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Indexability of 2D range search revisited: constant redundancy and weak indivisibility

Published:21 May 2012Publication History

ABSTRACT

In the 2D orthogonal range search problem, we want to preprocess a set of 2D points so that, given any axis-parallel query rectangle, we can report all the data points in the rectangle efficiently. This paper presents a lower bound on the query time that can be achieved by any external memory structure that stores a point at most r times, where r is a constant integer. Previous research has resolved the bound at two extremes: r = 1, and r being arbitrarily large. We, on the other hand, derive the explicit tradeoff at every specific r. A premise that lingers in existing studies is the so-called indivisibility assumption: all the information bits of a point are treated as an atom, i.e., they are always stored together in the same block. We partially remove this assumption by allowing a data structure to freely divide a point into individual bits stored in different blocks. The only assumption is that, those bits must be retrieved for reporting, as opposed to being computed -- we refer to this requirement as the weak indivisibility assumption. We also describe structures to show that our lower bound is tight up to only a small factor.

References

  1. A. Aggarwal and J. S. Vitter. The input/output complexity of sorting and related problems. Communications of the ACM (CACM), 31(9):1116--1127, 1988. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. L. Arge, V. Samoladas, and J. S. Vitter. On two-dimensional indexability and optimal range search indexing. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), pages 346--357, 1999. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. L. Arge and J. S. Vitter. Optimal external memory interval management. SIAM Journal of Computing, 32(6):1488--1508, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. N. Beckmann, H. Kriegel, R. Schneider, and B. Seeger. The R*-tree: An efficient and robust access method for points and rectangles. In Proceedings of ACM Management of Data (SIGMOD), pages 322--331, 1990. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. J. L. Bentley. Multidimensional binary search trees used for associative searching. Communications of the ACM (CACM), 18(9):509--517, 1975. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. B. Chazelle. Filtering search: A new approach to query-answering. SIAM Journal of Computing, 15(3):703--724, 1986. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. R. Grossi and G. F. Italiano. Efficient splitting and merging algorithms for order decomposable problems. Information and Computation, 154(1):1--33, 1999. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. A. Guttman. R-trees: a dynamic index structure for spatial searching. In Proceedings of ACM Management of Data (SIGMOD), pages 47--57, 1984. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. J. M. Hellerstein, E. Koutsoupias, D. P. Miranker, C. H. Papadimitriou, and V. Samoladas. On a model of indexability and its bounds for range queries. Journal of the ACM (JACM), 49(1):35--55, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. J. M. Hellerstein, E. Koutsoupias, and C. H. Papadimitriou. On the analysis of indexing schemes. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), pages 249--256, 1997. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. J. Iacono and M. Patrascu. Using hashing to solve the dictionary problem (in external memory). To appear in Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2012. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. K. V. R. Kanth and A. K. Singh. Optimal dynamic range searching in non-replicating index structures. In Proceedings of International Conference on Database Theory (ICDT), pages 257--276, 1999. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. E. Koutsoupias and D. S. Taylor. Tight bounds for 2-dimensional indexing schemes. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), pages 52--58, 1998. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. M. H. Overmars. The Design of Dynamic Data Structures. Springer-Verlag, 1987. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. O. Procopiuc, P. K. Agarwal, L. Arge, and J. S. Vitter. Bkd-tree: A dynamic scalable kd-tree. In Proceedings of Symposium on Advances in Spatial and Temporal Databases (SSTD), pages 46--65, 2003.Google ScholarGoogle ScholarCross RefCross Ref
  16. J. T. Robinson. The K-D-B-tree: A search structure for large multidimensional dynamic indexes. In Proceedings of ACM Management of Data (SIGMOD), pages 10--18, 1981. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. V. Samoladas and D. P. Miranker. A lower bound theorem for indexing schemes and its application to multidimensional range queries. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), pages 44--51, 1998. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. M. Streppel and K. Yi. Approximate range searching in external memory. Algorithmica, 59(2):115--128, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. S. Subramanian and S. Ramaswamy. The p-range tree: A new data structure for range searching in secondary memory. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 378--387, 1995. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. E. Verbin and Q. Zhang. The limits of buffering: a tight lower bound for dynamic membership in the external memory model. In Proceedings of ACM Symposium on Theory of Computing (STOC), pages 447--456, 2010. Google ScholarGoogle ScholarDigital LibraryDigital Library

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          cover image ACM Conferences
          PODS '12: Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI symposium on Principles of Database Systems
          May 2012
          332 pages
          ISBN:9781450312486
          DOI:10.1145/2213556

          Copyright © 2012 ACM

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          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 21 May 2012

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