Abstract
The selection problem has a long history, with deep roots in concrete complexity theory, and important practical applications. Nevertheless, some very basic questions remain unresolved. Included among these is the exact specification of the worst-case complexity of selecting the third largest of a set of n elements, a problem (originally formulated in terms of tennis tournaments) that dates back to 1883.
Inspired, in part, by the contributions of J. Ian Munro, to the selection problem as well as many other problems in concrete complexity, we revisit a question concerning the complexity of selecting the third largest element. The question was raised, and only partially solved, in the author’s Ph.D. thesis, at a time that marks the beginning of a long friendship with Ian, and research journeys with numerous parallel interests. In this paper, we settle one very specific instance of this question that is interesting, in part, because it constitutes (i) a new counterexample to a natural conjecture about the exact complexity of this problem, and (ii) what the author now believes is the only remaining counterexample.
The author hopes, in this modest way, to reflect his deep admiration for Ian’s many contributions to the theory, practice and appreciation of algorithm design and analysis.
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References
Aigner, M.: Selecting the top three elements. Discrete Applied Mathematics 4, 247–267 (1982)
Cardinal, J., Florini, S., Joret, G., Jungers, R.M., Munro, J.I.: An efficient algorithm for partial order production. SIAM J. on Computing 39, 2927–2940 (2010)
Cunto, W., Munro, J.I.: Average case selection. J. ACM 36, 270–279 (1989)
Cunto, W., Munro, J.I., Rey, M.: Selecting the median and two quartiles in a set of numbers. Software: Practice and Experience 22, 439–454 (1992)
Dobkin, D., Munro, J.I.: Optimal time minimal space selection algorithms. J. ACM 28, 454–461 (1981)
Eusterbrock, J.: Errata to “Selecting the top three elements” by M. Aigner: A result of a computer-assisted proof search. Discrete Applied Mathematics 41, 131–137 (1993)
Hadian, A., Sobel, M.: Selecting the i-th largest using binary errorless comparisons. Colloquia Mathematica Societatis Janos Bolyai 4, 585–599 (1969)
Kaligosi, K., Mehlhorn, K., Munro, J.I., Sanders, P.: Towards optimal multiple selection. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 103–114. Springer, Heidelberg (2005)
Kirkpatrick, D.G.: Topics in the complexity of combinatorial algorithms. Technical Report 74, Department of Computer Science, University of Toronto (1974)
Kirkpatrick, D.G.: A unified lower bound for selection and set partitioning problems. J. ACM 28, 150–165 (1981)
Kislitsyn, S.S.: On the selection of the k-th element of an ordered set by pairwise comparisons. Sibirsk. Mat. Zh. 5, 557–564 (1964) (in Russian)
Knuth, D.E.: Sorting and Searching. The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1973)
Knuth, D.E.: Sorting and Searching, 2nd edn. The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1998)
Knuth, D.E.: Private correspondence (July 1996)
Munro, J.I., Paterson, M.S.: Selection and sorting with limited storage. Theoretical Computer Science 12, 315–323 (1980)
Munro, J.I., Poblete, P.V.: A lower bound for determining the median. Technical Research Report CS-82-21, University of Waterloo (1982)
Schreier, J.: On tournament elimination systems. Mathesis Polska 7, 154–160 (1932) (in Polish)
Yao, F.F.: On lower bounds for selection problems. Technical Report MAC TR-121, M. I. T. (1974)
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Kirkpatrick, D. (2013). Closing a Long-Standing Complexity Gap for Selection: V 3(42) = 50. In: Brodnik, A., López-Ortiz, A., Raman, V., Viola, A. (eds) Space-Efficient Data Structures, Streams, and Algorithms. Lecture Notes in Computer Science, vol 8066. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40273-9_6
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DOI: https://doi.org/10.1007/978-3-642-40273-9_6
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