Succinct Color Searching in One Dimension

Authors Hicham El-Zein, J. Ian Munro, Yakov Nekrich

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Hicham El-Zein
J. Ian Munro
Yakov Nekrich

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Hicham El-Zein, J. Ian Munro, and Yakov Nekrich. Succinct Color Searching in One Dimension. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 30:1-30:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


In this paper we study succinct data structures for one-dimensional color reporting and color counting problems. We are given a set of n points with integer coordinates in the range [1,m] and every point is assigned a color from the set {1,...\sigma}. A color reporting query asks for the list of distinct colors that occur in a query interval [a,b] and a color counting query asks for the number of distinct colors in [a,b]. We describe a succinct data structure that answers approximate color counting queries in O(1) time and uses \mathcal{B}(n,m) + O(n) + o(\mathcal{B}(n,m)) bits, where \mathcal{B}(n,m) is the minimum number of bits required to represent an arbitrary set of size n from a universe of m elements. Thus we show, somewhat counterintuitively, that it is not necessary to store colors of points in order to answer approximate color counting queries. In the special case when points are in the rank space (i.e., when n=m), our data structure needs only O(n) bits. Also, we show that \Omega(n) bits are necessary in that case. Then we turn to succinct data structures for color reporting. We describe a data structure that uses \mathcal{B}(n,m) + nH_d(S) + o(\mathcal{B}(n,m)) + o(n\lg\sigma) bits and answers queries in O(k+1) time, where k is the number of colors in the answer, and nH_d(S) (d=\log_\sigma n) is the d-th order empirical entropy of the color sequence. Finally, we consider succinct color reporting under restricted updates. Our dynamic data structure uses nH_d(S)+o(n\lg\sigma) bits and supports queries in O(k+1) time.
  • Succinct Data Structures
  • Range Searching
  • Computational Geometry


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  1. Miklós Ajtai. A lower bound for finding predecessors in Yao’s cell probe model. Combinatorica, 8(3):235-247, 1988. Google Scholar
  2. Ricardo Baeza-Yates, Berthier Ribeiro-Neto, et al. Modern information retrieval, volume 463. ACM press New York, 1999. Google Scholar
  3. Paul Beame and Faith E Fich. Optimal bounds for the predecessor problem. In Proceedings of the 31st annual ACM symposium on Theory of computing, pages 295-304. ACM, 1999. Google Scholar
  4. Philip Bille, Gad M Landau, Rajeev Raman, Kunihiko Sadakane, Srinivasa Rao Satti, and Oren Weimann. Random access to grammar-compressed strings. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 373-389. Society for Industrial and Applied Mathematics, 2011. Google Scholar
  5. Prosenjit Bose, Eric Y. Chen, Meng He, Anil Maheshwari, and Pat Morin. Succinct geometric indexes supporting point location queries. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, pages 635-644, 2009. Google Scholar
  6. Panayiotis Bozanis, Nectarios Kitsios, Christos Makris, and Athanasios Tsakalidis. New upper bounds for generalized intersection searching problems. In Proceedings of the 22nd International Colloquium on Automata, Languages, and Programming, pages 464-474. Springer, 1995. Google Scholar
  7. Timothy M Chan and Bryan T Wilkinson. Adaptive and approximate orthogonal range counting. ACM Transactions on Algorithms (TALG), 12(4):45, 2016. Google Scholar
  8. Arash Farzan, J. Ian Munro, and Rajeev Raman. Succinct indices for range queries with applications to orthogonal range maxima. In Proceedings of the 39th International Colloquium on Automata, Languages, and Programming, Part I, pages 327-338, 2012. Google Scholar
  9. Johannes Fischer. Optimal succinctness for range minimum queries. In Proceedings of the 9th Latin American Symposium on Theoretical Informatics, pages 158-169. Springer, 2010. Google Scholar
  10. Michael L Fredman and Dan E Willard. Surpassing the information theoretic bound with fusion trees. Journal of Computer and System Sciences, 47(3):424-436, 1993. Google Scholar
  11. Travis Gagie and Juha Kärkkäinen. Counting colours in compressed strings. In Proceedings of the 22nd Annual Symposium on Combinatorial Pattern Matching, CPM 2011, pages 197-207, 2011. Google Scholar
  12. Travis Gagie, Juha Kärkkäinen, Gonzalo Navarro, and Simon J Puglisi. Colored range queries and document retrieval. Theoretical Computer Science, 483:36-50, 2013. Google Scholar
  13. Mayank Goswami, Allan Grønlund Jørgensen, Kasper Green Larsen, and Rasmus Pagh. Approximate range emptiness in constant time and optimal space. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 769-775, 2015. Google Scholar
  14. Roberto Grossi, Rajeev Raman, Srinivasa Rao Satti, and Rossano Venturini. Dynamic compressed strings with random access. In Proceedings of the 40th International Colloquium on Automata, Languages, and Programming, Part I, pages 504-515, 2013. Google Scholar
  15. Meng He. Succinct and implicit data structures for computational geometry. In Space-Efficient Data Structures, Streams, and Algorithms - Papers in Honor of J. Ian Munro on the Occasion of His 66th Birthday, pages 216-235, 2013. Google Scholar
  16. J Ian Munro and S Srinivasa Rao. Succinct representation of data structures. In Handbook of Data Structures and Applications, chapter 37. Chapman and Hall/CRC, 2004. Google Scholar
  17. Ravi Janardan and Mario Lopez. Generalized intersection searching problems. International Journal of Computational Geometry &Applications, 3(01):39-69, 1993. Google Scholar
  18. Christian W Mortensen. Generalized static orthogonal range searching in less space. Technical report, Citeseer, 2003. Google Scholar
  19. J Ian Munro. Tables. In International Conference on Foundations of Software Technology and Theoretical Computer Science, pages 37-42. Springer, 1996. Google Scholar
  20. S Muthukrishnan. Efficient algorithms for document retrieval problems. In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 657-666. Society for Industrial and Applied Mathematics, 2002. Google Scholar
  21. Yakov Nekrich. Efficient range searching for categorical and plain data. ACM Trans. Database Syst., 39(1):9:1-9:21, 2014. Google Scholar
  22. Yakov Nekrich and Jeffrey Scott Vitter. Optimal color range reporting in one dimension. In Proceedings of the 21st European Symposium on Algorithms, pages 743-754. Springer, 2013. Google Scholar
  23. Mihai Patrascu and Mikkel Thorup. Time-space trade-offs for predecessor search. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 232-240, 2006. Google Scholar
  24. Saladi Rahul. Approximate range counting revisited. arXiv preprint arXiv:1512.01713, 2015. Google Scholar
  25. Rajeev Raman, Venkatesh Raman, and S Srinivasa Rao. Succinct indexable dictionaries with applications to encoding k-ary trees and multisets. In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 233-242, 2002. Google Scholar
  26. Michiel Smid and Prosenjit Gupta. Further results on generalized intersection searching problems: counting, reporting, and dynamization. Technical report, Max-Planck-Institut für Informatik, 1992. Google Scholar
  27. Peter van Emde Boas. Preserving order in a forest in less than logarithmic time. In Proccedings of the 16th Annual Symposium on Foundations of Computer Science, pages 75-84. IEEE, 1975. Google Scholar