Document Open Access Logo

Query-Competitive Sorting with Uncertainty

Authors Magnús M. Halldórsson , Murilo Santos de Lima

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2019.7.pdf
  • Filesize: 0.51 MB
  • 15 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Theory and algorithms for application domains
Keywords
  • online algorithms
  • sorting
  • randomized algorithms
  • advice complexity
  • threshold tolerance graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We study the problem of sorting under incomplete information, when queries are used to resolve uncertainties. Each of n data items has an unknown value, which is known to lie in a given interval. We can pay a query cost to learn the actual value, and we may allow an error threshold in the sorting. The goal is to find a nearly-sorted permutation by performing a minimum-cost set of queries.
We show that an offline optimum query set can be found in polynomial time, and that both oblivious and adaptive problems have simple query-competitive algorithms. The query-competitiveness for the oblivious problem is n for uniform query costs, and unbounded for arbitrary costs; for the adaptive problem, the ratio is 2.
We then present a unified adaptive strategy for uniform query costs that yields: (i) a 3/2-query-competitive randomized algorithm; (ii) a 5/3-query-competitive deterministic algorithm if the dependency graph has no 2-components after some preprocessing, which has query-competitive ratio 3/2 + O(1/k) if the components obtained have size at least k; (iii) an exact algorithm if the intervals constitute a laminar family. The first two results have matching lower bounds, and we have a lower bound of 7/5 for large components.
We also show that the advice complexity of the adaptive problem is floor[n/2] if no error threshold is allowed, and ceil[n/3 * lg 3] for the general case.

Cite As Get BibTex

Magnús M. Halldórsson and Murilo Santos de Lima. Query-Competitive Sorting with Uncertainty. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.MFCS.2019.7

Author Details

Magnús M. Halldórsson
  • ICE-TCS, Department of Computer Science, Reykjavik University, Iceland
Murilo Santos de Lima
  • ICE-TCS, Department of Computer Science, Reykjavik University, Iceland

Funding

Partially supported by Icelandic Research Fund grant 174484-051.

References

  1. M. Ajtai, V. Feldman, A. Hassidim, and J. Nelson. Sorting and Selection with Imprecise Comparisons. ACM Transactions on Algorithms, 12(2):19:1-19:19, 2016. URL: https://doi.org/10.1145/2701427.
  2. L. Arantes, E. Bampis, A. V. Kononov, M. Letsios, G. Lucarelli, and P. Sens. Scheduling under uncertainty: A query-based approach. In IJCAI 2018: 27th International Joint Conference on Artificial Intelligence, pages 4646-4652, 2018. URL: https://doi.org/10.24963/ijcai.2018/646.
  3. H.-G. Beyer and B. Sendhoff. Robust optimization - a comprehensive survey. Computer Methods in Applied Mechanics and Engineering, 196(33-34):3190-3218, 2007. URL: https://doi.org/10.1016/j.cma.2007.03.003.
  4. J. R. Birge and F. Louveaux. Introduction to Stochastic Programming. Springer Series in Operations Research and Financial Engineering. Springer, 2011. Google Scholar
  5. A. Borodin and R. El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, 1998. Google Scholar
  6. J. Boyar, L. M. Favrholdt, C. Kudahl, K. S. Larsen, and J. W. Mikkelsen. Online Algorithms with Advice: A Survey. ACM Computing Surveys, 50(2), 2017. URL: https://doi.org/10.1145/3056461.
  7. R. Bruce, M. Hoffmann, D. Krizanc, and R. Raman. Efficient Update Strategies for Geometric Computing with Uncertainty. Theory of Computing Systems, 38(4):411-423, 2005. URL: https://doi.org/10.1007/s00224-004-1180-4.
  8. G. Charalambous and M. Hoffmann. Verification Problem of Maximal Points under Uncertainty. In T. Lecroq and L. Mouchard, editors, IWOCA 2013: 24th International Workshop on Combinatorial Algorithms, volume 8288 of Lecture Notes in Computer Science, pages 94-105. Springer Berlin Heidelberg, 2013. URL: https://doi.org/10.1007/978-3-642-45278-9_9.
  9. C. Dürr, T. Erlebach, N. Megow, and J. Meißner. Scheduling with Explorable Uncertainty. In A. R. Karlin, editor, ITCS 2018: 9th Innovations in Theoretical Computer Science Conference, volume 94 of Leibniz International Proceedings in Informatics, pages 30:1-30:14, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.30.
  10. T. Erlebach and M. Hoffmann. Query-competitive algorithms for computing with uncertainty. Bulletin of EATCS, 116:22-39, 2015. URL: http://bulletin.eatcs.org/index.php/beatcs/article/view/335.
  11. T. Erlebach, M. Hoffmann, and F. Kammer. Query-competitive algorithms for cheapest set problems under uncertainty. Theoretical Computer Science, 613:51-64, 2016. URL: https://doi.org/10.1016/j.tcs.2015.11.025.
  12. T. Erlebach, M. Hoffmann, D. Krizanc, M. Mihal'Ák, and R. Raman. Computing minimum spanning trees with uncertainty. In STACS'08: 25th International Symposium on Theoretical Aspects of Computer Science, pages 277-288, 2008. URL: https://arxiv.org/abs/0802.2855.
  13. T. Feder, R. Motwani, L. O'Callaghan, C. Olston, and R. Panigrahy. Computing shortest paths with uncertainty. Journal of Algorithms, 62(1):1-18, 2007. URL: https://doi.org/10.1016/j.jalgor.2004.07.005.
  14. T. Feder, R. Motwani, R. Panigrahy, C. Olston, and J. Widom. Computing the median with uncertainty. SIAM Journal on Computing, 32(2):538-547, 2003. URL: https://doi.org/10.1137/S0097539701395668.
  15. J. Focke, N. Megow, and J. Meißner. Minimum Spanning Tree under Explorable Uncertainty in Theory and Experiments. In C. S. Iliopoulos, S. P. Pissis, S. J. Puglisi, and R. Raman, editors, SEA 2017: 16th International Symposium on Experimental Algorithms, volume 75 of Leibniz International Proceedings in Informatics, pages 22:1-22:14, 2017. URL: https://doi.org/10.4230/LIPIcs.SEA.2017.22.
  16. F. Gavril. Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph. SIAM Journal on Computing, 1(2):180-187, 1972. URL: https://doi.org/10.1137/0201013.
  17. M. Goerigk, M. Gupta, J. Ide, A. Schöbel, and S. Sen. The robust knapsack problem with queries. Computers & Operations Research, 55:12-22, 2015. URL: https://doi.org/10.1016/j.cor.2014.09.010.
  18. M. Gupta, Y. Sabharwal, and S. Sen. The Update Complexity of Selection and Related Problems. Theory of Computing Systems, 59(1):112-132, 2016. URL: https://doi.org/10.1007/s00224-015-9664-y.
  19. C. A. R. Hoare. Quicksort. The Computer Journal, 5(1):10-16, 1962. URL: https://doi.org/10.1093/comjnl/5.1.10.
  20. S. Kahan. A model for data in motion. In STOC'91: 23rd Annual ACM Symposium on Theory of Computing, pages 265-277, 1991. URL: https://doi.org/10.1145/103418.103449.
  21. S. Khanna and W.-C. Tan. On computing functions with uncertainty. In PODS'01: 20th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pages 171-182, 2001. URL: https://doi.org/10.1145/375551.375577.
  22. N. Megow, J. Meißner, and M. Skutella. Randomization Helps Computing a Minimum Spanning Tree under Uncertainty. SIAM Journal on Computing, 46(4):1217-1240, 2017. URL: https://doi.org/10.1137/16M1088375.
  23. C. L. Monma, B. Reed, and W.T. Trotter Jr. Threshold Tolerance Graphs. Journal of Graph Theory, 12(3):343-362, 1988. URL: https://doi.org/10.1002/jgt.3190120307.
  24. C. Olston and J. Widom. Offering a Precision-Performance Tradeoff for Aggregation Queries over Replicated Data. In VLDB 2000: 26th International Conference on Very Large Data Bases, pages 144-155, 2000. URL: http://ilpubs.stanford.edu:8090/437/.
  25. I. O. Ryzhov and W. B. Powell. Information Collection for Linear Programs with Uncertain Objective Coefficients. SIAM Journal on Optimization, 22(4):1344-1368, 2012. URL: https://doi.org/10.1137/12086279X.
  26. A. Salah, K. Li, and K. Li. Lazy-Merge: A Novel Implementation for Indexed Parallel k-Way In-Place Merging. IEEE Transactions on Parallel and Distributed Systems, 27(7):2049-2061, 2015. URL: https://doi.org/10.1109/TPDS.2015.2475763.
  27. C. E. Shannon. A mathematical theory of communication. The Bell System Technical Journal, 27(3):379-423, 1948. Google Scholar
  28. Y. Yamaguchi and T. Maehara. Stochastic Packing Integer Programs with Few Queries. In SODA'18: Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 293-310, 2018. URL: https://doi.org/10.1137/1.9781611975031.21.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact