,15 16,21

9,10 11,12 17,18 19,20

Original labels

5


How to decrease the update cost Increase the interval size

Small interval size is easy to lead to re-labeling large interval size wastes a lot of values which causes the increase

of storage Use real (float-point) values [2]

float-point is represented in computer with a fixed number of bits which is similar to the representation of integer

As a result, there are a finite number of values between any two real values

[2] Toshiyuki Amagasa, Masatoshi Yoshikawa, Shunsuke Uemura: QRS: A Robust Numbering Scheme for XML Documents. ICDE 2003: 705-707

6

Related work – Prefix scheme

Prefix scheme Node u is an ancestor of v iff label(u) is a prefix of

label(v)

1 2 3 4 5

3.1 3.2 5.1 5.2

DeweyID [6] Integer based

Binary [4] Binary string based

0 10 110 1110 11110

110.0 110.10 11110.0 11110.10

[4] Edith Cohen, Haim Kaplan, Tova Milo: Labeling Dynamic XML Trees. PODS 2002: 271-281

[6] Igor Tatarinov, Stratis Viglas, Kevin S. Beyer, Jayavel Shanmugasundaram, Eugene J. Shekita, Chun Zhang: Storing and querying ordered XML using a relational database system. SIGMOD Conference 2002: 204-215

7


Update is better DeweyID Need to re-label all the sibling nodes after the inserted node and all the

descendants of these siblings

Insert a node

1

1 2 3 4 5

3.1 3.2 5.1 5.2

1 2 3 4 5

3.1 3.2 5.1 5.2

2 3 4 5 6

4.1 4.2 6.1 6.2

Original labels

8


Update is better Binary Same number of nodes to re-label as DeweyID

Insert a node

Original labels

0 10 110 1110 11110

110.0 110.10 11110.0 11110.10

0 0 10 110 1110 11110

110.0 110.10 11110.0 11110.10

10 110 1110 11110 111110

1110.0 1110.10 111110.0 111110.10

9

Related work – Prime scheme

Prime scheme [7] Node u is an ancestor of v iff label(v) mod label(u) = 0 The number above each node is the document order The right number is the label The numbers below each label is its parent_label and self_label

(prime number)

(11*23)(11*19)(5*17)(5*13)

9854(1*7)(1*2) (1*3) (1*5) (1*11)

7632

10

12 3 5 7 11

65 85 209 253

[7] Xiaodong Wu, Mong-Li Lee, Wynne Hsu: A Prime Number Labeling Scheme for Dynamic Ordered XML Trees. ICDE 2004: 66-78

10

Related work – Prime scheme

Update Based on Chinese Remainder Theorem (CRT) [3] And Simultaneous Congruence (SC) value The SC value is 56135603 such that

56135603 mod 2 = 1 (2 is the self_label, 1 is the document order) 56135603 mod 3 = 2, 56135603 mod 5 = 3 , 56135603 mod 13 = 4, ···, and 56135603 mod 23 = 9

Need not re-label the existing nodes, need to re-calculate the SC value

(1*29)

11

Use more SC values to prevent the single SC value become a very large number.

[3] James A. Anderson and James M. Bell, Number Theory with Application, Prentice-Hall, New Jersey, 1997.

(11*23)(11*19)(5*17)(5*13)

9854(1*7)(1*2) (1*3) (1*5) (1*11)

7632

10

12 3 5 7 11

65 85 209 253

Original labels

Insert a node

(11*23)(11*19)(5*17)(5*13)

9854(1*7)(1*2) (1*3) (1*5) (1*11)

7632

10

12 3 5 7 11

65 85 209 253(11*23) 253

(11*19)(5*17)(5*13)

10965(1*7)(1*2) (1*3) (1*5) (1*11)

8743

1

22 3 5 7 11

65 85 209

After insertion, the new SC value is 4071807264, such that

4071807264 mod 29 = 1,

4071807264 mod 2 = 2,

4071807264 mod 3 = 3,

4071807264 mod 5 = 4,

4071807264 mod 13 = 5,

···,

4071807264 mod 23 = 10.

11

Motivation Containment scheme is very bad in updating Prefix and prime schemes are better in updating which are called

dynamic labeling schemes, and we only compare the performance of dynamic labeling schemes in this paper.

Prefix schemes DeweyID and Binary both need to re-label the existing nodes

Prime need not re-label existing nodes, but need to re-calculate the SC values SC value are very large numbers Re-calculation is costly

Our objective Need not re-label existing nodes in updating Need not re-calculate any values in updating

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Our ImprovedBinary scheme How to label the XML tree based on our ImprovedBinary scheme

Root is empty First sibling self_label is “01” Last sibling self_label is “011” Case (a) left self_label size is less than or equal to right

self_label size The middle self_label is that we change the last character of the right

self_label to “0” and concatenate one more “1”. Case (b) left self_label size is larger than right self_label size

The middle self_label is that we directly concatenate one more “1” after the left self_label.

01 01001 0101 01011 011

0101.01 0101.011 011.01 011.011

13

Properties of our ImprovedBinary labels

Self_labels of siblings are lexically ordered 01 < 01001 < 0101 < 01011 < 011

Labels are lexically ordered when comparing the labels component by component 01 < 01001 < 0101 < 0101.01 < 0101.011 < 01011 < 011 <

011.01 < 011.011

01 01001 0101 01011 011

0101.01 0101.011 011.01 011.011

The component is the labels between two consecutive delimiters.

14

The formal labeling algorithm for our ImprovedBinary

Left self_label is smaller than or equal to the size of the right self_label, the self_label of the middle node is that we change the last character of the right self_label to “0” and concatenate one more “1”.

Otherwise, the self_label of the middle node is the left self_label concatenates “1”.

Algorithm 1: AssignMiddleSelfLabelInput: left self label self_label_L, and right self label self_label_ROutput: middle self label self_label_M, such that self_label_Lself_label_Mself_label_R lexically.begin1: calculate the size of self_label_L and the size of self_label_R2: if size(self_label_L) size(self_label_R)

then self_label_M = change the last character of self_label_R to “0”, and concatenate () one more “1”

3: else if size(self_label_L) > size(self_label_R) then self_label_M = self_label_L “1”

end

AssignMiddleSelfLabel algorithm.

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The formal labeling algorithm for our ImprovedBinary

SubLabeling is a recursive function

Self_label is an array

Labeling algorithm.

Algorithm 2: LabelingInput: XML documentOutput: Label of each nodebegin1: for all the sibling child nodes of each node of the XML document 2: for the first sibling node, self_label[1]=“01” //self_label is an array3: if the Number of Sibling nodes SN > 1 then self_label[SN]=“011” self_label=SubLabeling(self_label, 1, SN)4: label = prefix_label concatenates delimiter and each element of self_label array end

SubLabelingInput: self_label array, left element index of self_label array L, and right element index of self_label array ROutput: self_label arraybegin1: M = floor((L+R)/2) //M refers to the Mth element of self_label array2: if L+1<R then self_label[M]=AssignMiddleSelfLabel(self_label[L], self_label[R]) SubLabeling(self_label, L, M) SubLabeling(self_label, M, R)end

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Order-sensitive update for our ImprovedBinary

Case (1): Insert a node before the first sibling node. The self_label of the inserted node is that the last character of the first self_label is changed to “0” and is concatenated with one more “1”. Label of node of a will be 001

After insertion, the order is still kept. Label(a) < 01 lexically

a

c

01011

011.011

b d

011.01

01 01001 0101

0101.01 0101.011

011b d

c

17


Case (2): Insert a node at any position between the first and last sibling node. We use the AssignMiddleSelfLabel algorithm to assign the self_label of the new inserted node. Label of node of b will be 010011 Label of node of c will be 0101.0101

After insertion, the order is still kept. 01001 < label(b) < 0101 lexically 0101.01 < label(c) < 0101.011 lexically

a

c

01011

011.011

b d

011.01

01 01001 0101

0101.01 0101.011

011b d

c

18


Case (3): Insert a node after the last sibling node. The self_label of the inserted node is that the last self_label concatenates one more “1”. Label of node of d will be 0111

After insertion, the order is still kept. 011 < label(d) lexically

a

c

01011

011.011

b d

011.01

01 01001 0101

0101.01 0101.011

011b d

c

19


For all the three cases We need not re-label the exiting nodes We need not re-calculate any values to keep the document order

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Order-sensitive query1) position = i:

Selects the ith node within a context node set. 2) preceding-sibling or following-sibling:

Selects all the preceding (following) sibling nodes of the context node.

3) preceding or following:Selects all the nodes before (after) (in document order) the context node excluding any ancestors (descendants).

When inserting a node DeweyID and Binary need to re-label the existing nodes Prime needs to re-calculate the SC values

before they can process the order-sensitive queries.

21

Experimental result -- datasets

Datesets available at [8] The depths of real XMLs are usually not too high [5]

[5] Laurent Mignet, Denilson Barbosa, Pierangelo Veltri: The XML web: a first study. WWW 2003: 500-510

[8] The Niagara Project Experimental Data. Available at: http://www.cs.wisc.edu/niagara/data.html

Datasets Topics # of filesMax fan-out for a

fileMax depth for a

fileTotal # of nodes for

each dataset

D1 Bib 18 25 4 2111

D2 Club 12 47 3 2928

D3 Movie 490 38 4 26044

D4 Sigmod Record 988 26 6 39058

D5 Department 19 257 3 48542

D6 Actor 480 368 4 56769

D7 Company 24 529 4 161576

D8 Shakespeare’s play 37 434 5 179689

D9 NASA 1882 1188 6 370292

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Experimental results -- storage

Storage requirement ImprovedBinary has the smallest label size in each

dataset

40.612677

0

4

8

12

16

D1 D2 D3 D4 D5 D6 D7 D8 D9

Datasets

La

be

l S

ize

(1

,00

0,0

00

bit

s) DeweyID

Binary

Prime

ImprovedBinary

23

Experimental results -- query

Query performance ImprovedBinary works the

fastest for most of the queries, both ordered and un-ordered

2342

0

500

1000

1500

2000

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9

Queries

Re

sp

on

se

Tim

e (

ms

)

DeweyID

Binary

Prime

ImprovedBinary

Queries # of nodes returned

Q1 /play/act[4] 370

Q2 /play/act[5]//preceding::scene 6110

Q3 /play/act/scene/speech[2] 7300

Q4 /play/*/* 19380

Q5 /play/act//speech[3]/preceding-sibling::* 30930

Q6 /play//act[2]/following::speaker 184060

Q7 /play//scene/speech[6]/following-sibling::speech 267050

Q8 /play/act/scene/speech 309330

Q9 /play/*//line 1078330

24

Experimental results -- update

Update performance Here we study the update performance of the Hamlet XML file in

D8. The update performance of other XML files in D8 is similar. Hamlet has 5 acts, and we test the following six cases:

inserting an act before act[1] inserting an act between act[1] and act[2] inserting an act between act[2] and act[3] inserting an act between act[3] and act[4] inserting an act between act[4] and act[5] and inserting an act after act[5]

Acts are the child nodes of the root play in the Hamlet XML file

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Experimental results -- update Update performance

Number of nodes to re-label The number of SC values to re-calculate is counted for Prime, one SC

value is for every three nodes. One SC value for 4 or more nodes can not be calculated using 64 bits,

the long type value in java Our ImprovedBinary need not re-label any existing nodes and need not

re-calculate any values

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5 6

Insertion Cases

Nu

mb

er

of

No

de

s f

or

Re

-la

be

lin

g

or

Re

-ca

lcu

lati

on

DeweyID/Binary

Prime

ImprovedBinary

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Experimental results -- update Update performance

Time to re-label or re-calculate The re-calculation time for prime is more than 337 times larger

than the re-labeling time for DeweyID and Binary Our ImprovedBinary needs 0 milliseconds for the update

210036266203 53166924

0

10

20

30

40

1 2 3 4 5 6

Insertion Cases

Tim

e f

or

Re

-la

be

lin

g o

r R

e-c

alc

ula

tio

n (

ms

)

DeweyID

Binary

Prime

ImprovedBinary

The query time and re-labeling (re-calculation) time in this paper refer to the processing time only without including the I/O time.

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Conclustion

Our contribution We proposed a node labeling scheme called

ImprovedBinary to decrease the label update cost This scheme need not re-label any existing nodes Need not re-calculate any values

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Reference[1] Rakesh Agrawal, Alexander Borgida, H. V. Jagadish: Efficient Management of

Transitive Relationships in Large Data and Knowledge Bases. SIGMOD Conference 1989: 253-262

[2] Toshiyuki Amagasa, Masatoshi Yoshikawa, Shunsuke Uemura: QRS: A Robust Numbering Scheme for XML Documents. ICDE 2003: 705-707

[3] James A. Anderson and James M. Bell, Number Theory with Application, Prentice-Hall, New Jersey, 1997.

[4] Edith Cohen, Haim Kaplan, Tova Milo: Labeling Dynamic XML Trees. PODS 2002: 271-281

[5] Laurent Mignet, Denilson Barbosa, Pierangelo Veltri: The XML web: a first study. WWW 2003: 500-510

[6] Igor Tatarinov, Stratis Viglas, Kevin S. Beyer, Jayavel Shanmugasundaram, Eugene J. Shekita, Chun Zhang: Storing and querying ordered XML using a relational database system. SIGMOD Conference 2002: 204-215

[7] Xiaodong Wu, Mong-Li Lee, Wynne Hsu: A Prime Number Labeling Scheme for Dynamic Ordered XML Trees. ICDE 2004: 66-78

[8] The Niagara Project Experimental Data. Available at:http://www.cs.wisc.edu/niagara/data.html

An Improved Prefix Labeling Scheme: A Binary String Approach for Dynamic Ordered XML

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