A work-efficient algorithm for parallel unordered depth-first search

ABSTRACT

Advances in processing power and memory technology have made multicore computers an important platform for high-performance graph-search (or graph-traversal) algorithms. Since the introduction of multicore, much progress has been made to improve parallel breadth-first search. However, less attention has been given to algorithms for unordered or loosely ordered traversals.

We present a parallel algorithm for unordered depth-first-search on graphs. We prove that the algorithm is work efficient in a realistic algorithmic model that accounts for important scheduling costs. This work-efficiency result applies to all graphs, including those with high diameter and high out-degree vertices. The algorithmic techniques behind this result include a new data structure for representing the frontier of vertices in depth-first search, a new amortization technique for controlling excess parallelism, and an adaptation of the lazy-splitting technique to depth first search.

We validate the theoretical results with an implementation and experiments. The experiments show that the algorithm performs well on a range of graphs and that it can lead to significant improvements over comparable algorithms.

References

1. Stanford large network dataset collection. http://snap.stanford.edu/.
2. The 9^th dimacs implementation challenge, 2013. http://www.dis.uniroma1.it/challenge9/.
3. The 10^th dimacs implementation challenge, 2014. http://www.cc.gatech.edu/dimacs10/.
4. U. A. Acar, G. E. Blelloch, and R. D. Blumofe. The data locality of work stealing. Theory of Computing Systems (TOCS), 35(3):321--347, 2002.
5. U. A. Acar, A. Charguéraud, and M. Rainey. Oracle scheduling: Controlling granularity in implicitly parallel languages. In ACM SIGPLAN Conference on Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA), 2011.
6. U. A. Acar, A. Charguéraud, and M. Rainey. Scheduling parallel programs by work stealing with private deques. In PPoPP '13, 2013.
7. U. A. Acar, A. Charguéraud, and M. Rainey. Theory and practice of chunked sequences. In ESA 2014, volume 8737 of LNCS, pages 25--36. Springer Berlin Heidelberg, 2014.
8. V. Agarwal, F. Petrini, D. Pasetto, and D. A. Bader. Scalable graph exploration on multicore processors. In Conference on High Performance Computing Networking, Storage and Analysis, SC 2010, New Orleans, LA, USA, November 13--19, 2010, pages 1--11, 2010.
9. A. Aggarwal, R. J. Anderson, and M. Kao. Parallel depth-first search in general directed graphs. SIAM J. Comput., 19(2):397--409, 1990.
10. D. A. Bader and K. Madduri. Designing multithreaded algorithms for breadth-first search and st-connectivity on the cray MTA-2. In 2006 International Conference on Parallel Processing (ICPP 2006), 14--18 August 2006, Columbus, Ohio, USA, pages 523--530, 2006.
11. S. Beamer, K. Asanović, and D. Patterson. Direction-optimizing breadth-first search. In SC '12, pages 12:1--12:10, Los Alamitos, CA, USA, 2012. IEEE.
12. R. Berrendorf and M. Makulla. Level-synchronous parallel breadth-first search algorithms for multicore and multiprocessor systems. In FC '14, pages 26--31, 2014
13. G. E. Blelloch, P. Cheng, and P. B. Gibbons. Room synchronizations. In SPAA '01, pages 122--133. ACM, 2001.
14. G. E. Blelloch, J. T. Fineman, P. B. Gibbons, and J. Shun. Internally deterministic parallel algorithms can be fast. In PPoPP '12, pages 181--192, New York, NY, USA, 2012. ACM.
15. R. D. Blumofe and C. E. Leiserson. Scheduling multithreaded computations by work stealing. J. ACM, 46:720--748, Sept. 1999.
16. D. Chakrabarti, Y. Zhan, and C. Faloutsos. R-mat: A recursive model for graph mining. In SIAM SDM, 2004.
17. D. Chase and Y. Lev. Dynamic circular work-stealing deque. In SPAA '05, pages 21--28, 2005.
18. C.-Y. Cher, A. L. Hosking, and T. Vijaykumar. Software prefetching for mark-sweep garbage collection: hardware analysis and software redesign. In ASPLOS '04, volume 38, pages 199--210. ACM, 2004.
19. J. Chhugani, N. Satish, C. Kim, J. Sewall, and P. Dubey. Fast and efficient graph traversal algorithm for cpus: Maximizing single-node efficiency. In IPDPS '12, pages 378--389. IEEE, 2012.
20. G. Cong, S. B. Kodali, S. Krishnamoorthy, D. Lea, V. A. Saraswat, and T. Wen. Solving large, irregular graph problems using adaptive work-stealing. In ICPP, pages 536--545, 2008.
21. T. A. Davis. University of florida sparse matrix collection, 2010. Available at http://www.cise.ufl.edu/research/sparse/matrices/.
22. T. Endo, K. Taura, and A. Yonezawa. A scalable mark-sweep garbage collector on large-scale shared-memory machines. In SC '97, pages 48--48. IEEE, 1997.
23. C. H. Flood, D. Detlefs, N. Shavit, and X. Zhang. Parallel garbage collection for shared memory multiprocessors. In JVM '01, 2001.
24. T. Hagerup. Planar depth-first search in o(log n) parallel time. SIAM J. Comput., 19(4):678--704, 1990.
25. Harshvardhan, A. Fidel, N. M. Amato, and L. Rauchwerger. KLA: A new algorithmic paradigm for parallel graph computations. In PACT '14, pages 27--38, New York, NY, USA, 2014. ACM.
26. D. Hendler and N. Shavit. Non-blocking steal-half work queues. In PODC '02, pages 280--289, 2002.
27. X. Huang, S. M. Blackburn, K. S. McKinley, J. E. B. Moss, Z. Wang, and P. Cheng. The garbage collection advantage: improving program locality. In Proceedings of the 19th Annual ACM SIGPLAN Conference on Object-Oriented Programming, Systems, Languages, and Applications, OOPSLA 2004, October 24--28, 2004, Vancouver, BC, Canada, pages 69--80, 2004.
28. Intel. Cilk Plus. http://www.cilkplus.org/.
29. R. Jones, A. Hosking, and E. Moss. The garbage collection handbook: the art of automatic memory management. Chapman & Hall/CRC, 2011.
30. V. Kumar and V. Rao. Parallel depth first search. part ii. analysis. International Journal of Parallel Programming, 16(6):501--519, 1987.
31. H. Kwak, C. Lee, H. Park, and S. Moon. What is twitter, a social network or a news media? In WWW '10, pages 591--600. ACM, 2010.
32. C. E. Leiserson and T. B. Schardl. A work-efficient parallel breadth-first search algorithm. SPAA '10, pages 303--314, New York, NY, USA, 2010. ACM.
33. A. Mislove, M. Marcon, K. P. Gummadi, P. Druschel, and B. Bhattacharjee. Measurement and analysis of online social networks. In SIGCOMM '07, pages 29--42. ACM, 2007.
34. D. Mizell and K. J. Maschhoff. Early experiences with large-scale cray XMT systems. In 23rd IEEE International Symposium on Parallel and Distributed Processing, IPDPS 2009, Rome, Italy, May 23--29, 2009, pages 1--9, 2009.
35. D. Nguyen, A. Lenharth, and K. Pingali. A lightweight infrastructure for graph analytics. In SOSP '13, pages 456--471. ACM, 2013.
36. M. Patwary, P. Refsnes, and F. Manne. Multi-core spanning forest algorithms using the disjoint-set data structure. In Parallel Distributed Processing Symposium (IPDPS), 2012 IEEE 26th International, pages 827--835, May 2012.
37. M. J. Quinn and N. Deo. Parallel graph algorithms. ACM Comput. Surv., 16(3):319--348, 1984.
38. V. Rao and V. Kumar. Parallel depth first search. part i. implementation. IJPP, 16(6):479--499, 1987.
39. E. Reghbati and D. G. Corneil. Parallel computations in graph theory. SIAM J. Comput., 7(2):230--237, 1978.
40. E. Reghbati (Arjomandi) and D. Corneil. Parallel computations in graph theory. SIAM JoC, 7(2):230--237, 1978.
41. J. H. Reif. Depth-first search is inherently sequential. Inf. Process. Lett., 20(5):229--234, 1985.
42. V. A. Saraswat, P. Kambadur, S. B. Kodali, D. Grove, and S. Krishnamoorthy. Lifeline-based global load balancing. In C. Cascaval and P.-C. Yew, editors, PPOPP, pages 201--212. ACM, 2011.
43. A. E. Sariyüce, K. Kaya, E. Saule, and U. V. Çatalyürek. Betweenness centrality on gpus and heterogeneous architectures. In Proceedings of the 6th Workshop on General Purpose Processor Using Graphics Processing Units, GPGPU-6, pages 76--85, New York, NY, USA, 2013. ACM.
44. E. Saule and Ü. V. Çatalyürek. An early evaluation of the scalability of graph algorithms on the intel MIC architecture. In 26th IEEE International Parallel and Distributed Processing Symposium Workshops & PhD Forum, IPDPS 2012, Shanghai, China, May 21--25, 2012, pages 1629--1639, 2012
45. J. Shun and G. E. Blelloch. Ligra: a lightweight graph processing framework for shared memory. In PPOPP '13, pages 135--146, New York, NY, USA, 2013. ACM.
46. J. Shun, L. Dhulipala, and G. Blelloch. A simple and practical linear-work parallel algorithm for connectivity. In Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '14, pages 143--153, 2014.
47. F. Siebert. Concurrent, parallel, real-time garbage-collection. In ACM Sigplan Notices, volume 45, pages 11--20. ACM, 2010.
48. A. Tzannes, G. C. Caragea, U. Vishkin, and R. Barua. Lazy scheduling: A runtime adaptive scheduler for declarative parallelism. TOPLAS, 36(3):10:1--10:51, Sept. 2014.
49. C. Wilson, B. Boe, A. Sala, K. P. Puttaswamy, and B. Y. Zhao. User interactions in social networks and their implications. In EUROSYS '09, pages 205--218. Acm, 2009.
50. Y. Xia and V. K. Prasanna. Topologically adaptive parallel breadth-first search on multicore processors. In IASTED '09, volume 668, page 91, 2009.

Index Terms

• Computer systems organization

  • Dependable and fault-tolerant systems and networks

• General and reference

  • Cross-computing tools and techniques

    • Performance

• Networks

  • Network performance evaluation

• Theory of computation

  • Design and analysis of algorithms

    • Data structures design and analysis

      • Sorting and searching

  • Models of computation

    • Concurrency

      • Parallel computing models