Tolerant Algorithms

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 – find the maximum of n numbers in O(n) time, – solve 2D linear programming in O(n log n) time, – approximately sort n numbers in O(n) time such that each number’s position deviates from its true rank by at most O(log n) positions, – find an element in a sorted array in O(log n log log n) 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.