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Transcript of QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok...
![Page 1: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/1.jpg)
QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates
Changqing Li, Tok Wang Ling
![Page 2: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/2.jpg)
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Outline
• Background and related work
• Our QED encoding
• Completely avoid re-labeling in XML updates based on our QED
• Experiments
• Conclusion
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Background
• Three main categories of labeling schemes to process XML queries– Containment labeling scheme [Zhang et al SIGMOD01
etc.]
– Prefix labeling scheme [Tatarinov et al SIGMOD02 etc.]
– Prime number labeling scheme [Wu et al ICDE04]
![Page 4: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/4.jpg)
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(1) Containment Scheme
• “start”, “end”, and “level”
• Determine ancestor-descendant and parent-child relationships based on the containment property
1,16,1
2,3,2 4,9,2 10,13,2 14,15,2
5,6,3 7,8,3 11,12,3
“5,6,3” is a descendant of “1,16,1” because interval [5,6] is contained in interval [1,16]
“5,6,3” is a child of “4,9,2” because interval [5,6] is contained in interval [4,9], and levels 3-2=1
![Page 5: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/5.jpg)
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(1) Containment Scheme, Containment is bad to process updates
• Need to re-label all the ancestor nodes and all the nodes after the inserted node in document order
1,16,1
4,9,22,3,2 10,13,2 14,15,2
5,6,3 7,8,3 11,12,3
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(1) Containment Scheme, Containment is bad to process updates
• Need to re-label all the ancestor nodes and all the nodes after the inserted node in document order
2,3,2
1,18,1
4,9,2 10,11,2 12,15,2 16,17,2
5,6,3 7,8,3 13,14,3
• All the red color numbers need to be changed, very expensive
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(1) Containment Scheme, Approaches to solve the update problem
• Increase the interval size and leave some values unused [Li et al VLDB01]– When unused values are used up, have to re-bel
• Use float-point value [Amagasa et al ICDE03]– Float-point value represented in a computer with a fixed number of
bits– Due to float-point precision, have to re-label
• They both can not completely avoid re-labeling
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(2) Prefix Scheme
• Determine ancestor-descendant and parent-child relationships based on the prefix property
41 2 3
2.1 2.2 3.1
“2.1” is a descendant of the root, because the label of the root is empty which is a prefix of “2.1”
“2.1” is a child of “2” because “2” is an immediate prefix of “2.1”, i.e. when removing “2” from the left side of “2.1”, “2.1” has no other prefixes.
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(2) Prefix Scheme,Prefix is bad to process order-sensitive updates
• To maintain the document order when updates are performed ---- order-sensitive updates
• Need to re-label all the sibling nodes after the inserted node and all the descendants of these siblings
421 3
2.1 2.2 3.1
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(2) Prefix Scheme,Prefix is bad to process order-sensitive updates
• To maintain the document order when updates are performed ---- order-sensitive updates
• Need to re-label all the sibling nodes after the inserted node and all the descendants of these siblings
1 2 3 4 5
2.1 2.2 4.1
• All the red color numbers need to be changed, very expensive
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(2) Prefix Scheme,Approaches to solve the update problem
• OrdPath [O'Neil et al SIGMOD04]
– At the beginning, use odd numbers only
1 3 5 7
3.1 3.3 5.1
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3.1
b d
a
(2) Prefix Scheme,Approaches to solve the update problem
• OrdPath [O'Neil et al SIGMOD04]
– In insertion, use even number together with odd numbers
1 3 5
3.1
Label of node a “-1”
Label of node b “6.1”
Label of node c “6.3”
Label of node d “6.2.1” 5.13.3
7c
• All are at the same level, bad
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(2) Prefix Scheme,Problems of OrdPath
• Nodes a, b, and c are at the same level, but their labels “-1”, “6.1”, and “6.3” do not look like this; need more time to determine this; will decrease the query performance
• Waste half numbers (even numbers); will make label size increase
• Need to calculate the even number between two odd numbers; update cost not cheap
• Use a fixed length size to indicate the size of a label, the fixed length size field will eventually encounter the overflow problem when a lot of nodes are inserted, so OrdPath can not completely avoid re-labeling
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(3) Prime scheme
• Based on a top-down approach, each node is given a unique prime number (self_label) and the label of each node is the product of its parent node’s label (parent_label) and its own self_label.
• Query – Use the modular and division operations to determine the
ancestor-descendant and ordering relationships, which are very expensive
• Update– When nodes are inserted into the XML tree, needs to re-calculate
the SC values, which is much more expensive than re-labeling
• Details can be found in [Wu et al ICDE04]
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Our QED encoding
• Dynamic Quaternary Encoding (QED)
• Four quaternary numbers “0”, “1”, “2” and “3” are used in the code and each number is stored with two bits, i.e. “00”, “01”, “10” and “11”.
• The quaternary number “0” is used as the separator, and only “1”, “2”, and “3” are used in the QED encoding.– Compare QED codes based on the lexicographical
order
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Example about QED
• We show how to encode 16 numbers; we choose 16 because the total “start” and “end” values in the containment scheme is 16; this is only an example
• Any other number is ok to be encoded by our QED• Every time encode the (1/3)th and (2/3)th numbers
between two numbers– “0” is the separator, and only “1”, “2”, and “3” appear in the QED
codes, so (1/3)th and (2/3)th
1,16,1
2,3,2 4,9,2 10,13,2 14,15,2
5,6,3 7,8,3 11,12,3
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Example about QED
Decimal number FixedLength VarLength QED Position
1 00001 1
2 00010 10
3 00011 11
4 00100 100
5 00101 101
6 00110 110 2 (1/3)th position = 6 = round(0+(17-0)/3)
7 00111 111
8 01000 1000
9 01001 1001
10 01010 1010
11 01011 1011 3 (2/3)th position = 11 = round(0+(17-0)*2/3)
12 01100 1100
13 01101 1101
14 01110 1110
15 01111 1111
16 10000 10000
0
17
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Example about QED
Decimal number FixedLength VarLength QED Position
1 00001 1
2 00010 10 12 (1/3)th position = 2 = round(0+(6-0)/3)
3 00011 11
4 00100 100 13 (2/3)th position = 4 = round(0+(6-0)*2/3)
5 00101 101
6 00110 110 2 (1/3)th position = 6 = round(0+(17-0)/3)
7 00111 111
8 01000 1000 22 (1/3)th position = 8 = round(6+(11-6)/3)
9 01001 1001 23 (2/3)th position = 9 = round(6+(11-6)*2/3)
10 01010 1010
11 01011 1011 3 (2/3)th position = 11 = round(0+(17-0)*2/3)
12 01100 1100
13 01101 1101 32 (1/3)th position = 13 = round(11+(17-11)/3)
14 01110 1110
15 01111 1111 33 (2/3)th position = 15 = round(0+(17-11)*2/3)
16 10000 10000
0
17
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Example about QED
Decimal number FixedLength VarLength QED Position
1 00001 1 112 (1/3)th position = 1 = round(0+(2-0)/3)
2 00010 10 12 (1/3)th position = 2 = round(0+(6-0)/3)
3 00011 11 122 (1/3)th position = 3 = round(2+(4-2)/3)
4 00100 100 13 (2/3)th position = 4 = round(0+(6-0)*2/3)
5 00101 101 132 (1/3)th position = 5 = round(4+(6-4)/3)
6 00110 110 2 (1/3)th position = 6 = round(0+(17-0)/3)
7 00111 111 212 (1/3)th position = 7 = round(6+(8-6)/3)
8 01000 1000 22 (1/3)th position = 8 = round(6+(11-6)/3)
9 01001 1001 23 (2/3)th position = 9 = round(6+(11-6)*2/3)
10 01010 1010 232 (1/3)th position = 10 = round(9+(11-9)/3)
11 01011 1011 3 (2/3)th position = 11 = round(0+(17-0)*2/3)
12 01100 1100 312 (1/3)th position = 12 = round(11+(13-11)/3)
13 01101 1101 32 (1/3)th position = 13 = round(11+(17-11)/3)
14 01110 1110 322 (1/3)th position = 14 = round(13+(15-13)/3)
15 01111 1111 33 (2/3)th position = 15 = round(0+(17-11)*2/3)
16 10000 10000 332 (1/3)th position = 16 = round(15+(17-15)/3)
0
17
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Overflow problem of other methods
• In the previous page, we can see that the FixedLenth codes are stored with length 5, i.e. the length of each code is 5 bits
• When a lot of codes are inserted, the length 5 is not large enough, all the FixedLength codes need to be changed.
• For the VarLength codes, we also need to store the length of each VarLength code, e.g., the length of “10000” is 5. We need to store this 5 using fixed length of bits (“101”; 3 bits). The sizes of other codes should also be stored using fixed length of bits (3 bits).
• When a lot of codes are inserted, this size of the size field 3 is not large enough, then all the codes must be changed
• This is called the overflow problem.
![Page 21: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/21.jpg)
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Our QED use “0” to separate different codes ---- will never encounter the overflow problem
• For the QED codes “112”, “12”, and “122” etc. in the table, they are separated with “0”
• Stored as “11201201220”, based on the separator “0”, we can separate different codes
• “0” will never encounter the overflow problem
• Our QED encoding can help to completely avoid the re-labeling
![Page 22: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/22.jpg)
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Lexicographical order for our QED
• Our QED compares codes based on the lexicographical order
• The QED codes in the table are lexicographically ordered from top to bottom. – E.g., “132” < “2” lexicographically because the
comparison is from left to right, and the 1st symbol of “132” is “1”, while the 1st symbol of “2” is “2”.
– Another example, “23” < “232” lexicographically because “23” is a prefix of “232”.
![Page 23: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/23.jpg)
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(a) Applying QED encoding to the containment scheme
• Replace the “start” and “end” values “1” to “16” with our QED codes
• A QED encoding based on containment scheme is formed
• Compare labels based on lexicographical order
112,332
12,122 13,23 232,32 322,33
132,2 212,22 3,312
• Note that we drop the level values from the right graph just for a clear presentation
![Page 24: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/24.jpg)
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(b) Applying QED encoding to the prefix scheme
• The root has 4 children. To encode 4 numbers based on our QED, the codes will be “12”, “2”, “3” and “32”.
• Similarly if there are 2 siblings, their self_labels (last component, e.g., “3” in “2.3” is the self_label) are “2” and “3”.
• If there is only 1 sibling, its self_label is “2”.
3212 2 3
2.2 2.3 3.2
![Page 25: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/25.jpg)
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(b) Processing the delimiters of the prefix scheme based on our QED
• For the prefix scheme, the delimiter “.” can not be stored together with the numbers in the implementation to separate different components.
• For our QED encoding, we use the following approach to process the delimiters. – We use one “0” as the delimiter to separate different
components of a prefix label• e.g. separate “12” and “3” in “12.3”; the delimiter “0” is
equivalent to the “.”; “12.3” is stored as “1203” in the implementation;
– use two consecutive separators “00” as the separator to separate different labels
• e.g. “1202001203” represents 2 labels, i.e. “1202” and “1203”.
![Page 26: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/26.jpg)
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Algorithm for insertion based on QED
Algorithm: GetInsertedCodeInput: Left_Code, Right_CodeOutput: Inserted_Code, such that Left_Code < Inserted_Code < Right_Code lexicographically.
1: get the sizes of Left_Code and Right_Code2: if size(Left_Code) < size(Right_Code) //Case (1)3: then Inserted_Code = (the Right_Code with the last4: symbol changed to “1”) concatenate “2”5: else if size(Left_Code) > size(Right_Code)6: if the last symbol of Left_Code is “2” //Case (2)7: then Inserted_Code = the Left_Code with the8: last symbol changed from “2” to “3”9: else if the last symbol of Left_Code is “3” //Case (3)10: then Inserted_Code = Left_Code concatenate “2”11: else if size(Left_Code) = size(Right_Code) //Case (4)12: then Inserted_Code = Left_Code concatenate “2”
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XML updates based on our QED–containment
• When we insert a node as shown in the below figure• We should insert two QED codes between “23” and “232”
– First create the “start” value• i.e. a code between “23” and “232”, the new code is “2312”; • see Case (1) of the GetInsertedCode algorithm;
– Then create the “end” value• i.e. a code between “2312” and “232”, the new code is “2313”; • see Case (2) of the GetInsertedCode algorithm;
• “23” < “2312” < “2313” < “232” lexicographically, we need not re-label any existing nodes.
112,332
13,2312,122 232,32 322,33
132,2 212,22 3,312
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XML updates based on our QED – based on prefix scheme
• When we insert a node as shown in the below figure• We should insert one QED code between “2” and “3”
– The new QED code between “2” and “3” is “22”;
– see Case (4) of the GetInsertedCode algorithm;
• “2” < “22” < “3” lexicographically, we need not re-label any existing nodes, but we can keep the order.
12 2 22 3 32
202 203 302
![Page 29: QED: A Novel Quaternary Encoding to Completely Avoid Re-labeling in XML Updates Changqing Li,Tok Wang Ling.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649f415503460f94c60573/html5/thumbnails/29.jpg)
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Experimental results – Experimental setup
• We mainly report the results in updates• We select the Hamlet file in Shakespeare’s play
dataset• Intermittent updates
– Hamlet file has 5 act elements, 6 insertion cases, i.e. before act[1], between act[1] and act[2], …, between act[4] and act[5], and after act[5].
• Uniformly frequent updates– Insertions happens randomly at different places of the
Hamlet file
• Skewed frequent updates– Insertions always happen at a fixed place of the Hamlet file
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Experimental results – intermittent updates
• Prime needs to re-calculate less SC values, but its re-calculation time is very large
• Theorem. Our QED never needs to re-label any existing nodes
• The update time of our QED is much smaller
• The update performance differences among OrdPath, Float-point, and our QED can be seen in the next page
• Note that QED represents both the QED encoding and the QED-containment scheme, QED-PREFIX represents the scheme when we apply QED encoding to the prefix scheme.
0
1000
2000
3000
4000
5000
6000
7000
1 2 3 4 5 6
Insertion cases
Nu
mb
er o
f n
od
es t
o r
e-la
bel
in
h
ori
zon
tal
up
dat
e
Prime
OrdPath1
OrdPath2
QED-PREFIX
Float-point
FixedLength
VarLength
QED
3.6165.2296.1486.841 2.093
0
0.01
0.02
0.03
0.04
1 2 3 4 5 6
Insertion cases
Ho
rizo
nta
l u
pd
ate
tim
e (s
eco
nd
s)
Prime
OrdPath1
OrdPath2
QED-PREFIX
Float-point
FixedLength
VarLength
QED
(a) Number of nodes to re-label
(b) Time to re-label
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Experimental results – uniformly frequent updates
• When uniformly frequent updates are performed,– The update time of OrdPath
and Float-Point is much larger (more than 386 times) than the time required by our QED approaches
• Our QED encoding only needs to modify the last 2 bits of the neighbor label, which is very cheap
• Both OrdPath and Float-point can not completely avoid re-labeling
0
100
200
300
400
500
600
0 150000 300000 450000
Number of nodes inserted
Up
dat
e ti
me
for
bal
ance
d t
iny
inse
rtio
ns
(sec
on
ds)
OrderPath1
OrderPath2
QED-PREFIX
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7000
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9000
0 150000 300000 450000
Number of nodes inserted
Up
dat
e ti
me
for
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ance
d t
iny
inse
rtio
ns
(sec
on
ds)
Float-point
QED
(a) OrdPath1&2 vs QED-PREFIX
(b) Float-point vs QED
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Experimental results – skewed frequent updates
• When skewed frequent updates are performed,– The update time of OrdPath and
Float-Point is much larger (more than 8126 times) than the time required by our QED approaches
• The very large update time makes OrdPath and Float-point unsuitable to answer queries in the frequent insertion environment.
• Our QED still works the best to answer queries in the environment that frequent insertions are executed
0
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8
10
12
0 50 100 150 200
Number of nodes inserted
Up
dat
e ti
me
wit
h r
e-la
bel
ing
fo
r sk
ewed
tin
y in
sert
ion
s (s
eco
nd
s)
OrdPath1
OrdPath2
QED-PREFIX
(a) OrdPath1&2 vs QED-PREFIX
(b) Float-point vs QED
0
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Number of nodes inserted
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dat
e ti
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wit
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e-la
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fo
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ewed
tin
y in
sert
ion
s (s
eco
nd
s)Float-point
QED
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Conclusion
• We propose the QED encoding
• QED can be applied broadly to different labeling schemes
• QED can completely avoid re-labeling in XML updates