Brief Announcement: Parallel Dynamic Tree Contraction via Self-Adjusting Computation

ABSTRACT

Dynamic algorithms are used to compute a property of some data while the data undergoes changes over time. Many dynamic algorithms have been proposed but nearly all are sequential. In this paper, we present our ongoing work on designing a parallel algorithm for the dynamic trees problem, which requires computing a property of a forest as the forest undergoes changes. Our algorithm allows insertion and/or deletion of both vertices and edges anywhere in the input and performs updates in parallel. We obtain our algorithm by applying a dynamization technique called self-adjusting computation to the classic algorithm of Miller and Reif for tree contraction.

References

1. Umut A. Acar, Guy E. Blelloch, Matthias Blume, Robert Harper, and Kanat Tangwongsan. An experimental analysis of self-adjusting computation. ACM Trans. Prog. Lang. Sys., 32(1):3:1--53, 2009.
2. Umut A. Acar, Guy E. Blelloch, and Robert Harper. Adaptive functional programming. ACM Trans. Prog. Lang. Sys., 28(6):990--1034, 2006.
3. Umut A. Acar, Guy E. Blelloch, Robert Harper, Jorge L. Vittes, and Maverick Woo. Dynamizing static algorithms with applications to dynamic trees and history independence. In ACM-SIAM Symposium on Discrete Algorithms, pages 531--540, 2004.
4. Umut A. Acar, Guy E. Blelloch, and Jorge L. Vittes. An experimental analysis of change propagation in dynamic trees. In Workshop on Algorithm Engineering and Experimentation, 2005.
5. Umut A. Acar, Arthur Charguéraud, and Mike Rainey. Scheduling parallel programs by work stealing with private deques. In PPoPP '13, 2013.
6. Umut A. Acar, Andrew Cotter, Benoît Hudson, and Duru Türkoălu. Parallelism in dynamic well-spaced point sets. In Proceedings of the 23rd ACM Symposium on Parallelism in Algorithms and Architectures, 2011.
7. Umut A. Acar, Andrew Cotter, Benoît Hudson, and Duru Türkoălu. Dynamic well-spaced point sets. Journal of Computational Geometry: Theory and Applications, 2013.
8. Umut A. Acar, Benoît Hudson, and Duru Türkoălu. Kinetic mesh-refinement in 2D. In SCG '11: Proceedings of the 27th Annual Symposium on Computational Geometry, 2011.
9. Umut A. Acar, Alexander Ihler, Ramgopal Mettu, and Özgür Sümer. Adaptive Bayesian inference. In Neural Information Processing Systems (NIPS), 2007.
10. Stephen Alstrup, Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Maintaining information in fully-dynamic trees with top trees, 2003. The Computing Research Repository (CoRR)[cs.DS/0310065].
11. Jon Louis Bentley and James B. Saxe. Decomposable searching problems I: Static-to-dynamic transformation. Journal of Algorithms, 1(4):301--358, 1980.
12. Pramod Bhatotia, Pedro Fonseca, Umut A. Acar, Björn B. Brandenburg, and Rodrigo Rodrigues. iThreads: A threading library for parallel incremental computation. In Proceedings of the Twentieth International Conference on Architectural Support for Programming Languages and Operating Systems, ASPLOS '15, pages 645--659, 2015.
13. Pramod Bhatotia, Alexander Wieder, Rodrigo Rodrigues, Umut A. Acar, and Rafael Pasquini. Incoop: MapReduce for incremental computations. In ACM Symposium on Cloud Computing, 2011.
14. Sebastian Burckhardt, Daan Leijen, Caitlin Sadowski, Jaeheon Yi, and Thomas Ball. Two for the price of one: A model for parallel and incremental computation. In ACM SIGPLAN Conference on Object-Oriented Programming, Systems, Languages, and Applications, 2011.
15. Yan Chen, Umut A. Acar, and Kanat Tangwongsan. Functional programming for dynamic and large data with self-adjusting computation. In International Conference on Functional Programming (ICFP '14), pages 227--240, Sep 2014.
16. Y.-J. Chiang and R. Tamassia. Dynamic algorithms in computational geometry. Proceedings of the IEEE, 80(9):1412--1434, 1992.
17. Camil Demetrescu, Irene Finocchi, and Giuseppe F. Italiano. Handbook on Data Structures and Applications, chapter 36: Dynamic Graphs. CRC Press, 2005.
18. Krzysztof Diks and Torben Hagerup. More general parallel tree contraction: Register allocation and broadcasting in a tree. Theoretical Computer Science, 203(1):3 -- 29, 1998.
19. David Eppstein and Zvi Galil. Parallel algorithmic techniques for combinational computation. Annual review of computer science, 3(1):233--283, 1988.
20. Greg N. Frederickson. A data structure for dynamically maintaining rooted trees. Journal Algorithms, 24(1):37--65, 1997.
21. Matteo Frigo, Charles E. Leiserson, and Keith H. Randall. The implementation of the Cilk-5 multithreaded language. In PLDI, pages 212--223, 1998.
22. Matthew Hammer, Umut A. Acar, Mohan Rajagopalan, and Anwar Ghuloum. A proposal for parallel self-adjusting computation. In DAMP '07: Declarative Aspects of Multicore Programming, 2007.
23. Matthew A. Hammer, Umut A. Acar, and Yan Chen. CEAL: a C-based language for self-adjusting computation. In ACM SIGPLAN Conference on Programming Language Design and Implementation, 2009.
24. Monika R. Henzinger and Valerie King. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. Journal of the ACM, 46(4):502--516, 1999.
25. Joseph Jaja. An introduction to parallel algorithms. Addison Wesley Longman Publishing Company, 1992.
26. Richard M. Karp and Vijaya Ramachandran. Parallel algorithms for shared-memory machines. In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (A), pages 869--942. 1990.
27. Ruy Ley-Wild, Umut A. Acar, and Matthew Fluet. A cost semantics for self-adjusting computation. In Proceedings of the 26th Annual ACM Symposium on Principles of Programming Languages, 2009.
28. Gary L. Miller and John H. Reif. Parallel tree contraction, part I: Fundamentals. Advances in Computing Research, 5:47--72, 1989.
29. Gary L. Miller and John H. Reif. Parallel tree contraction, part 2: Further applications. SIAM Journal on Computing, 20(6):1128--1147, 1991.
30. Ketan Mulmuley. Randomized multidimensional search trees: Lazy balancing and dynamic shuffling (extended abstract). In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, pages 180--196, 1991.
31. Mark H. Overmars. The Design of Dynamic Data Structures. Springer, 1983.
32. Shaunak Pawagi and Owen Kaser. Optimal parallel algorithms for multiple updates of minimum spanning trees. Algorithmica, 9:357--381, 1993.
33. G. Ramalingam and T. Reps. A categorized bibliography on incremental computation. In Principles of Programming Languages, pages 502--510, 1993.
34. S. Rao Kosaraju and Arthur L. Delcher. Optimal parallel evaluation of tree-structured computations by raking (extended abstract), pages 101--110. 1988.
35. John H. Reif and Stephen R. Tate. Dynamic parallel tree contraction (extended abstract). In SPAA, pages 114--121, 1994.
36. Julian Shun, Yan Gu, Guy E. Blelloch, Jeremy T. Fineman, and Phillip B. Gibbons. Sequential random permutation, list contraction and tree contraction are highly parallel. In Proceedings of the Twenty-sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pages 431--448, 2015.
37. Daniel D. Sleator and Robert Endre Tarjan. A data structure for dynamic trees. Journal of Computer and System Sciences, 26(3):362--391, 1983.
38. Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652--686, 1985.
39. Özgür Sümer, Umut A. Acar, Alexander Ihler, and Ramgopal Mettu. Adaptive exact inference in graphical models. Journal of Machine Learning, 8:180--186, 2011.
40. Robert Tarjan and Renato Werneck. Self-adjusting top trees. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2005.
41. Robert E. Tarjan. Dynamic trees as search trees via euler tours, applied to the network simplex algorithm. Mathematical Programming, 78:167--177, 1997.
42. Robert E. Tarjan and Renato F. Werneck. Dynamic trees in practice. J. Exp. Algorithmics, 14:5:4.5--5:4.23, January 2010.

Index Terms

• Theory of computation

  • Design and analysis of algorithms

    • Algorithm design techniques

    • Graph algorithms analysis

      • Dynamic graph algorithms

    • Parallel algorithms