1. An Improved Prefix Labeling Scheme: A Binary String Approach for Dynamic Ordered XML Changqing LiTok Wang Ling Department of Computer Science School of Computing National University of Singapore

2. 2 Overview Related work and motivation Labeling with our ImprovedBinary scheme Order-sensitive Update and Query Experimental results Conclustion

3. 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 4,56,1112,1314,19 7,89,1015,1617,18

4. 4 2,3 1,20 2,3 4,56,1112,1314,19 7,89,1015,1617,18 Related work – Containment scheme 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 4,56,1112,1314,19 7,89,1015,1617,18 Insert a node 1,22 4,5 6,78,1314,1516,21 9,1011,1217,1819,20 Original labels

5. 5 Related work – Containment scheme 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. 6 Related work – Prefix scheme Prefix scheme Node u is an ancestor of v iff label(u) is a prefix of label(v) 1 2345 3.13.25.15.2 DeweyID [6] Integer based Binary [4] Binary string based 0 10110111011110 110.0110.1011110.011110.10 [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. 7 Related work – Prefix scheme 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 2345 3.13.25.15.2 1 2345 3.13.25.15.2 2 3456 4.14.26.16.2 Original labels

8. 8 Related work – Prefix scheme Update is better Binary Same number of nodes to re-label as DeweyID Insert a node Original labels 0 10110111011110 110.0110.1011110.011110.10 0 0 10110111011110 110.0110.1011110.011110.10 10 110111011110111110 1110.01110.10111110.0111110.10

9. 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 1 0 1 2 35711 6585209 253 [7]Xiaodong Wu, Mong-Li Lee, Wynne Hsu: A Prime Number Labeling Scheme for Dynamic Ordered XML Trees. ICDE 2004: 66-78

10. 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) 1 1 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 1 0 1 2 35711 6585209 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 1 0 1 2 35711 6585209 253 (11*23) 253 (11*19)(5*17)(5*13) 10965 (1*7) (1*2) (1*3)(1*5)(1*11) 8743 1 2 2 35711 6585209 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. 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

12. 12 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 01001010101011011 0101.010101.011011.01011.011

13. 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 01001010101011011 0101.010101.011011.01011.011 The component is the labels between two consecutive delimiters.

14. 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: AssignMiddleSelfLabel Input: left self label self_label_L, and right self label self_label_R Output: middle self label self_label_M, such that self_label_Lself_label_Mself_label_R lexically. begin 1: calculate the size of self_label_L and the size of self_label_R 2: 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.

15. 15 The formal labeling algorithm for our ImprovedBinary SubLabeling is a recursive function Self_label is an array Labeling algorithm. Algorithm 2: Labeling Input: XML document Output: Label of each node begin 1: 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 array 3: 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 SubLabeling Input: self_label array, left element index of self_label array L, and right element index of self_label array R Output: self_label array begin 1: M = floor((L+R)/2) //M refers to the M th element of self_label array 2: 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

16. 16 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 bd 011.01 01 010010101 0101.010101.011 011 bd c

17. 17 Order-sensitive update for our ImprovedBinary 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 bd 011.01 01 010010101 0101.010101.011 011 bd c

18. 18 Order-sensitive update for our ImprovedBinary 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 bd 011.01 01 010010101 0101.010101.011 011 bd c

19. 19 Order-sensitive update for our ImprovedBinary 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

20. 20 Order-sensitive query 1) position = i: Selects the i th 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. 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 DatasetsTopics# of files Max fan-out for a file Max depth for a file Total # of nodes for each dataset D1Bib182542111 D2Club124732928 D3Movie49038426044 D4Sigmod Record98826639058 D5Department19257348542 D6Actor480368456769 D7Company245294161576 D8Shakespeare’s play374345179689 D9NASA188211886370292

22. 22 Experimental results -- storage Storage requirement ImprovedBinary has the smallest label size in each dataset

23. 23 Experimental results -- query Query performance ImprovedBinary works the fastest for most of the queries, both ordered and un-ordered Queries# of nodes returned Q1/play/act[4]370 Q2/play/act[5]//preceding::scene6110 Q3/play/act/scene/speech[2]7300 Q4/play/*/*19380 Q5/play/act//speech[3]/preceding-sibling::*30930 Q6/play//act[2]/following::speaker184060 Q7/play//scene/speech[6]/following-sibling::speech267050 Q8/play/act/scene/speech309330 Q9/play/*//line1078330

24. 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

25. 25 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

26. 26 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 The query time and re-labeling (re-calculation) time in this paper refer to the processing time only without including the I/O time.

27. 27 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

28. 28 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