Document Open Access Logo

Multiparty Selection

Authors Ke Chen , Adrian Dumitrescu

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2020.42.pdf
  • Filesize: 497 kB
  • 13 pages

Document Identifiers

Author Details

Ke Chen
  • Department of Computer Science, University of Wisconsin-Milwaukee, WI, USA
Adrian Dumitrescu
  • Department of Computer Science, University of Wisconsin-Milwaukee, WI, USA

Cite AsGet BibTex

Ke Chen and Adrian Dumitrescu. Multiparty Selection. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.42

Abstract

Given a sequence A of n numbers and an integer (target) parameter 1 ≤ i ≤ n, the (exact) selection problem is that of finding the i-th smallest element in A. An element is said to be (i,j)-mediocre if it is neither among the top i nor among the bottom j elements of S. The approximate selection problem is that of finding an (i,j)-mediocre element for some given i,j; as such, this variant allows the algorithm to return any element in a prescribed range. In the first part, we revisit the selection problem in the two-party model introduced by Andrew Yao (1979) and then extend our study of exact selection to the multiparty model. In the second part, we deduce some communication complexity benefits that arise in approximate selection. In particular, we present a deterministic protocol for finding an approximate median among k players.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • approximate selection
  • mediocre element
  • comparison algorithm
  • i-th order statistic
  • tournaments
  • quantiles
  • communication complexity

Metrics

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

References

  1. Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. Data Structures and Algorithms. Addison-Wesley, 1983. Google Scholar
  2. Miklós Ajtai, János Komlós, William L. Steiger, and Endre Szemerédi. Optimal parallel selection has complexity o(log log n). Journal of Computer and System Sciences, 38(1):125-133, 1989. URL: https://doi.org/10.1016/0022-0000(89)90035-4.
  3. Andrei Alexandrescu. Fast deterministic selection. In Costas S. Iliopoulos, Solon P. Pissis, Simon J. Puglisi, and Rajeev Raman, editors, Proceedings of the 16th International Symposium on Experimental Algorithms, SEA 2017, London, UK, June 21-23, 2017, volume 75 of LIPIcs, pages 24:1-24:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.SEA.2017.24.
  4. Sara Baase. Computer algorithms - introduction to design and analysis. Addison-Wesley, 1988. Google Scholar
  5. Sebastiano Battiato, Domenico Cantone, Dario Catalano, Gianluca Cincotti, and Micha Hofri. An efficient algorithm for the approximate median selection problem. In Gian Carlo Bongiovanni, Giorgio Gambosi, and Rossella Petreschi, editors, Proceedings of the 4th Italian Conference on Algorithms and Complexity, CIAC 2000, Rome, Italy, March 2000, volume 1767 of Lecture Notes in Computer Science, pages 226-238. Springer, 2000. URL: https://doi.org/10.1007/3-540-46521-9_19.
  6. Samuel W. Bent and John W. John. Finding the median requires 2n comparisons. In Robert Sedgewick, editor, Proceedings of the 17th Annual ACM Symposium on Theory of Computing, STOC 1985, Providence, Rhode Island, USA, May 6-8, 1985, pages 213-216. ACM, 1985. URL: https://doi.org/10.1145/22145.22169.
  7. Manuel Blum, Robert W. Floyd, Vaughan R. Pratt, Ronald L. Rivest, and Robert Endre Tarjan. Time bounds for selection. Journal of Computer and System Sciences, 7(4):448-461, 1973. URL: https://doi.org/10.1016/S0022-0000(73)80033-9.
  8. Ke Chen and Adrian Dumitrescu. Selection algorithms with small groups. International Journal of Foundations of Computer Science, 31(3):355-369, 2020. URL: https://doi.org/10.1142/s0129054120500136.
  9. Francis Y. L. Chin and Hing fung Ting. An improved algorithm for finding the median distributively. Algorithmica, 2:235-249, 1987. URL: https://doi.org/10.1007/BF01840361.
  10. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, 3rd Edition. MIT Press, 2009. URL: http://mitpress.mit.edu/books/introduction-algorithms.
  11. Walter Cunto and J. Ian Munro. Average case selection. Journal of ACM, 36(2):270-279, 1989. URL: https://doi.org/10.1145/62044.62047.
  12. Sanjoy Dasgupta, Christos H. Papadimitriou, and Umesh V. Vazirani. Algorithms. McGraw-Hill, 2008. Google Scholar
  13. Dorit Dor, Johan Håstad, Staffan Ulfberg, and Uri Zwick. On lower bounds for selecting the median. SIAM Journal on Discrete Mathematics, 14(3):299-311, 2001. URL: https://doi.org/10.1137/S0895480196309481.
  14. Dorit Dor and Uri Zwick. Finding the α n-th largest element. Combinatorica, 16(1):41-58, 1996. URL: https://doi.org/10.1007/BF01300126.
  15. Dorit Dor and Uri Zwick. Selecting the median. SIAM Journal on Computing, 28(5):1722-1758, 1999. URL: https://doi.org/10.1137/S0097539795288611.
  16. Adrian Dumitrescu. Finding a mediocre player. In Pinar Heggernes, editor, Proceedings of the 11th International Conference on Algorithms and Complexity, CIAC 2019, Rome, Italy, May 27-29, 2019, volume 11485 of Lecture Notes in Computer Science, pages 212-223. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-17402-6_18.
  17. Adrian Dumitrescu. A selectable sloppy heap. Algorithms, special issue on efficient data structures, 12(3):58, 2019. URL: https://doi.org/10.3390/a12030058.
  18. Stefan Edelkamp and Armin Weiß. Worst-case efficient sorting with quickmergesort. In Stephen G. Kobourov and Henning Meyerhenke, editors, Proceedings of the 21st Workshop on Algorithm Engineering and Experiments, ALENEX 2019, San Diego, CA, USA, January 7-8, 2019, pages 1-14. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975499.1.
  19. Robert W. Floyd and Ronald L. Rivest. Expected time bounds for selection. Communications of ACM, 18(3):165-172, 1975. URL: https://doi.org/10.1145/360680.360691.
  20. Frank Fussenegger and Harold N. Gabow. A counting approach to lower bounds for selection problems. Journal of ACM, 26(2):227-238, 1979. URL: https://doi.org/10.1145/322123.322128.
  21. Abdollah Hadian and Milton Sobel. Selecting the t-th largest using binary errorless comparisons. Technical Report No. 121, School of Statistics, University of Minnesota, 1969. URL: http://hdl.handle.net/11299/199105.
  22. Charles Antony Richard Hoare. Algorithm 63: Partition and algorithm 65: Find. Communications of ACM, 4(7):321-322, 1961. URL: https://doi.org/10.1145/366622.366647.
  23. Laurent Hyafil. Bounds for selection. SIAM Journal on Computing, 5(1):109-114, 1976. URL: https://doi.org/10.1137/0205010.
  24. John W. John. A new lower bound for the set-partitioning problem. SIAM Journal on Computing, 17(4):640-647, 1988. URL: https://doi.org/10.1137/0217040.
  25. Haim Kaplan, László Kozma, Or Zamir, and Uri Zwick. Selection from heaps, row-sorted matrices, and x+y using soft heaps. In Jeremy T. Fineman and Michael Mitzenmacher, editors, Proceedings of the 2nd Symposium on Simplicity in Algorithms, SOSA 2019, San Diego, CA, USA, January 8-9, 2019, volume 69 of OASICS, pages 5:1-5:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/OASIcs.SOSA.2019.5.
  26. David G. Kirkpatrick. A unified lower bound for selection and set partitioning problems. Journal of ACM, 28(1):150-165, 1981. URL: https://doi.org/10.1145/322234.322245.
  27. Jon M. Kleinberg and Éva Tardos. Algorithm design. Addison-Wesley, 2006. Google Scholar
  28. Donald E. Knuth. The art of computer programming, Volume III: Sorting and Searching, 2nd Edition. Addison-Wesley, 1998. URL: https://www.worldcat.org/oclc/312994415.
  29. Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, 1997. Google Scholar
  30. Eyal Kushilevitz, Noam Nisan, and Bill Gasarch. Errata of communication complexity. URL: http://www.cs.technion.ac.il/~eyalk/book.html.
  31. S. L. Mantzaris. On "an improved algorithm for finding the median distributively". Algorithmica, 10(6):501-504, 1993. URL: https://doi.org/10.1007/BF01891834.
  32. Conrado Martínez and Salvador Roura. Optimal sampling strategies in quicksort and quickselect. SIAM Journal on Computing, 31(3):683-705, 2001. URL: https://doi.org/10.1137/S0097539700382108.
  33. Catherine C. McGeoch and J. Doug Tygar. Optimal sampling strategies for quicksort. Random Structures & Algorithms, 7(4):287-300, 1995. URL: https://doi.org/10.1002/rsa.3240070403.
  34. Mike Paterson. Progress in selection. In Rolf G. Karlsson and Andrzej Lingas, editors, Proceedings of the 5th Scandinavian Workshop on Algorithm Theory SWAT 1996, Reykjavík, Iceland, July 3-5, 1996, volume 1097 of Lecture Notes in Computer Science, pages 368-379. Springer, 1996. URL: https://doi.org/10.1007/3-540-61422-2_146.
  35. Anup Rao and Amir Yehudayoff. Communication complexity and applications. Cambridge University Press, 2020. URL: https://doi.org/10.1017/9781108671644.
  36. Michael Rodeh. Finding the median distributively. Journal of Computer and System Sciences, 24(2):162-166, 1982. URL: https://doi.org/10.1016/0022-0000(82)90045-9.
  37. Arnold Schönhage, Mike Paterson, and Nicholas Pippenger. Finding the median. Journal of Computer and System Sciences, 13(2):184-199, 1976. URL: https://doi.org/10.1016/S0022-0000(76)80029-3.
  38. Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In Michael J. Fischer, Richard A. DeMillo, Nancy A. Lynch, Walter A. Burkhard, and Alfred V. Aho, editors, Proceedings of the 11th Annual ACM Symposium on Theory of Computing, STOC 1979, Atlanta, Georgia, USA, April 30 - May 2, 1979, pages 209-213. ACM, 1979. URL: https://doi.org/10.1145/800135.804414.
  39. Frances F. Yao. On lower bounds for selection problems. Technical Report MAC TR-121, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA, 1974. Google Scholar
  40. Chee-Keng Yap. New upper bounds for selection. Communications of ACM, 19(9):501-508, 1976. URL: https://doi.org/10.1145/360336.360339.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact