Document Open Access Logo

Approximation Strategies for Generalized Binary Search in Weighted Trees

Authors Dariusz Dereniowski, Adrian Kosowski, Przemyslaw Uznanski, Mengchuan Zou

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2017.84.pdf
  • Filesize: 0.65 MB
  • 14 pages

Document Identifiers

Author Details

Dariusz Dereniowski
Adrian Kosowski
Przemyslaw Uznanski
Mengchuan Zou

Cite AsGet BibTex

Dariusz Dereniowski, Adrian Kosowski, Przemyslaw Uznanski, and Mengchuan Zou. Approximation Strategies for Generalized Binary Search in Weighted Trees. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 84:1-84:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ICALP.2017.84

Abstract

We consider the following generalization of the binary search problem. A search strategy is required to locate an unknown target node t in a given tree T. Upon querying a node v of the tree, the strategy receives as a reply an indication of the connected component of T\{v} containing the target t. The cost of querying each node is given by a known non-negative weight function, and the considered objective is to minimize the total query cost for a worst-case choice of the target. Designing an optimal strategy for a weighted tree search instance is known to be strongly NP-hard, in contrast to the unweighted variant of the problem which can be solved optimally in linear time. Here, we show that weighted tree search admits a quasi-polynomial time approximation scheme (QPTAS): for any 0 < epsilon < 1, there exists a (1+epsilon)-approximation strategy with a computation time of n^O(log n / epsilon^2). Thus, the problem is not APX-hard, unless NP is contained in DTIME(n^O(log n)). By applying a generic reduction, we obtain as a corollary that the studied problem admits a polynomial-time O(sqrt(log n))-approximation. This improves previous tilde-O(log n)-approximation approaches, where the tilde-O-notation disregards O(poly log log n)-factors.
Keywords
  • Approximation Algorithm
  • Adaptive Algorithm
  • Graph Search
  • Binary Search
  • Vertex Ranking
  • Trees

Metrics

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

References

  1. Esther M. Arkin, Henk Meijer, Joseph S. B. Mitchell, David Rappaport, and Steven Skiena. Decision trees for geometric models. Int. J. Comput. Geometry Appl., 8(3):343-364, 1998. URL: http://dx.doi.org/10.1142/S0218195998000175.
  2. Yosi Ben-Asher and Eitan Farchi. The cost of searching in general trees versus complete binary trees. Technical report, Technical report, 1997. Google Scholar
  3. Ferdinando Cicalese, Tobias Jacobs, Eduardo Sany Laber, and Marco Molinaro. On the complexity of searching in trees and partially ordered structures. Theor. Comput. Sci., 412(50):6879-6896, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2011.08.042.
  4. Ferdinando Cicalese, Tobias Jacobs, Eduardo Sany Laber, and Caio Dias Valentim. The binary identification problem for weighted trees. Theor. Comput. Sci., 459:100-112, 2012. URL: http://dx.doi.org/10.1016/j.tcs.2012.06.023.
  5. Ferdinando Cicalese, Balázs Keszegh, Bernard Lidický, Dömötör Pálvölgyi, and Tomás Valla. On the tree search problem with non-uniform costs. Theor. Comput. Sci., 647:22-32, 2016. URL: http://dx.doi.org/10.1016/j.tcs.2016.07.019.
  6. Dariusz Dereniowski. Edge ranking of weighted trees. Discrete Applied Mathematics, 154(8):1198-1209, 2006. URL: http://dx.doi.org/10.1016/j.dam.2005.11.005.
  7. Dariusz Dereniowski. Edge ranking and searching in partial orders. Discrete Applied Mathematics, 156(13):2493-2500, 2008. URL: http://dx.doi.org/10.1016/j.dam.2008.03.007.
  8. Dariusz Dereniowski and Marek Kubale. Efficient parallel query processing by graph ranking. Fundam. Inform., 69(3):273-285, 2006. Google Scholar
  9. Dariusz Dereniowski and Adam Nadolski. Vertex rankings of chordal graphs and weighted trees. Inf. Process. Lett., 98(3):96-100, 2006. URL: http://dx.doi.org/10.1016/j.ipl.2005.12.006.
  10. Ehsan Emamjomeh-Zadeh, David Kempe, and Vikrant Singhal. Deterministic and probabilistic binary search in graphs. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 519-532, 2016. URL: http://dx.doi.org/10.1145/2897518.2897656.
  11. Archontia C. Giannopoulou, Paul Hunter, and Dimitrios M. Thilikos. Lifo-search: A min-max theorem and a searching game for cycle-rank and tree-depth. Discrete Applied Mathematics, 160(15):2089-2097, 2012. URL: http://dx.doi.org/10.1016/j.dam.2012.03.015.
  12. Brent Heeringa, Marius Catalin Iordan, and Louis Theran. Searching in dynamic tree-like partial orders. In Algorithms and Data Structures - 12th International Symposium, WADS 2011, New York, NY, USA, August 15-17, 2011. Proceedings, pages 512-523, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22300-6_43.
  13. Ananth V. Iyer, H. Donald Ratliff, and Gopalakrishnan Vijayan. Optimal node ranking of trees. Inf. Process. Lett., 28(5):225-229, 1988. URL: http://dx.doi.org/10.1016/0020-0190(88)90194-9.
  14. Ananth V. Iyer, H. Donald Ratliff, and Gopalakrishnan Vijayan. Parallel assembly of modular products - an analysis. Technical report, Technical Report 88-86, Georgia Institute of Technology, 1988. Google Scholar
  15. Meir Katchalski, William McCuaig, and Suzanne M. Seager. Ordered colourings. Discrete Mathematics, 142(1-3):141-154, 1995. URL: http://dx.doi.org/10.1016/0012-365X(93)E0216-Q.
  16. Eduardo Sany Laber, Ruy Luiz Milidiú, and Artur Alves Pessoa. On binary searching with nonuniform costs. SIAM J. Comput., 31(4):1022-1047, 2002. URL: http://dx.doi.org/10.1137/S0097539700381991.
  17. Eduardo Sany Laber and Marco Molinaro. An approximation algorithm for binary searching in trees. Algorithmica, 59(4):601-620, 2011. URL: http://dx.doi.org/10.1007/s00453-009-9325-0.
  18. Eduardo Sany Laber and Loana Tito Nogueira. Fast searching in trees. Electronic Notes in Discrete Mathematics, 7:90-93, 2001. URL: http://dx.doi.org/10.1016/S1571-0653(04)00232-X.
  19. Eduardo Sany Laber and Loana Tito Nogueira. On the hardness of the minimum height decision tree problem. Discrete Applied Mathematics, 144(1-2):209-212, 2004. URL: http://dx.doi.org/10.1016/j.dam.2004.06.002.
  20. Tak Wah Lam and Fung Ling Yue. Optimal edge ranking of trees in linear time. Algorithmica, 30(1):12-33, 2001. URL: http://dx.doi.org/10.1007/s004530010076.
  21. Nathan Linial and Michael E. Saks. Searching ordered structures. J. Algorithms, 6(1):86-103, 1985. URL: http://dx.doi.org/10.1016/0196-6774(85)90020-3.
  22. Joseph W. H. Liu. Computational models and task scheduling for parallel sparse cholesky factorization. Parallel Computing, 3(4):327-342, 1986. URL: http://dx.doi.org/10.1016/0167-8191(86)90014-1.
  23. Joseph W. H. Liu. The role of elimination trees in sparse factorization. SIAM. J. Matrix Anal. & Appl., 11(1):134-172, 1990. Google Scholar
  24. Shay Mozes, Krzysztof Onak, and Oren Weimann. Finding an optimal tree searching strategy in linear time. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008, pages 1096-1105, 2008. Google Scholar
  25. Jaroslav Nesetril and Patrice Ossona de Mendez. Tree-depth, subgraph coloring and homomorphism bounds. Eur. J. Comb., 27(6):1022-1041, 2006. URL: http://dx.doi.org/10.1016/j.ejc.2005.01.010.
  26. Krzysztof Onak and Pawel Parys. Generalization of binary search: Searching in trees and forest-like partial orders. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 379-388, 2006. URL: http://dx.doi.org/10.1109/FOCS.2006.32.
  27. Alex Pothen. The complexity of optimal elimination trees. Technical report, Technical Report CS-88-13, Pennsylvannia State University, 1988. Google Scholar
  28. Alejandro A. Schäffer. Optimal node ranking of trees in linear time. Inf. Process. Lett., 33(2):91-96, 1989. URL: http://dx.doi.org/10.1016/0020-0190(89)90161-0.
  29. George Steiner. Searching in 2-dimensional partial orders. J. Algorithms, 8(1):95-105, 1987. URL: http://dx.doi.org/10.1016/0196-6774(87)90029-0.
  30. Jayme Luiz Szwarcfiter, Gonzalo Navarro, Ricardo A. Baeza-Yates, Joísa de S. Oliveira, Walter Cunto, and Nivio Ziviani. Optimal binary search trees with costs depending on the access paths. Theor. Comput. Sci., 290(3):1799-1814, 2003. URL: http://dx.doi.org/10.1016/S0304-3975(02)00084-1.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact