Abstract
Assume we are interested in solving a computational task, e.g., sorting n numbers, and we only have access to an unreliable primitive operation, for example, comparison between two numbers. Suppose that each primitive operation fails with probability at most p and that repeating it is not helpful, as it will result in the same outcome. Can we still approximately solve our task with probability 1 − f(p) for a function f that goes to 0 as p goes to 0? While previous work studied sorting in this model, we believe this model is also relevant for other problems. We
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find the maximum of n numbers in O(n) time,
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solve 2D linear programming in O(n logn) time,
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approximately sort n numbers in O(n 2) time such that each number’s position deviates from its true rank by at most O(logn) positions,
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find an element in a sorted array in O(logn loglogn) time.
Our sorting result can be seen as an alternative to a previous result of Braverman and Mossel (SODA, 2008) who employed the same model. While we do not construct the maximum likelihood permutation, we achieve similar accuracy with a substantially faster running time.
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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. LNCS, vol. 5555, pp. 37–48. Springer, Heidelberg (2009)
Blum, M., Luby, M., Rubinfeld, R.: Self-Testing/Correcting with Applications to Numerical Problems. JCSS 47(3), 549–595 (1993)
Braverman, M., Mossel, E.: Noisy Sorting Without Resampling. In: Proc. 19th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA 2008), pp. 268–276 (2008)
Clarkson, K.L.: Las Vegas Algorithms for Linear and Integer Programming when the Dimension is Small. J. ACM 42(2), 488–499 (1995)
Feige, U., Peleg, D., Raghavan, P., Upfal, E.: Computing with Unreliable Information. In: Proceedings 22nd STOC 1990, pp. 128–137 (1990)
Finocchi, I., Grandoni, F., Italiano, G.: Resilient Dictionaries. ACM Transactions on Algorithms 6(1), 1–19 (2009)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading (1994)
Karp, D., Kleinberg, R.: Noisy Binary Search and Applications. In: 18th SODA, pp. 881–890 (2007)
Megiddo, N.: Linear Programming in Linear Time When the Dimension Is Fixed. J. ACM 31(1), 114–127 (1984)
Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)
Pelc, A.: Searching Games with Errors - Fifty Years of Coping with Liars. Theoretical Computer Science 270(1-2), 71–109 (2002)
Schirra, S.: Robustness and Precision Issues in Geometric Computation. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 597–632. Elsevier, Amsterdam (2000)
Seidel, R.: Small-Dimensional Linear Programming and Convex Hulls Made Easy. Discrete & Computational Geometry (6), 423–434 (1991)
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Klein, R., Penninger, R., Sohler, C., Woodruff, D.P. (2011). Tolerant Algorithms. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_62
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DOI: https://doi.org/10.1007/978-3-642-23719-5_62
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