The Range 1 Query (R1Q) Problem

  • Michael A. Bender
  • Rezaul A. Chowdhury
  • Pramod Ganapathi
  • Samuel McCauley
  • Yuan Tang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8591)


We define the range 1 query (R1Q) problem as follows. Given a d-dimensional (d ≥ 1) input bit matrix A, preprocess A so that for any given region \(\mathcal{R}\) of A, one can efficiently answer queries asking if \(\mathcal{R}\) contains a 1 or not. We consider both orthogonal and non-orthogonal shapes for \(\mathcal{R}\) including rectangles, axis-parallel right-triangles, certain types of polygons, and spheres. We provide space-efficient deterministic and randomized algorithms with constant query times (in constant dimensions) for solving the problem in the word RAM model. The space usage in bits is sublinear, linear, or near linear in the size of A, depending on the algorithm.


R1Q range query range emptiness randomized rectangular orthogonal non-orthogonal triangular polygonal circular spherical 


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  1. 1.
    Agarwal, P.K., Erickson, J.: Geometric range searching and its relatives. Contemporary Mathematics 223, 1–56 (1999)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Amir, A., Fischer, J., Lewenstein, M.: Two-dimensional range minimum queries. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, vol. 4580, pp. 286–294. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Bender, M.A., Farach-Colton, M.: The lca problem revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000)Google Scholar
  4. 4.
    Bender, M.A., Farach-Colton, M., Pemmasani, G., Skiena, S., Sumazin, P.: Lowest common ancestors in trees and directed acyclic graphs. Journal of Algorithms 57(2), 75–94 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Berkman, O., Vishkin, U.: Recursive star-tree parallel data structure. SIAM Journal on Computing 22(2), 221–242 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Brodal, G.S., Davoodi, P., Rao, S.S.: On space efficient two dimensional range minimum data structures. Algorithmica 63(4), 815–830 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chazelle, B., Rosenberg, B.: Computing partial sums in multidimensional arrays. In: SoCG, pp. 131–139. ACM (1989)Google Scholar
  8. 8.
    Cormode, G., Muthukrishnan, S.: An improved data stream summary: The count-min sketch and its applications. Journal of Algorithms 55(1), 58–75 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    De Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational geometry. Springer (2008)Google Scholar
  10. 10.
    Fischer, J.: Optimal succinctness for range minimum queries. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 158–169. Springer, Heidelberg (2010)Google Scholar
  11. 11.
    Fischer, J., Heun, V.: A new succinct representation of rmq-information and improvements in the enhanced suffix array. In: Chen, B., Paterson, M., Zhang, G. (eds.) ESCAPE 2007. LNCS, vol. 4614, pp. 459–470. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Fischer, J., Heun, V., Stiihler, H.: Practical entropy-bounded schemes for o(1)-range minimum queries. In: Data Compression Conference, pp. 272–281. IEEE (2008)Google Scholar
  13. 13.
    Golynski, A.: Optimal lower bounds for rank and select indexes. TCS 387(3), 348–359 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    González, R., Grabowski, S., Mäkinen, V., Navarro, G.: Practical implementation of rank and select queries. In: Poster Proc. WEA, pp. 27–38 (2005)Google Scholar
  15. 15.
    Navarro, G., Nekrich, Y., Russo, L.: Space-efficient data-analysis queries on grids. TCS (2012)Google Scholar
  16. 16.
    Overmars, M.H.: Efficient data structures for range searching on a grid. Journal of Algorithms 9(2), 254–275 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Sadakane, K.: Compressed suffix trees with full functionality. Theory of Computing Systems 41(4), 589–607 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Sadakane, K.: Succinct data structures for flexible text retrieval systems. JDA 5(1), 12–22 (2007)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Sharir, M., Shaul, H.: Semialgebraic range reporting and emptiness searching with applications. SIAM Journal on Computing 40(4), 1045–1074 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Tang, Y., Chowdhury, R., Kuszmaul, B.C., Luk, C.K., Leiserson, C.E.: The Pochoir stencil compiler. In: SPAA, pp. 117–128. ACM (2011)Google Scholar
  21. 21.
    Yao, A.C.: Space-time tradeoff for answering range queries. In: STOC, pp. 128–136. ACM (1982)Google Scholar
  22. 22.
    Yuan, H., Atallah, M.J.: Data structures for range minimum queries in multidimensional arrays. In: SODA, pp. 150–160 (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Michael A. Bender
    • 1
    • 2
  • Rezaul A. Chowdhury
    • 1
  • Pramod Ganapathi
    • 1
  • Samuel McCauley
    • 1
  • Yuan Tang
    • 3
  1. 1.Department of Computer ScienceStony Brook UniversityStony BrookUSA
  2. 2.Tokutek, Inc.USA
  3. 3.Software SchoolFudan UniversityShanghaiChina

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