Guy E. Blelloch
Yan Gu
Julian Shun
Yihan Sun
ABSTRACT
The randomized incremental convex hull algorithm is one of the most practical and important geometric algorithms in the literature. Due to its simplicity, and the fact that many points or facets can be added independently, it is also widely used in parallel convex hull implementations. However, to date there have been no non-trivial theoretical bounds on the parallelism available in these implementations. In this paper, we provide a strong theoretical analysis showing that the standard incremental algorithm is inherently parallel. In particular, we show that for n points in any constant dimension, the algorithm has O(log n) dependence depth with high probability. This leads to a simple work-optimal parallel algorithm with polylogarithmic span with high probability.
Our key technical contribution is a new definition and analysis of the configuration dependence graph extending the traditional configuration space, which allows for asynchrony in adding configurations. To capture the "true" dependence between configurations, we define the support set of configuration c to be the set of already added configurations that it depends on. We show that for problems where the size of the support set can be bounded by a constant, the depth of the configuration dependence graph is shallow (O(log n) with high probability for input size n). In addition to convex hull, our approach also extends to several related problems, including half-space intersection and finding the intersection of a set of unit circles. We believe that the configuration dependence graph and its analysis is a general idea that could potentially be applied to more problems.
- Umut Acar, Guy E. Blelloch, and Robert Blumofe. The data locality of work stealing. Theory of Computing Systems (TOCS), 35(3):321--347, 2002.Google Scholar
- Kunal Agrawal, Jeremy T. Fineman, Kefu Lu, Brendan Sheridan, Jim Sukha, and Robert Utterback. Provably good scheduling for parallel programs that use data structures through implicit batching. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2014.Google ScholarDigital Library
- Dan Alistarh, Trevor Brown, Justin Kopinsky, and Giorgi Nadiradze. Relaxed schedulers can efficiently parallelize iterative algorithms. In ACM Symposium on Principles of Distributed Computing (PODC), pages 377--386, 2018.Google ScholarDigital Library
- Dan Alistarh, Giorgi Nadiradze, and Nikita Koval. Efficiency guarantees for parallel incremental algorithms under relaxed schedulers. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 145--154, 2019.Google ScholarDigital Library
- Nancy M Amato, Michael T Goodrich, and Edgar A Ramos. Parallel algorithms for higher-dimensional convex hulls. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 683--694. IEEE, 1994.Google ScholarDigital Library
- Nancy M. Amato and Franco P. Preparata. The parallel 3D convex hull problem revisited. International Journal of Computational Geometry & Applications, 2(02):163--173, 1992.Google ScholarCross Ref
- Mikhail J. Atallah, Richard Cole, and Michael T. Goodrich. Cascading divide-and-conquer: A technique for designing parallel algorithms. SIAM J. on Computing, 18(3):499--532, June 1989.Google ScholarDigital Library
- Mikhail J. Atallah and Michael T. Goodrich. Efficient parallel solutions to some geometric problems. Journal of Parallel and Distributed Computing, 3(4):492 -- 507, 1986.Google ScholarDigital Library
- Mikhail J. Atallah and Michael T. Goodrich. Parallel algorithms for some functions of two convex polygons. Algorithmica, 3(4):535--548, 1988.Google ScholarDigital Library
- Brad Barber. Qhull. http://www.qhull.org/html/index.htm, 2015.Google Scholar
- Naama Ben-David, Guy E. Blelloch, Jeremy T. Fineman, Phillip B. Gibbons, Yan Gu, Charles McGuffey, and Julian Shun. Parallel algorithms for asymmetric read-write costs. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 145--156, 2016.Google ScholarDigital Library
- Guy E. Blelloch, Daniel Ferizovic, and Yihan Sun. Just join for parallel ordered sets. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 253--264. ACM, 2016.Google ScholarDigital Library
- Guy E. Blelloch, Jeremy T. Fineman, Yan Gu, and Yihan Sun. Optimal (randomized) parallel algorithms in the binary-forking model. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2020.Google ScholarDigital Library
- Guy E. Blelloch, Jeremy T. Fineman, and Julian Shun. Greedy sequential maximal independent set and matching are parallel on average. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 308--317, 2012.Google ScholarDigital Library
- Guy E. Blelloch, Phillip B. Gibbons, and Harsha Vardhan Simhadri. Low depth cache-oblivious algorithms. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2010.Google ScholarDigital Library
- Guy E. Blelloch and Yan Gu. Improved parallel cache-oblivious algorithms on dynamic programming. SIAM Symposium on Algorithmic Principles of Computer Systems (APOCS), 2020.Google Scholar
- Guy E. Blelloch, Yan Gu, Julian Shun, and Yihan Sun. Parallelism in randomized incremental algorithms. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 467--478, 2016.Google ScholarDigital Library
- Guy E. Blelloch, Yan Gu, Julian Shun, and Yihan Sun. Parallel write-efficient algorithms and data structures for computational geometry. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA). ACM, 2018.Google ScholarDigital Library
- Jean-Daniel Boissonnat, Olivier Devillers, René Schott, Monique Teillaud, and Mariette Yvinec. Applications of random sampling to on-line algorithms in computational geometry. Discrete & Computational Geometry, 8(1):51--71, Jul 1992.Google ScholarDigital Library
- Jean-Daniel Boissonnat and Monique Teillaud. On the randomized construction of the delaunay tree. Theoretical Computer Science, 112(2):339--354, 1993.Google ScholarDigital Library
- Jean-Daniel Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, 1998.Google ScholarCross Ref
- Richard P. Brent. The parallel evaluation of general arithmetic expressions. J. ACM, 21(2):201--206, April 1974.Google ScholarDigital Library
- Zoran Budimlić, Vincent Cavé, Raghavan Raman, Jun Shirako, Saug nak Tacs irlar, Jisheng Zhao, and Vivek Sarkar. The design and implementation of the habanero-java parallel programming language. In Symposium on Object-oriented Programming, Systems, Languages and Applications (OOPSLA), pages 185--186, 2011.Google ScholarDigital Library
- Matt Campbell. Miconvexhull. https://designengrlab.github.io/MIConvexHull/, 2017.Google Scholar
- Philippe Charles, Christian Grothoff, Vijay Saraswat, Christopher Donawa, Allan Kielstra, Kemal Ebcioglu, Christoph Von Praun, and Vivek Sarkar. X10: an object-oriented approach to non-uniform cluster computing. In Symposium on Object-oriented Programming, Systems, Languages and Applications (OOPSLA), volume 40, pages 519--538, 2005.Google Scholar
- A. Chow. Parallel Algorithms for Geometric Problems. PhD thesis, Department of Computer Science, University of Illinois, Urbana-Champaign, December 1981.Google Scholar
- Marcelo Cintra, Diego R. Llanos, and Belén Palop. Speculative parallelization of a randomized incremental convex hull algorithm. In International Conference on Computational Science and Its Applications, pages 188--197, 2004.Google ScholarCross Ref
- Kenneth L. Clarkson and Peter W. Shor. Applications of random sampling in computational geometry, II. Discrete & Computational Geometry, 4(5):387--421, 1989.Google ScholarDigital Library
- Richard Cole and Vijaya Ramachandran. Resource oblivious sorting on multicores. ACM Transactions on Parallel Computing (TOPC), 3(4), 2017.Google Scholar
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein.Introduction to Algorithms (3. ed.). MIT Press, 2009.Google Scholar
- Norm Dadoun and David G. Kirkpatrick. Parallel construction of subdivision hierarchies. J. Computer and System Sciences, 39(2):153--165, 1989.Google ScholarDigital Library
- Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 2008.Google ScholarCross Ref
- Laxman Dhulipala, Guy E. Blelloch, and Julian Shun. Theoretically efficient parallel graph algorithms can be fast and scalable. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2018.Google ScholarDigital Library
- Pedro Diaz, Diego R. Llanos, and Belen Palop. Parallelizing 2D-convex hulls on clusters: Sorting matters. Jornadas De Paralelismo, 2004.Google Scholar
- Herbert Edelsbrunner. Geometry and Topology for Mesh Generation. Cambridge University Press, 2006.Google Scholar
- Manuela Fischer and Andreas Noever. Tight analysis of parallel randomized greedy mis. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2152--2160, 2018.Google ScholarCross Ref
- Matteo Frigo, Charles E Leiserson, and Keith H Randall. The implementation of the cilk-5 multithreaded language. Proceedings of the SIGPLAN Symposium on Programming Language Design and Implementation, 33(5):212--223, 1998.Google Scholar
- Mingcen Gao, Thanh-Tung Cao, Ashwin Nanjappa, Tiow-Seng Tan, and Zhiyong Huang. gHull: A GPU algorithm for 3D convex hull. ACM Transactions on Mathematical Software, 40(1):3:1--3:19, October 2013.Google ScholarDigital Library
- J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 698--710, 1991.Google ScholarDigital Library
- Arturo Gonzalez-Escribano, Diego R. Llanos, David Orden, and Belen Palop. Parallelization alternatives and their performance for the convex hull problem. Applied Mathematical Modelling, 30(7):563 -- 577, 2006.Google ScholarCross Ref
- Michael T. Goodrich, Yossi Matias, and Uzi Vishkin. Optimal parallel approximation for prefix sums and integer sorting. In ACM-SIAM Symposium on Discrete Algorithms (SODA), 1994.Google Scholar
- Neelima Gupta and Sandeep Sen. Faster output-sensitive parallel algorithms for 3D convex hulls and vector maxima. Journal of Parallel and Distributed Computing, 63(4):488--500, April 2003.Google ScholarDigital Library
- William Hasenplaugh, Tim Kaler, Tao B Schardl, and Charles E Leiserson. Ordering heuristics for parallel graph coloring. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 166--177, 2014.Google ScholarDigital Library
- https://www.threadingbuildingblocks.org.Google Scholar
- Joseph JáJá. An Introduction to Parallel Algorithms. Addison Wesley, Reading, MA, 1992.Google ScholarDigital Library
- http://docs.oracle.com/javase/tutorial/essential/concurrency/forkjoin.html.Google Scholar
- Diego R. Llanos, David Orden, and Belen Palop. Meseta: A new scheduling strategy for speculative parallelization of randomized incremental algorithms. International Conference on Parallel Processing Workshops, pages 121--128, 2005.Google ScholarDigital Library
- Mikola Lysenko. incremental-convex-hull. https://github.com/mikolalysenko/incremental-convex-hull, 2014.Google Scholar
- Russ Miller and Quentin F. Stout. Efficient parallel convex hull algorithms. IEEE Trans. on Comput., 37(12):1605--1618, December 1988.Google ScholarDigital Library
- Ketan Mulmuley. Computational geometry - an introduction through randomized algorithms. Prentice Hall, 1994.Google Scholar
- Xinghao Pan, Dimitris Papailiopoulos, Samet Oymak, Benjamin Recht, Kannan Ramchandran, and Michael I. Jordan. Parallel correlation clustering on big graphs. In Advances in Neural Information Processing Systems (NIPS), pages 82--90, 2015.Google Scholar
- John H. Reif and Sandeep Sen. Optimal randomized parallel algorithms for computational geometry. Algorithmica, 7(1--6):91--117, 1992.Google Scholar
- Raimund Seidel. Small-dimensional linear programming and convex hulls made easy. Discrete & Computational Geometry, 6(3):423--434, 1991.Google ScholarDigital Library
- Raimund Seidel. Backwards analysis of randomized geometric algorithms. In New Trends in Discrete and Computational Geometry, pages 37--67. Springer-Verlag, 1993.Google ScholarCross Ref
- 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 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 431--448, 2015.Google ScholarDigital Library
- Ayal Stein, Eran Geva, and Jihad El-Sana. Applications of geometry processing: CudaHull: Fast parallel 3D convex hull on the GPU. Comput. Graph., 36(4):265--271, June 2012.Google ScholarDigital Library
- https://msdn.microsoft.com/en-us/library/dd460717 %28v=vs.110%29.aspx.Google Scholar
- The CGAL Project. CGAL User and Reference Manual. CGAL Editorial Board, 4.14 edition, 2019.Google Scholar
- Stanley Tzeng and John D. Owens. Finding convex hulls using quickhull on the GPU. CoRR, abs/1201.2936, 2012.Google Scholar
- Uzi Vishkin. Thinking in parallel: Some basic data-parallel algorithms and techniques, 2010. Course notes, University of Maryland.Google Scholar
Index Terms
- Randomized Incremental Convex Hull is Highly Parallel
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