1. ICDCS 2006 Efficient Incremental Optimal Chain Partition of Distributed Program Traces Selma Ikiz Vijay K. Garg Parallel and Distributed Systems Laboratory

2. Outline  Introduction and Motivation  Problem definition  Outline two previous algorithms  Offline vs Incremental -> Experimental Results  New Incremental algorithm  Experimental Results  Concluding Remarks

3. Software testing and debugging  Commercial software has large number of components – Verification Formal proof of correctness is not feasible Predicate detection (Runtime verification) – simulation & formal methods – Debugging Large number of states – Abstraction or grouping of states/processes

4. Distributed Computation as Partial order set  Partial-order models – Poset (X,P) : X is a set P is antisymmetric, reflexive, and transitive binary relation on X – Lamport 1978: “happened- before” relation f 1 → e 3  c(f 1 ) < c(e 3 ) – Fidge 1991 & Mattern 1989 : vector-clocks f 1 → e 3  c(f 1 ) < c(e 3 ) P1P1 P2P2 P3P3 (1,0,0)(2,2,0)(3,2,0) (0,1,0)(0,2,0)(0,3,0) (0,2,1)(3,2,2) e1e1 e2e2 e3e3 f1f1 f2f2 f3f3 g1g1 g2g2

5. Optimal chain partition of a poset  Width – the size of the largest antichain (a subset of poset whose every distinct pair is mutually incomparable) – A poset cannot be partition into k chains if k < width(P) [R.P. Dilworth]  Debugging – Visualization  Testing & Analyzing – Bounded sum predicates (x 1 + x 2 + x 3 < k) – Mutual exclusion violation

6. Mutual exclusion violation example P1P1 (1,0,0)(2,2,0)(3,2,0) P2P2 (0,1,0)(0,3,0) P3P3 (0,2,1) (3,2,2) Critical event C1C1 C2C2 C3C3 (1,0,0) (3,2,0) (0,2,0) (0,3,0) (0,2,0) (3,2,2)

7. Problem definition & Previous algorithms  Problem definition: – Given a chain partition of P, C = {C 1,...,C N } into N disjoint chains, rearrange these chains into a chain partition with the fewest number of chains.  Previous algorithms that answer the question given k chains whether it is possible to partition it into k-1 chains. – Bogart and Magagnosc BM – Tomlinson and Garg TG

8. Bogart & Magagnosc  A sequence of elements a 0,b 0,a 1,b 1,…,a s,b s is a reducing sequence if – a 0 is the least element of some chain, – b i is the immediate predecessor of ai+1 in some chain, – all b i ’s are distinct, – for all i: a i > b i in the partial order, – b s is the greatest element of its chain. C1C1 C2C2 (1,0,0) (2,0,0) (3,0,0) (0,1,0) (0,2,0) (0,3,0) (3,4,0) (3,5,0) C3C3 (0,0,1) (0,0,2) (0,0,3) (0,3,7) (0,3,8) C4C4 (0,0,4) (0,0,5) (0,0,6)

9. Tomlinson and Garg (1,0,0) (2,0,0) (3,0,0) (0,1,0) (0,2,0) (0,3,0) (3,4,0) (3,5,0) (0,0,1) (0,0,2) (0,0,3) (0,3,7) (0,3,8) Output Chains (0,0,4) (0,0,5) (0,0,6) Input Chains ┴┴┴(0,0,1)

10. Tomlinson and Garg (1,0,0) (2,0,0) (3,0,0) (0,1,0) (0,2,0) (0,3,0) (3,4,0) (3,5,0) (0,0,3) (0,3,7) (0,3,8) Output Chains (0,0,4) (0,0,5) (0,0,6) Input Chains ┴┴ (0,0,1) (0,0,2) (0,0,3) (0,0,1) (0,3,7) (0,3,8) (0,1,0) (0,2,0) (0,3,0) (3,4,0) (3,5,0)

11. Tomlinson and Garg (1,0,0) (2,0,0) (3,0,0) Output Chains (0,0,4) (0,0,5) (0,0,6) Input Chains ┴┴ (0,0,2) (0,0,3) (0,0,1) (0,3,7) (0,3,8) (0,2,0) (0,3,0) (3,4,0) (3,5,0) (0,1,0) ┴(0,3,0) (0,2,0) (0,1,0) (3,4,0) (3,5,0)

12. Tomlinson and Garg (2,0,0) (3,0,0) Output Chains (0,0,4) (0,0,5) (0,0,6) Input Chains (0,0,2) (0,0,3) (0,0,1) (0,3,7) (0,3,8) (0,3,0) (0,2,0) (0,1,0) (3,4,0) (3,5,0) (1,0,0) (3,0,0) (1,0,0) (2,0,0)

13. Tomlinson and Garg Output Chains (0,0,4) (0,0,5) (0,0,6) Input Chains (0,0,2) (0,0,3) (0,0,1) (0,3,7) (0,3,8) (0,3,0) (0,2,0) (0,1,0) (3,4,0) (3,5,0) (1,0,0) (2,0,0) (3,0,0) ┴

14. Tomlinson and Garg Output Chains (0,0,4) (0,0,5) (0,0,6) Input Chains (0,0,2) (0,0,3) (0,0,1) (0,3,7) (0,3,8) (0,3,0) (0,2,0) (0,1,0) (3,4,0) (3,5,0) (1,0,0) (2,0,0) (3,0,0) ┴┴┴

15. Offline vs Incremental  Offline – Call BM or TG algorithm until there is no further reduction in the number of chains  Incremental (Naïve) – Place the new element into a new chain, and call the BM or TG algorithm

16. Linear extension hypothesis [Bouchitte & Rampon]  Let f be the new event, then we assume that all the events happed before f are already arrived and processed. f

17. Memory Usage

18. Running times

19. Idea behind Chain Partitioner  New element – No effect – Updates the maximum antichain – Expands the maximum antichain   Given the maximum antichain, it’s sufficient to compare the new element with the maximum antichain Maximum antichain

20. CP example (1,2,1)(1,2,3) (1,2,0)(2,3,0)(2,4,1) (2,0,0)(3,0,0)4,3,0) (0,1,0) (1,0,0) Linearization: (1,0,0), (0,1,0),(1,2,0),(2,0,0),(1,2,1),(2,3,0),(3,0,0),(1,2,3),(2,4,1),(4,3,0)

21. CP example (1,2,1)(1,2,3) (1,2,0)(2,3,0)(2,4,1) (2,0,0)(3,0,0)4,3,0) (0,1,0) (1,0,0) (1,0,0), (0,1,0),(1,2,0),(2,0,0),(1,2,1),(2,3,0),(3,0,0),(1,2,3),(2,4,1),(4,3,0) H 0 : { } W 0 : { } (1,0,0) (0,1,0) (1,2,0) (2,0,0) H: {} O: {} Q: {} (1,0,0) (0,1,0) (1,2,0) (2,0,0) ┴ ┴ ┴ ┴ k = 3 CP Merge

22. CP example (1,2,1)(1,2,3) (1,2,0)(2,3,0)(2,4,1) (2,0,0)(3,0,0)4,3,0) (0,1,0) (1,0,0) (1,0,0), (0,1,0),(1,2,0),(2,0,0),(1,2,1),(2,3,0),(3,0,0),(1,2,3),(2,4,1),(4,3,0) H 4 : { } W 4 : { } (1,0,0) (0,1,0) (1,2,0) (2,0,0) H: {} O: {} Q: {} (1,0,0) (0,1,0) (1,2,0) (2,0,0) ┴ ┴ ┴ ┴(1,0,0) (1,2,0) k = 3 CP Merge

23. CP example (1,2,1)(1,2,3) (1,2,0)(2,3,0)(2,4,1) (2,0,0)(3,0,0)4,3,0) (0,1,0) (1,0,0) (1,0,0), (0,1,0),(1,2,0),(2,0,0),(1,2,1),(2,3,0),(3,0,0),(1,2,3),(2,4,1),(4,3,0) H 4 : { } W 4 : { } (1,0,0) (0,1,0) (1,2,0) (2,0,0) H: {} O: {} Q: {} (0,1,0) (2,0,0) ┴ ┴ ┴(1,0,0) (1,2,0) (0,1,0) k = 3 CP Merge

24. CP example (1,2,1)(1,2,3) (1,2,0)(2,3,0)(2,4,1) (2,0,0)(3,0,0)4,3,0) (0,1,0) (1,0,0) (1,0,0), (0,1,0),(1,2,0),(2,0,0),(1,2,1),(2,3,0),(3,0,0),(1,2,3),(2,4,1),(4,3,0) H 4 : { } W 4 : { } (1,0,0) (0,1,0) (1,2,0) (2,0,0) H: {} O: {} Q: {} (2,0,0) (1,0,0) (1,2,0) (0,1,0) k = 3 CP Merge

25. CP example (1,2,1)(1,2,3) (1,2,0)(2,3,0)(2,4,1) (2,0,0)(3,0,0)4,3,0) (0,1,0) (1,0,0) (1,0,0), (0,1,0),(1,2,0),(2,0,0),(1,2,1),(2,3,0),(3,0,0),(1,2,3),(2,4,1),(4,3,0) H 4 : { } W 4 : { } (2,0,0) (1,0,0) (1,2,0) (0,1,0) (1,2,1) (2,3,0) (3,0,0) H: {} O: {} Q: {} k = 3 (2,0,0) (1,2,0) (1,2,1) (2,3,0) (3,0,0) ┴ ┴ ┴ ┴ CP Merge

26. CP example (1,2,1)(1,2,3) (1,2,0)(2,3,0)(2,4,1) (2,0,0)(3,0,0)4,3,0) (0,1,0) (1,0,0) (1,0,0), (0,1,0),(1,2,0),(2,0,0),(1,2,1),(2,3,0),(3,0,0),(1,2,3),(2,4,1),(4,3,0) H 4 : { } W 4 : { } (2,0,0) (1,0,0) (1,2,0) (0,1,0) (1,2,1) (2,3,0) (3,0,0) H: {} O: {} Q: {} k = 3 (2,0,0) (1,2,0) (1,2,1) (2,3,0) (3,0,0) ┴ ┴ ┴ ┴ (2,3,0) (2,0,0) CP Merge

27. CP example (1,2,1)(1,2,3) (1,2,0)(2,3,0)(2,4,1) (2,0,0)(3,0,0)4,3,0) (0,1,0) (1,0,0) (1,0,0), (0,1,0),(1,2,0),(2,0,0),(1,2,1),(2,3,0),(3,0,0),(1,2,3),(2,4,1),(4,3,0) H 0 : { } W 0 : { } (2,0,0) (1,0,0) (1,2,0) (0,1,0) (1,2,1) (2,3,0) (3,0,0) H: {} O: {} Q: {} k = 3 (1,2,0) (1,2,1) (3,0,0) ┴ ┴ ┴ (2,3,0) (2,0,0) (1,2,0) (1,2,1) CP Merge

28. CP example (1,2,1)(1,2,3) (1,2,0)(2,3,0)(2,4,1) (2,0,0)(3,0,0)4,3,0) (0,1,0) (1,0,0) (1,0,0), (0,1,0),(1,2,0),(2,0,0),(1,2,1),(2,3,0),(3,0,0),(1,2,3),(2,4,1),(4,3,0) H 7 : { } W 7 : { } (1,0,0) (0,1,0) H: {} O: {} Q: {} k = 3 (1,2,1) (3,0,0) (2,3,0) (2,0,0) (1,2,0) (1,2,3) (2,4,1) 4,3,0) CP Merge

29. Running times

30. Test suites  7 new test suites are created.  Each test suite contains 18 different test cases – initial partition vary from 10 to 450 chains, – size vary from 100 to 70,000  We used a fixed vectorclock size (10)  Reducing factor = (N-w)/N – where N is the size of the initial partition, and w is the width of the poset.  Posets are randomly created according to a given width and size.  Test suites differ in their reducing factor.

31. Reducing factor effect

32. Average run time per event

33. Comparison with related work

34. Concluding Remarks  Partitioning a distributed computation – Under the linear extension hypothesis pruning the work space (without any significant extra cost ) improves the performance of the incremental algorithm  Main limitation: – x 1 + x 2 + x 3 < k (efficient only for small k)  A decentralized algorithm  Integrating with computation slicing

35. Questions ?

36. Problem definition & Previous algorithms  Problem definition: – Given a chain partition of P, C = {C 1,...,C N } into N disjoint chains, rearrange these chains into a chain partition with the fewest number of chains.  Previous algorithms that answer the question given k chains whether it is possible to partition it into k-1 chains. – Bogart and Magagnosc BM – Tomlinson and Garg TG

37. Trace Model: Total Order vs Partial Order  Total order: interleaving of events in a trace – relevant tools: Temporal Rover [Drusinsky 00], Java-MaC [Kim, Kannan, Lee, Sokolsky, and Viswanathan 01], jPaX [Havelund and Rosu 01] Partial order: Lamport’s happened-before model – e.g., jMPaX [Sen, Rosu, and Agha 03] Total order: + low computational complexity Partial order: + suitable for concurrent and distributed programs + encodes exponential number of total orders ) captures bugs that may not be found with a total order