Abstract
The multiple selection problem asks for the elements of rank r 1, r 2, ..., r k from a linearly ordered set of n elements. Let B denote the information theoretic lower bound on the number of element comparisons needed for multiple selection. We first show that a variant of multiple quickselect — a well known, simple, and practical generalization of quicksort — solves this problem with \(B+\mathcal{O}(n)\) expected comparisons. We then develop a deterministic divide-and-conquer algorithm that solves the problem in \(\mathcal{O}(B)\) time and \(B+o(B)+\mathcal{O}(n)\) element comparisons.
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References
Blelloch, G.E., Leiserson, C.E., Maggs, B.M., Plaxton, C.G., Smith, S.J., Zagha, M.: A comparison of sorting algorithms for the connection machine CM-2. In: 3rd ACM Symposium on Parallel Algorithms and Architectures, pp. 3–16 (1991)
Blum, M., Floyd, R.W., Pratt, V.R., Rivest, R.L., Tarjan, R.E.: Time bounds for selection. Journal of Computer and System Sciences 7(4), 448–461 (1973)
Chambers, J.: Partial sorting (algorithm 410). Communications of the ACM 14, 357–358 (1971)
Cunto, W., Munro, J.I.: Average case selection. J. ACM 36(2), 270–279 (1989)
Dobkin, D.P., Munro, J.I.: Optimal time minimal space selection algorithms. Journal of the ACM 28(3), 454–461 (1981)
Dor, D., Zwick, U.: Selecting the median. In: SODA: ACM-SIAM Symposium on Discrete Algorithms (1995)
Floyd, R.W., Rivest, R.L.: Expected time bounds for selection. Commun. ACM 18(3), 165–172 (1975)
Hoare, C.A.R.: Find (algorithm 65). Communications of the ACM 4(7), 321–322 (1961)
Ford Jr., L.R., Johnson, S.B.: A tournament problem. AMM 66(5), 387–389 (1959)
Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)
Panholzer, A.: Analysis of multiple quickselect variants. Theor. Comput. Sci. 302(1-3), 45–91 (2003)
Pohl, I.: A sorting problem and its complexity. Commun. ACM 15(6), 462–464 (1972)
Prodinger, H.: Multiple quickselect - Hoare’s find algorithm for several elements. Information Processing Letters 56, 123–129 (1995)
Schönhage, A., Paterson, M., Pippenger, N.: Finding the median. J. Comput. Syst. Sci. 13, 184–199 (1976)
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Kaligosi, K., Mehlhorn, K., Munro, J.I., Sanders, P. (2005). Towards Optimal Multiple Selection. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_9
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DOI: https://doi.org/10.1007/11523468_9
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