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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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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