Selection in the Presence of Memory Faults, with Applications to In-place Resilient Sorting

The selection problem, where one wishes to locate the k smallest element in an unsorted array of size n, is one of the basic problems studied in computer science. The main focus of this work is designing algorithms for solving the selection problem in the presence of memory faults. These can happen as the result of cosmic rays, alpha particles, or hardware failures. Specifically, the computational model assumed here is a faulty variant of the RAM model (abbreviated as FRAM ), which was introduced by Finocchi and Italiano [FI04]. In this model, the content of memory cells might get corrupted adversarially during the execution, and the algorithm cannot distinguish between corrupted cells and uncorrupted cells. The model assumes a constant number of reliable memory cells that never become corrupted, and an upper bound δ on the number of corruptions that may occur, which is given as an auxiliary input to the algorithm. An output element is correct if it has rank between k − α and k + α in the input array, where α is the number of corruptions that occurred during the execution of the algorithm. An algorithm is called resilient if it always outputs a correct answer. The main contribution of this work is a deterministic resilient selection algorithm with optimal O(n) worst-case running time. Interestingly, the running time does not depend on the number of faults, and the algorithm does not need to know δ. As part of the solution, several techniques that allow to sometimes use non-tail recursion algorithms in the FRAMmodel are developed. Notice that using recursive algorithms in this model is problematic, as the stack might be too large to fit in reliable memory. The aforementioned resilient selection algorithm can be used to improve the complexity bounds for resilient k-d trees developed by Gieseke, Moruz and Vahrenhold [GMV10]. Specifically, the time complexity for constructing a k-d tree is improved from O(n log n+ δ) to O(n log n). Besides the deterministic algorithm, a randomized resilient selection algorithm is developed, which is simpler than the deterministic one, and has O(n+α) expected time complexity and O(1) space complexity (i.e., is in-place). This algorithm is used to develop the first resilient sorting algorithm that is in-place and achieves optimal O(n log n+αδ) expected running time.