Permuting and Batched Geometric Lower Bounds in the I/O Model
We study permuting and batched orthogonal geometric reporting problems in the External Memory Model (EM), assuming indivisibility of the input records.
Our main results are twofold. First, we prove a general simulation result that essentially shows that any permutation algorithm (resp. duplicate removal algorithm) that does alpha*N/B I/Os (resp. to remove a fraction of the existing duplicates) can be simulated with an algorithm that does alpha phases where each phase reads and writes each element once, but using a factor alpha smaller block size.
Second, we prove two lower bounds for batched rectangle stabbing and batched orthogonal range reporting queries. Assuming a short cache, we prove very high lower bounds that currently are not possible with the existing techniques under the tall cache assumption.
I/O Model
Batched Geometric Queries
Lower Bounds
Permuting
2:1-2:13
Regular Paper
Peyman
Afshani
Peyman Afshani
Ingo
van Duijn
Ingo van Duijn
10.4230/LIPIcs.ESA.2017.2
Peyman Afshani. Improved pointer machine and I/O lower bounds for simplex range reporting and related problems. In Symposium on Computational Geometry (SoCG), pages 339-346, 2012.
Peyman Afshani, Lars Arge, and Kasper Dalgaard Larsen. Orthogonal range reporting in three and higher dimensions. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 149-158, 2009.
Peyman Afshani, Lars Arge, and Kasper Dalgaard Larsen. Orthogonal range reporting: query lower bounds, optimal structures in 3-d, and higher-dimensional improvements. In Symposium on Computational Geometry (SoCG), pages 240-246, 2010.
Peyman Afshani, Gerth Stolting Brodal, and Norbert Zeh. Ordered and unordered top-k range reporting in large data sets. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 390-400, 2011.
Peyman Afshani and Nodari Sitchinava. I/O-efficient range minima queries. In Scandinavian Workshop on Algorithms Theory, pages 1-12, 2014.
Peyman Afshani and Norbert Zeh. Lower bounds for sorted geometric queries in the I/O model. In ESA 12: Proceedings of the 20th Annual European Symposium, pages 48-59, 2012.
Alok Aggarwal and Jeffrey Scott Vitter. The input/output complexity of sorting and related problems. Communications of the ACM (CACM), 31(9):1116-1127, 1988.
L. Arge. External memory data structures. In J. Abello, P. M. Pardalos, and M. G. C. Resende, editors, Handbook of Massive Data Sets, pages 313-358. Kluwer Academic Publishers, 2002.
Lars Arge. Efficient external-memory data structures and applications. PhD thesis, Aarhus University, 1996.
Lars Arge. The buffer tree: A technique for designing batched external data structures. Algorithmica, 37(1):1-24, 2003.
Lars Arge, Octavian Procopiuc, Sridhar Ramaswamy, Torsten Suel, and Jeffrey Scott Vitter. Theory and practice of I/O efficient algorithms for multidimensional batched searching problems. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 685-694, 1998.
Lars Arge, Vasilis Samoladas, and Ke Yi. Optimal external-memory planar point enclosure. In Proceedings of European Symposium on Algorithms (ESA), pages 40-52, 2004.
Lars Arge, Darren Erik Vengroff, and Jeffrey Scott Vitter. External-memory algorithms for processing line segments in geographic information systems. In Proceedings of European Symposium on Algorithms (ESA), pages 295-310. Springer, 1995.
Bernard Chazelle. Lower bounds for orthogonal range searching: I. the reporting case. Journal of the ACM (JACM), 37(2):200-212, 1990.
Yi-Jen Chiang, Michael T. Goodrich, Edward F. Grove, Roberto Tamassia, Darren Erik Vengroff, and Jeffrey Scott Vitter. External-memory graph algorithms. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 139-149, 1995.
T. H. Cormen. Fast permuting on disk arrays. Journal of Parallel and Distributed Computing, 17(1):41-57, 1993.
Andreas Crauser, Paolo Ferragina, Kurt Mehlhorn, Ulrich Meyer, and Edgar A Ramos. I/O-optimal computation of segment intersections. External Memory Algorithms and Visualization, pages 131-138, 1999.
Andreas Crauser, Paolo Ferragina, Kurt Mehlhorn, Ulrich Meyer, and Edgar A. Ramos. Randomized external-memory algorithms for line segment intersection and other geometric problems. International Journal of Computational Geometry &Applications, 11(03):305-337, 2001.
Robert W. Floyd. Permuting information in idealized two-level storage. In Complexity of computer computations, pages 105-109. Springer, 1972.
Michael T. Goodrich, Jyh-Jong Tsay, Darren Erik Vengroff, and Jeffrey Scott Vitter. External-memory computational geometry. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 714-723, 1993.
Gero Griener. Sparse Matrix Computations and their I/O Complexity. PhD thesis, Technische Universität München, 2012.
Joseph M. Hellerstein, Elias Koutsoupias, Daniel P. Miranker, Christos H. Papadimitriou, and Vasilis Samoladas. On a model of indexability and its bounds for range queries. Journal of the ACM (JACM), 49(1):35-55, 2002.
Hong T. Kung. Computational models for parallel computers. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 326(1591):357-371, 1988.
J. S. Vitter. Algorithms and data structures for external memory. Foundations and Trends in Theoretical Computer Science, 2(4):305-474, 2008. URL: http://dx.doi.org/10.1561/0400000014.
http://dx.doi.org/10.1561/0400000014
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