1. Structural Joins: A Primitive for Efficient XML Query Pattern Matching
Al Khalifa et al., ICDE 2002

2. Element Numbering
(documentId, startpos:endpos, level)

3. Join Conditions Using Numbering
(D1, S1:E1, L1) (D2, S2:E2, L2)
Ancestor-Descendant
D1 = D2, S1 < S2 < E2 < E1
Parent-Child
D1 = D2, S1 < S2 < E2 < E1, L1 + 1 = L2

4. Tree pattern >> Structural Relationship

5. Structural Join Input Output 2 algorithms presented 2 element lists
Ancestor and descendant; parent and child
Sorted by start position
Output
Pairs of ancestor/descendant or parent/child
Sorted by first or second element
2 algorithms presented
With and without stacks
Both with ordering by ancestor and by descendant

6. Example of results Ancestor Descendant Parent/child
1,20
Ancestor
2,11
12,19
Descendant
3,10
13,18
4,5
6,7
8,9
14,15
16,17
Parent/child
Interval representation 1 20 2 11 12 19

7. Tree Merge Join ordered by ancestor

8. TREE
1,26,
2,3
4,13
14,15
16,23
24,25
Skip descendants with START < ancestor.start
FOR each ancestor
Check/output descendants
until START > ancestor.end
5,12
17,22
6,7
8,9
10,11
18,19
20,21
4,13
14,15
5,12
16,23
17,22
4,13
skip
loop
skip
loop
5,12
14,15
skip
no match
6,7
8,9
10,11
18,19
20,21
2,3
24,25
Results: [4,13+6,7][4,13+8,9][4,13+10,11]
Results: [5,13+6,7][5,13+8,9][5,13+10,11] …

9. Tree Merge Join ordered by descendant

10. TREE
1,26,
2,3
4,13
14,15
16,23
24,25
Skip ancestors with END < descendant.start
FOR each descendant
Check/output ancestors
until START > descendant.end
5,12
17,22
6,7
8,9
10,11
18,19
20,21
Results: [6,7+4,13][6,7+5,12]
[8,9+4,13][8,9+5,12] …
skip
6,7
skip
8,9
2,3
no match
4,13
14,15
5,12
16,23
17,22
2,3
6,7
8,9
10,11
18,19
20,21
24,25

11. Complexity For ancestor-descendant relationships:
Tree-Merge-Anc time complexity optimal
May be quadratic, but proportional to output size
But can have poor IO performance
For parent-child relationships
Tree merge cost may still be quadratic, but output size can only be linear
Tree-Merge-Desc can be quadratic in output size

12. Worst-Case Examples a1 has the whole d list as descendants
a2 has from d2 to d2n-1 as descendants
and so on
Which means: practically quadratic performance (each ancestor has to check the whole descendant list)

13. Worst-Case Examples
Equivalent situation considering when considering Tree-Merge-Desc

14. Stack-Tree Algorithm Basic idea: depth first traversal of XML tree
Linear time with stack size = depth of tree
All ancestor-descendant relationships appear on stack during traversal
Traverse the lists only once
Main problem: do not want to traverse the whole database, just nodes in A-list/D-list

15. Stack-Tree-Desc

16. TREE Print in order of descendants
Keep ancestors in the same path in a stack
When descendant comes, it is descendant of the whole stack, then print them
Pop from stack when a different path is processed
e.g. when 14,15 comes, both previous ancestors are popped
1,26,
2,3
4,13
14,15
16,23
24,25
5,12
17,22
6,7
8,9
10,11
18,19
20,21
4,13
5,12
14,15
16,23
17,22
Results: [4,13+6,7]
4,13
Print 8,9 with the whole stack:
[4,13+8,9] [4,13+5,12]
5,12
Results: [4,13+6,7] [5,12+6,7]
4,13
5,12
Results: [4,13+6,7] [5,12+6,7]
4,13
skip
POP!!
and keep going
stack
6,7
8,9
10,11
18,19
20,21
2,3
24,25
stack

17. Example of Stack-Tree-Desc Execution

18. Stack-Tree-Anc
Basic problem: results from a particular descendant cannot be output immediately
Later descendants may match earlier ancestor
Solution: keep lists of matching descendant nodes with each stack node
Self-list
Descendants that match this node
Add descendant node to self-lists of all matching ancestor nodes
Inherit list
Inherited from descendants already popped from stack, to be output after self-list matches are output

20. Stack-Tree Analysis Stack-Tree-Desc Stack-Tree-Anc
Time complexity (for anc-desc and par-child)
O(|Alist| + |Dlist| + |OutputList|)
IO Complexity (for anc-desc and par-child)
O(|Alist|/B + |Dlist|/B + |OutputList|/B)
Where B is blocking factor
Stack-Tree-Anc
Requires careful handling of lists
Complexity is same as for Desc case

21. Performance Study