1. N-Way Model Merging CSC2125 Project
Julia Rubin
December 19, 2012

2. Model merging is used to:
Reconcile complimentary views of different stakeholders

3. Model merging is used to:
Reconcile complimentary views of different stakeholders
Unite changes made in distinct branches of an SCM system

4. Model merging is used to:
Reconcile complimentary views of different stakeholders
Unite changes made in distinct branches of an SCM system
Combine distinct program variants into a single-copy product line representation

5. Consists of three steps:
Compare – a heuristic that calculates the similarity degree for input elements
Compare x M1 y z M2 ? ?

6. Consists of three steps:
Compare – a heuristic that calculates the similarity degree for input elements
Match – a heuristic that decides which elements are similar, based on the results of compare
Match x M1 y z M2

7. Consists of three steps:
Compare – a heuristic that calculates the similarity degree for input elements
Match – a heuristic that decides which elements are similar, based on the results of compare
Merge – an algorithm that combines matched elements in a prescribed way
Union-merge, agreement-based, annotative, …
Merge
M1+M2 xy z

8. Merging is extensively studied
UML class diagram
UML state machines
Matlab/Simulink models
Verilog …
Existing works focus on two input models!

9. What if we want to compose n inputs?

10. Do it in a pair-wise manner, one by one!
What if we want to compose n inputs?
Do it in a pair-wise manner, one by one!

11. Example 0.66 M2 M1 M3 Square Square position:point position:point
center:point
Square
Square
position:point
width:int
Square
center:point
width:int
Circle
position:point
radius:int M3

12. Example
0.66 M2 M1
position:point
center:point
Square
Square
position:point
width:int
Elements of a pair share one attribute and have one proprietary attribute each
Square
center:point
width:int
Circle
position:point
radius:int M3

13. Example 0.66 M2 M1 0.66 0.66 0.66 0.66 M3 Square Square position:point
center:point
Square
Square
position:point
width:int
0.66
0.66
0.66
0.66
Square
center:point
width:int
Circle
position:point
radius:int M3

14. Pair-wise similar (no matter how we define compare)
Example
0.66 M2 M1
position:point
center:point
Square
Square
position:point
width:int
Pair-wise similar (no matter how we define compare)
0.66
0.66
0.66
0.66
Square
center:point
width:int
Circle
position:point
radius:int M3

15. Example 0.66 M2 M1 0.66 0.66 0.66 0.66 M3 Square Square position:point
center:point
Square
Square
position:point
width:int
0.66
0.66
0.66
0.66
Square
center:point
width:int
Circle
position:point
radius:int M3

16. Example: the first merge
M1+M3 M2
Square [M3]
center:point
width:int
Square/Circle
position:point [M1, M3]
center:point [M1]
radius:int [M3]
Square
position:point
width:int

17. Example: the first merge
M1+M3 M2
Square [M3]
center:point
width:int
Square/Circle
position:point [M1, M3]
center:point [M1]
radius:int [M3]
Square
position:point
width:int
0.62
0.66

18. Example: the first merge
M1+M3 M2
Square [M3]
center:point
width:int
Square/Circle
position:point [M1, M3]
center:point [M1]
radius:int [M3]
Square
position:point
width:int
0.62
0.66

19. Example: the second merge
M1+ M2 + M3
Square/Circle
position:point [M1, M3]
center:point [M1]
radius:int [M3]
Square
position:point [M2]
center:point [M3]
width:int [M2,M3]

20. Example: the second merge
M1+ M2 + M3
Square/Circle
position:point [M1, M3]
center:point [M1]
radius:int [M3]
Square
position:point [M2]
center:point [M3]
width:int [M2,M3]
Circle
position:point
radius:int
Square
center:point
width:int M3 M2 M1

21. Example: the second merge
M1+ M2 + M3
Square/Circle
position:point [M1, M3]
center:point [M1]
radius:int [M3]
Square
position:point [M2]
center:point [M3]
width:int [M2,M3]
Circle
position:point
radius:int
Square
center:point
width:int M3 M2 M1
“Outlier”

22. Example: the second merge
M1+ M2 + M3
Square/Circle
position:point [M1, M3]
center:point [M1]
radius:int [M3]
Square
position:point [M2]
center:point [M3]
width:int [M2,M3]
position:point
center:point
Square
Square
position:point
width:int M1 M2
Square
center:point
width:int
Circle
position:point
radius:int M3

23. Example: the second merge
M1+ M2 + M3
Square/Circle
position:point [M1, M3]
center:point [M1]
radius:int [M3]
Square
position:point [M2]
center:point [M3]
width:int [M2,M3]
Circle [M3]
position:point
radius:int
Square
position:point [M1,M2]
center:point [M1,M3]
width:int [M2,M3]
position:point
center:point
Square
Square
position:point
width:int M1 M2
Square
center:point
width:int
Circle
position:point
radius:int M3

24. Example: the second merge
M1+ M2 + M3
Circle [M3]
position:point
radius:int
Square
position:point [M1,M2]
center:point [M1,M3]
width:int [M2,M3]
Taking the global view is better
Square/Circle
position:point [M1, M3]
center:point [M1]
radius:int [M3]
Square
position:point [M2]
center:point [M3]
width:int [M2,M3]

25. Maybe some “smart” heuristic on the order of merging models?
position:point
center:point
Square
Square
position:point
width:int M1 M2
Square
center:point
width:int
Circle
position:point
radius:int M3

26. Maybe some “smart” heuristic on the order of merging models?
Selecting M1 and M1 first would eliminate the problem
position:point
center:point
Square
Square
position:point
width:int M1 M2
Square
center:point
width:int
Circle
position:point
radius:int M3

27. Maybe some “smart” heuristic on the order of merging models?
Selecting M1 and M1 first would eliminate the problem
position:point
center:point
Square
Square
position:point
width:int
Does not work in the general case  M1 M2
Square
center:point
width:int
Circle
position:point
radius:int M3

28. Earlier Example – an Abstract Form
Circle
position:point
radius:int
Square
center:point
width:int B C A D M3

29. Earlier Example M1 M2
A B A C M3 B C A D

30. Example 2 M2 M1 M3 A C J K I M E F A B I J E H F G B C A D I K E G
case2compare.pdf B C A D I K E G M3 30

31. Example 2 M2 M1 M3 A C J K I M E F A B I J E H F G B C A D I K E G
case2compare.pdf B C A D I K E G M3 31

32. Example 2 M2 M1 M3 A C J K I M E F A B I J E H F G B C A D I K E G
case2compare.pdf B C A D I K E G M3 32

33. Example 2 M2 M1 M3 A C J K I M E F A B I J E H F G B C A D I K E G
case2compare.pdf B C A D I K E G M3 33

34. Example 2 M2 M1 M3 A C J K I M E F A B I J E H F G B C A D I K E G
case2compare.pdf B C A D I K E G M3 34

35. Models are pair-wise similar
Example 2 A C J K I M E F M2 M1 A B I J
Models are pair-wise similar E H F G
case2compare.pdf
Without seeing the “global picture”, it is impossible to know which match is the best B C A D I K E G M3 35

36. Example 2 M2 M1 M3 A C J K I M E F A B I J E H F G B C A D I K E G
case2compare.pdf B C A D I K E G M3 36

37. Do it in a pair-wise manner, one by one!
What if we want to compose n inputs?
Do it in a pair-wise manner, one by one!
Look at n models simultaneously – n-way merge!

38. Outline Why model merging? Why n-way merge?
Formal definition of n-way merge (using Alloy)
Data representation
The compare operator
The match operator
The merge operator
Issues with the implementation
Next steps

39. 1. Data Representation In general: In this work:
a model is a typed graph
In this work:
all elements are of the same type
e.g., UML classes
relevant information about elements is represented by properties A-Z
e.g., UML class attributes and methods

41. Each element of the tuple comes from some InputModel
2. The Compare Operator
Each element of the tuple comes from some InputModel

42. There are no two elements from the same model
2. The Compare Operator
There are no two elements from the same model

43. union is a set of all properties of all tuple elements
2. The Compare Operator
union is a set of all properties of all tuple elements

44. 2. The Compare Operator Weight is
Alloy does not support floating point numbers. Thus, we limit the precision to one decimal digit and do not normalize by the number of models

45. Example M2 M1 M3 Square Square position:point position:point
center:point
Square
Square
position:point
width:int
Square
center:point
width:int
Circle
position:point
radius:int M3

46. Example (2+2+2) /3=2 M2 M1 M3 position center width Square Square
position:point
center:point
Square
Square
position:point
width:int
Square
center:point
width:int
Circle
position:point
radius:int M3

47. Example (2+2) /3=1.33 M2 M1 M3 position center width Square Square
position:point
center:point
Square
Square
position:point
width:int
Square
center:point
width:int
Circle
position:point
radius:int M3

48. position center radius
Example
(2+2)
/3=1.33
position center radius M2 M1
position:point
center:point
Square
Square
position:point
width:int
Square
center:point
width:int
Circle
position:point
radius:int M3

49. position center width radius
Example
(2+2+2)
/4=1.5
position center width radius M2 M1
position:point
center:point
Square
Square
position:point
width:int
Square
center:point
width:int
Circle
position:point
radius:int M3

50. Disjoint: no element appears in two tuples
3. The Match Operator
Disjoint: no element appears in two tuples

51. 3. The Match Operator
Maximal: not possible to add additional tuples without violating the disjoint constraint

52. Weight: an average weight of the tuples
3. The Match Operator
Weight: an average weight of the tuples

53. 3. The Match Operator
Pick the solution with the maximal weight. If there are multiple, pick the one that has the smallest number of tuples

54. Example

55. 3. The Merge Operator
Each input element is either part of a tuple or in the result by itself

56. There is an element in the result for each tuple
3. The Merge Operator
There is an element in the result for each tuple

57. There are no additional elements in the result
3. The Merge Operator
There are no additional elements in the result

58. Example

59. In Summary Compare: computes weight for each tuple of elements
Match: picks a maximal disjoint set of tuples with the largest average weight
Merge: unifies matched tuples and copy the remaining elements
as is

60. Outline Why model merging? Why n-way merge?
Formal definition of n-way merge (using Alloy)
Data representation
The compare operator
The match operator
The merge operator
Issues with the implementation
Next steps

61. Exponential number of maximal disjoint sets
In Summary
Compare: computes weight for each tuple of elements
Match: picks a maximal disjoint set of tuples with the largest average weight
Merge: unifies matched tuples and copy the remaining elements
as is
Exponential number of tuples: (m1+1)*(m2+1)*..*(mn+1) - (m1+m2+..+mn+1)
Exponential number of maximal disjoint sets

62. Any implementation that is based on - enumerating the tuples and - considering all possible solution will not scale 

63. Open Issues and Next Steps
Devise an efficient n-way merge algorithm
Either exact or approximate
Q: What are the properties of compare / match that can be used for optimization?
If implementing an approximation:
Theoretically show its approximation factor
Compare to the pair-wise approach
Pair-wise is at least by 11/15 worse than optimal, so we should do better than that

64. Open Issues and Next Steps
Devise an efficient n-way merge algorithm
Either exact or approximate
Q: What are the properties of compare / match that can be used for optimization?
If implementing an approximation:
Theoretically show its approximation factor
Compare to the pair-wise approach
Empirically compare n-way and pairwise approaches
On a number of large scale examples
Q: How to obtain real-life examples of a reasonable size (vs. randomly generated)

65. Summary
Redefined model merging for the case of n input models: n-way model merging
Showed on two examples that n-way merging produces better results than a pair-wise approach
Provided a formal definition and a naïve implementation of n-way merging using the Alloy Analyzer
Discussed open issues and next steps

66. Thank You!