skip to main content
research-article

The Power of an Example: Hidden Set Size Approximation Using Group Queries and Conditional Sampling

Published:14 June 2016Publication History
Skip Abstract Section

Abstract

We study a basic problem of approximating the size of an unknown set S in a known universe U. We consider two versions of the problem. In both versions, the algorithm can specify subsets TU. In the first version, which we refer to as the group query or subset query version, the algorithm is told whether TS is nonempty. In the second version, which we refer to as the subset sampling version, if TS is nonempty, then the algorithm receives a uniformly selected element from TS. We study the difference between these two versions in both the case that the algorithm is adaptive and the case in which it is nonadaptive. Our main focus is on a natural family of allowed subsets, which correspond to intervals, as well as variants of this family.

References

  1. P. K. Agarwal and J. Erickson. 1999. Geometric range searching and its relatives. In Advances in Discrete and Computational Geometry. American Mathematical Society, 1--56.Google ScholarGoogle Scholar
  2. A. Anagnostopoulos, A. Z. Broder, and D. Carmel. 2006. Sampling search-engine results. World Wide Web 9, 4, 397--429. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. K. Bharat and A. Z. Broder. 1998. A technique for measuring the relative size and overlap of public web search engines. Computer Networks 30, 1--7, 379--388. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. A. Z. Broder, M. Fontoura, V. Josifovski, R. Kumar, R. Motwani, S. U. Nabar, R. Panigrahy, A. Tomkins, and Y. Xu. 2006. Estimating corpus size via queries. In Proceedings of the 15th ACM International Conference on Information and Knowledge Management (CIKM’06). 594--603. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. C. Cannone, D. Ron, and R. Servedio. 2014. Testing equivalence between probability distributions using conditional samples. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’14). 1174--1192. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. S. Chakraborty, E. Fischer, Y. Goldhirsh, and A. Matsliah. 2013. On the power of conditional samples in distribution testing. In Proceedings of the 2nd Symposium on Innovations in Theoretical Computer Science (ITCS’13). 561--580. DOI:http://dx.doi.org/10.1145/2422436.2422497 Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. R. Dorfman. 1943. The detection of defective members of large populations. The Annals of Mathematical Statistics 14, 4, 436--440. DOI:http://dx.doi.org/10.1214/aoms/1177731363Google ScholarGoogle ScholarCross RefCross Ref
  8. D. Du and F. Hwang. 1993. Combinatorial Group Testing and Its Applications. World Scientific, Singapore.Google ScholarGoogle Scholar
  9. P. Indyk, H. Q. Ngo, and A. Rudra. 2010. Efficiently decodable non-adaptive group testing. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’10). 1126--1142. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. S. Raskhodnikova, D. Ron, A. Shpilka, and A. Smith. 2009. Strong lower bonds for approximating distributions support size and the distinct elements problem. SIAM Journal on Computing 39, 3, 813--842. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. L. J. Stockmeyer. 1983. The complexity of approximate counting (preliminary version). In Proceedings of the 15th Annual ACM Symposium on Theory of Computing (STOC’83). 118--126. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. L. J. Stockmeyer. 1985. On approximation algorithms for #P. SIAM Journal on Computing 14, 4, 849--861.Google ScholarGoogle ScholarCross RefCross Ref
  13. G. Valiant and P. Valiant. 2011. Estimating the unseen: An n/log (n)-sample estimator for entropy and support size, shown optimal via new CLTs. In Proceedings of the 43rd Annual ACM Symposium on the Theory of Computing (STOC’11). 685--694. See also ECCC TR10-179 and TR10-180. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. P. Valiant. 2011. Testing symmetric properties of distributions. SIAM Journal on Computing 40, 6, 1927--1968. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. A. Yao. 1977. Probabilistic computations: Toward a unified measure of complexity. In Proceedings of the 18th IEEE Symposium on Foundations of Computer Science (FOCS'77). 222--227. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. The Power of an Example: Hidden Set Size Approximation Using Group Queries and Conditional Sampling

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 8, Issue 4
        July 2016
        97 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/2956681
        Issue’s Table of Contents

        Copyright © 2016 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 14 June 2016
        • Accepted: 1 April 2016
        • Revised: 1 February 2016
        • Received: 1 October 2014
        Published in toct Volume 8, Issue 4

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article
        • Research
        • Refereed

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!