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

TL;DR

This paper introduces both deterministic and randomized algorithms for the selection problem in faulty memory models, achieving optimal or near-optimal time complexities and enabling in-place resilient sorting.

Contribution

It presents the first deterministic resilient selection algorithm with optimal O(n) worst-case time and a simpler randomized in-place resilient selection with expected O(n) time, improving resilient data structures.

Findings

Deterministic resilient selection runs in O(n) worst-case time, independent of faults.

Randomized resilient selection achieves expected O(n) time and is in-place.

Resilient sorting algorithm is in-place with optimal O(n log n + elta) expected time.

Abstract

The selection problem, where one wishes to locate the $k^{th}$ 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. In this model, the content of memory cells might get corrupted adversarially during the execution, and the algorithm is given an upper bound $\delta$ on the number of corruptions that may occur. 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 $\delta$. The aforementioned resilient selection algorithm can be used to improve the complexity bounds for resilient $k$-d trees developed by Gieseke, Moruz and Vahrenhold. Specifically, the time complexity for constructing a $k$-d tree is improved from $O(n\log^2 n + \delta^2)$ 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 $O(n + \alpha)$ 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 $O(n\log n + \alpha\delta)$ expected running time.