Result-Sensitive Binary Search with Noisy Information

We describe new algorithms for the predecessor problem in the Noisy Comparison Model. In this problem, given a sorted list L of n (distinct) elements and a query q, we seek the predecessor of q in L: denoted by u, the largest element less than or equal to q. In the Noisy Comparison Model, the result of a comparison between two elements is non-deterministic. Moreover, multiple comparisons of the same pair of elements might have different results: each is generated independently, and is correct with probability p > 1/2. Given an overall error tolerance Q, the cost of an algorithm is measured by the total number of noisy comparisons; these must guarantee the predecessor is returned with probability at least 1 - Q. Feige et al. showed that predecessor queries can be answered by a modified binary search with Theta(log (n/Q)) noisy comparisons. We design result-sensitive algorithms for answering predecessor queries. The query cost is related to the index, k, of the predecessor u in L. Our first algorithm answers predecessor queries with O(log ((log^{*(c)} n)/Q) + log (k/Q)) noisy comparisons, for an arbitrarily large constant c. The function log^{*(c)} n iterates c times the iterated-logarithm function, log^* n. Our second algorithm is a genuinely result-sensitive algorithm whose expected query cost is bounded by O(log (k/Q)), and is guaranteed to terminate after at most O(log((log n)/Q)) noisy comparisons. Our results strictly improve the state-of-the-art bounds when k is in omega(1) intersected with o(n^epsilon), where epsilon > 0 is some constant. Moreover, we show that our result-sensitive algorithms immediately improve not only predecessor-query algorithms, but also binary-search-like algorithms for solving key applications

Error Detection in Number-Theoretic and Algebraic Algorithms

CPU's are unreliable: at any point in a computation, a bit may be altered with some (small) probability. This probability may seem negligible, but for large calculations (i.e., months of CPU time), the likelihood of an error being introduced becomes increasingly significant. Relying on this fact, this thesis defines a statistical measure called robustness, and measures the robustness of several number-theoretic and algebraic algorithms. Consider an algorithm A that implements function f, such that f has range O and algorithm A has range O' where O⊆O'. That is, the algorithm may produce results which are not in the possible range of the function. Specifically, given an algorithm A and a function f, this thesis classifies the output of A into one of three categories: 1. Correct and feasible -- the algorithm computes the correct result, 2. Incorrect and feasible -- the algorithm computes an incorrect result and this output is in O, 3. Incorrect and infeasible -- the algorithm computes an incorrect result and output is in O'\O. Using probabilistic measures, we apply this classification scheme to quantify the robustness of algorithms for computing primality (i.e., the Lucas-Lehmer and Pepin tests), group order and quadratic residues. Moreover, we show that typically, there will be an "error threshold" above which the algorithm is unreliable (that is, it will rarely give the correct result)