1. Bounding Space Usage of Conservative Garbage Collectors Ohad Shacham December 2002 Based on work by Hans-J. Boehm

2. Garbage Collector Mechanism that allows automatic recycling of unreachable memory objects Convenience – the programmer does not need to deallocate memory. Safety – don’t reuse reachable memory objects

3. Conservative GC Garbage collector that tolerates ambiguous pointers Ambiguous pointer – location which may or may not be a pointer CGC treated ambiguous pointer whose value is valid object address as they were pointers

4. Why do we use CGC? C/C++ programs Compilers that generate C code Compiler/Collector interface can be much simpler Facilitate language interoperability Etc…

5. CGC Problems Can’t safely update a pointer to a moved object (unless we are sure that the pointer is unambiguous) Bad for programs with a large number of very short lived objects Retaining unreferenced memory as a result of misidentified pointers Integer that misidentified as a pointer to a large Data Structure

6. Embarrassing Failure Scenario headtail

7. Embarrassing Failure Scenario headtail

8. Embarrassing Failure Scenario headtail

9. Embarrassing Failure Scenario headtail

10. Embarrassing Failure Scenario headtail

11. Embarrassing Failure Scenario headtail

12. Embarrassing Failure Scenario headtail

13. Embarrassing Failure Scenario headtail - - False Reference Unbounded space retaining How can it be fixed? Assigning NULL to the next field in the list element being removed from the queue

14. Our Purpose We know that conservatively collected programs can retain large amount of memory due to a particularly misidentified pointer We want to bound space usage independently of the particular misidentified pointers that arise during a particular execution

15. Motivation Provide intuitive explanations why CGC behaves reasonably well in practice Provide better characteristic of what can provoke failure of CGC Make CGC safe for environments that required hard space bound

16. How we’ll satisfy our motivation? Prove mathematical bound on space consumption Suggest a testing technique for identifying potential unbounded growth without executing the program and identifying misidentified pointers

17. Reachability Reachable Object – An object that can be reached by a real pointer (Y) Backward Reachable Objects – A set of Objects that are backward reachable from a real pointer {Y,Z,L,K} XY - - root Y L K Z X - - root D

18. Backward Forward Reachability Backward Forward Reachable Object – An object that is reachable through an anchor X and X is backward reachable (Y) X Y root

19. Strongly GC Robust Data Structure is Strongly GC robust iff all objects that are backward forward reachable from a root through a single anchor were at some point reachable through a root

20. Who doesn’t satisfy Strongly GC Robust criterion? Our FIFO queue example head tail headtail But these objects weren’t reachable at the same time head

21. Who does satisfy Strongly GC Robust criterion? Purely functional programs e.g. Stack head

22. Bounding the Space Consumption Set of Objects S that is reachable from an unreachable object x, were at some point backward forward reachable through object x X root

23. So what do we know? Set of Objects S that is reachable from an unreachable object x, was at some point backward forward reachable through object x Objects that are backward forward reachable in Strongly GC robust Data Structures were at some point reachable In programs that use only Strongly GC robust DS A Set of objects S that is reachable from an unreachable Object x was at some point reachable

24. Bounding the Space Consumption If a program uses only Strongly GC robust DS and the number of misidentified pointers in bounded by N, then the extra space retaining by CGC is bounded by N * maximal amount of live memory

25. What did we want to Satisfy? Prove mathematical bound on space consumption Suggest a testing technique for identifying potential unbounded growth without executing the program and identifying misidentified pointers

26. Observation The number of reachable objects from an unreachable object x that are not reachable from a proper root grows only if some objects became unreachable from a proper roots.

27. Observation X Reachable Objects Unreachable Objects Proper Roots

28. Observation X Reachable Objects Unreachable Objects Proper Roots False Reference

29. Theorem If the number of objects reachable from an unreachable object x but not reachable from proper roots grows without bound, then the length of the longest simple path from x through unreachable objects to a reachable object also grow without bounds

30. Proof According to our observation a path to a reachable object can grow only if a reachable object became unreachable We have two cases: 1. The newly unreachable object is on a path to a reachable object 2. The newly unreachable object is not on a path to a reachable object

31. Proof 1. We have two cases 1. The object is on a simple path 2. The object is not on a simple path False Referencex y w

32. So What do we know? Number of objects reachable from x but unreachable From proper roots grows without bound the length of the longest simple path From x through unreachable objects to a reachable object grows without bound

33. Therefore Number of objects reachable from x but unreachable From proper roots grows with bound the length of the longest simple path From x through unreachable objects to a reachable object grows with bound

34. Weakly GC Robust A Data Structure is weakly GC robust if the length of simple paths through unreachable objects ending at an object in the Data Structure remains bounded for any execution

35. Who does satisfy Weakly GC Robust ? Our FIFO queue with restriction that only bounded number of objects are ever removed headtail - - Bounded False Reference

36. Who doesn’t satisfy Weakly GC Robust ? Our FIFO queue without the previous restriction headtail - - Unbounded False Reference

37. Strongly VS Weakly Strongly Precise bound on the cost of misidentification pointer Max (live memory) Weakly Reason about bonded Vs unbounded space loss

38. Testing Algorithm 1. Building a backward reachability graph 2. Marking all the reachable objects 3. Performing DFS on the graph from 1 and counting the height of each reachable object. Reporting the maximum height 4. Attach to each reachable object its height 5. Discard unreachable objects and the auxiliary DS from 1

39. Stage 1 a c g i b f e n k d h l j m a b c d e f g j h i l n k m

40. Stage 2 a c g i b f e n k d h l j m a b c d e f g j h i l n k m

41. Stage 3 a c g i b f e n k d h l j m a b c d e f g j h i l n k m 0 1 2 3 4 0 1 2 3

42. Stage 4 a c g i b f e n k d h l j m a b c d e f g j h i l n k m 0 1 2 3 4 0 1 2 3 4 3

43. Stage 5 a g f n k j m 4 3

44. Results In most of the tests the maximum counter stabilized in tens or hundreds In our FIFO queue the counter passed the million

45. Conclusions In Strongly GC robust DS the space retaining is bounded by the maximum of live memory for each misidentified pointer Conjecture – All the DS, except for the straightforward implementation of a singly linked queue and infinite data structure that relay on lazy evaluation are weakly GC robust