Peyman Afshani
ABSTRACT
We consider a number of range reporting problems in two and three dimensions and prove lower bounds on the amount of space required by any cache-oblivious data structure for these problems that achieves an optimal query bound of O(logBN + K/B) block transfers in the worst case, where K is the size of the query output.
The problems we study are three-sided range reporting, 3-d dominance reporting, and 3-d halfspace range reporting. We prove that, in order to achieve the above query bound or even a bound of O((logB N)c (1 + K/B)), for any constant c > 0, the structure has to use Ohmega(N (log log N)ε) space, where ε > 0 is a constant that depends on c and on the constant hidden in the big-Oh notation of the query bound.
Our result has a number of interesting consequences. The first one is a new type of separation between the I/O model and the cache-oblivious model, as I/O-efficient data structures with the optimal query bound and using linear or O(N logAST N) space are known for the above problems. The second consequence is the non-existence of a linear-space cache-oblivious persistent B-tree with worst-case optimal 1-d range reporting queries.
- P. Afshani. On dominance reporting in 3D. In ESA 16th, volume 5193 of Lecture Notes in Computer Science, pages 41--51. Springer-Verlag, 2008. Google Scholar
Digital Library
- P. Afshani and T. M. Chan. Optimal halfspace range reporting in three dimensions. In SODA 20th, pages 180--186, 2009. Google ScholarDigital Library
- P. Afshani, C. Hamilton, and N. Zeh. A general approach for cache-oblivious range reporting and approximate range counting. In SOCG 25th, 2009. Google ScholarDigital Library
- P. K. Agarwal, L. Arge, A. Danner, and B. Holland-Minkley. Cache-oblivious data structures for orthogonal range searching. In SOCG 19th, pages 237--245, 2003. Google ScholarDigital Library
- P. K. Agarwal, L. Arge, J. Erickson, P. G. Franciosa, and J. S. Vitter. Efficient searching with linear constraints. Journal of Computer and System Sciences, 61:194--216, 2000. Google ScholarDigital Library
- A. Aggarwal and J. S. Vitter. The input/output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, 1988. Google ScholarDigital Library
- L. Arge, G. S. Brodal, R. Fagerberg, and M. Laustsen. Cache-oblivious planar orthogonal range searching and counting. In SOCG 21st, pages 160--169, 2005. Google ScholarDigital Library
- L. Arge, M. de Berg, and H. J. Haverkort. Cache-oblivious R-trees. In Proceedings of the 21st ACM Symposium on Computational Geometry, pages 170--179, 2005. Google ScholarDigital Library
- L. Arge, M. de Berg, H. J. Haverkort, and K. Yi. The priority R-tree: A practically efficient and worst-case optimal R-tree. In Proceedings of the ACM SIGMOD International Conference on Management of Data, pages 347--358, 2004. Google ScholarDigital Library
- L. Arge, V. Samoladas, and J. S. Vitter. On two-dimensional indexability and optimal range search indexing. In Proceedings of the 18th Symposium on Principles of Database Systems, pages 346--357, 1999. Google ScholarDigital Library
- L. Arge and N. Zeh. Simple and semi-dynamic structures for cache-oblivious orthogonal range searching. In SOCG 22nd, pages 158--166, 2006. Google ScholarDigital Library
- M. A. Bender, G. S. Brodal, R. Fagerberg, D. Ge, S. He, H. Hu, J. Iacono, and A. López-Ortiz. The cost of cache-oblivious searching. In FOCS 44th, pages 271--282, 2003. Google ScholarDigital Library
- G. S. Brodal and R. Fagerberg. On the limits of cache-obliviousness. In STOC 35th, pages 307--315, 2003. Google ScholarDigital Library
- T. M. Chan. Random sampling, halfspace range reporting, and construction of (łe k)-levels in three dimensions. In FOCS 39th, pages 586--595, 1998. Google ScholarDigital Library
- T. M. Chan. Random sampling, halfspace range reporting, and construction of (łe k)-levels in three dimensions. SIAM Journal on Computing, 30(2):561--575, 2000. Google ScholarDigital Library
- M. Frigo, C. E. Leiserson, H. Prokop, and S. Ramachandran. Cache-oblivious algorithms. In FOCS 40th, pages 285--397, 1999. Google ScholarDigital Library
- R. Grossi and G. F. Italiano. Efficient cross-tree for external memory. In J. Abello and J. S. Vitter, editors, External Memory Algorithms and Visualization, pages 87--106. American Mathematical Society, 1999. Google ScholarDigital Library
- R. Grossi and G. F. Italiano. Efficient splitting and merging algorithms for order decomposable problems. Information and Computation, 154(1):1--33, 1999. Google ScholarDigital Library
- K. V. R. Kanth and A. K. Singh. Optimal dynamic range searching in non-replicated index structures. In Proceedings of the International Conference on Database Theory, volume 1540 of Lecture Notes in Computer Science, pages 257--276. Springer-Verlag, 1999. Google ScholarDigital Library
- C. Makris and A. Tsakalidis. Algorithms for three-dimensional dominance searching in linear space. ıpl, 66(6):277--283, 1998. Google ScholarDigital Library
- E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14(2):257--76, 1985.Google ScholarCross Ref
- O. Procopiuc, P. K. Agarwal, L. Arge, and J. S. Vitter. Bkd-tree: A dynamic scalable kd-tree. In Proceedings of the 8th International Symposium on Advances in Spatial and Temporal Databases, volume 2750 of Lecture Notes in Computer Science, pages 46--65. Springer-Verlag, 2003.Google ScholarCross Ref
- E. A. Ramos. On range reporting, ray shooting and k-level construction. In SOCG 15th, pages 390--399, 1999. Google ScholarDigital Library
- J. Robinson. The K-D-B tree: A search structure for large dimensional dynamic indexes. In Proceedings of the SIGMOD International Conference on Management of Data, pages 10--18, 1981. Google ScholarDigital Library
- D. E. Vengroff and J. S. Vitter. Efficient 3-D range searching in external memory. In STOC 28th, pages 192--201, 1996. Google ScholarDigital Library
- J. S. Vitter. External memory algorithms and data structures: dealing with massive data. ACM Computing Surveys, 33(2), 2001. Updated version at http://www.cs.purdue.edu/jsv/Papers/Vit.IO_survey.pdf Google ScholarDigital Library
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- Cache-oblivious range reporting with optimal queries requires superlinear space
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