Abstract
We consider the fundamental sorting and selection problems on a list of elements that are not necessarily from a totally ordered set. Here relation between elements are determined by ‘equality’ comparisons whose outcome is \(=\) when the two elements being compared are equal and \(\ne \) otherwise. We determine the complexity of sorting (finding the frequency of every element), finding mode and other frequently occurring elements using only \(=, \ne \) comparisons. We show that \(\Omega (n^2/m)\) comparisons are necessary and this many comparisons are sufficient to find an element that appears at least m times. This is in sharp contrast to the bound of \(\Theta (n \log (n /m))\) bound in the model where comparisons are \(<, =, >\) or \(\le , >\).
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References
Ajtai, M., Feldman, V., Hassidim, A., Nelson, J.: Sorting and selection with imprecise comparisons. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 37–48. Springer, Heidelberg (2009)
Alonso, L., Reingold, E.M., Schott, R.: The Average-Case Complexity of Determining the Majority. SIAM Journal on Computing 26, 1–14 (1997)
Alonso, L., Reingold, E.M., Schott, R.: Determining the majority. Information Processing Letters 47, 253–255 (1993)
Alonso, L., Reingold, E.M., Schott, R.: Analysis of Boyer and Moore’s MJRTY. Information Processing Letters 113, 495–497 (2013)
Bollobaś, B.: Extremal Graph Theory. Academic Press (1978)
Boyer, R.S., Moore, J.S.: MJRTY - A fast majority vote algorithm. In: Boyer, R.S. (ed.) Automated Reasoning: Essays in Honor of Woody Bledsoe. Automated Reasoning Series, pp. 105–117. Kluwer (1991)
Dobkin, D.P., Munro, J.I.: Determining the Mode. Theoretical Computer Science 12, 255–263 (1980)
Fischer, M.J., Salzberg, S.L.: Solution to problem 81–5. Journal of Algorithms 3, 376–379 (1982)
Munro, J.I., Spira, P.M.: Sorting and Searching in Multisets. SIAM Journal on Computing 5(1), 1–8 (1976)
Misra, J., Gries, D.: Finding Repeated Elements. Science of Computing Programming 2(2), 143–152 (1982)
Reingold, E.M.: On the optimality of some set algorithms. Journal of the ACM 19(4), 649–659 (1972)
Saks, M.E., Werman, M.: On computing majority by comparisons. Combinatorica 11(4), 383–387 (1991)
Simpson, J.A., Weiner, E.S.C.: The Oxford English Dictionary, vol. XVI. Clarendon Press (1989)
Turań, P.: On an extremal problem in graph theory (in Hungarian). Mat. Fiz. Lapk 48, 436–452 (1941)
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Jayapaul, V., Munro, J.I., Raman, V., Satti, S.R. (2015). Sorting and Selection with Equality Comparisons. In: Dehne, F., Sack, JR., Stege, U. (eds) Algorithms and Data Structures. WADS 2015. Lecture Notes in Computer Science(), vol 9214. Springer, Cham. https://doi.org/10.1007/978-3-319-21840-3_36
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DOI: https://doi.org/10.1007/978-3-319-21840-3_36
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