Robust Max Selection

TL;DR

This paper introduces a new model for algorithm design under unreliable information, focusing on identifying the uncorrupted maximum element in a list with corrupted elements, and provides bounds and algorithms for this problem.

Contribution

The paper defines a novel model for unreliable comparisons, establishes lower bounds, and proposes algorithms with near-optimal query complexities for finding the uncorrupted maximum.

Findings

Any algorithm must output a set of size at least min{n, 2k+1} to guarantee inclusion of the uncorrupted maximum.

Deterministic algorithms require Θ(nk) comparisons to find such a set.

A randomized 2-stage algorithm achieves O(n + k polylog k) comparisons, nearly matching the lower bounds.

Abstract

We introduce a new model to study algorithm design under unreliable information, and apply this model for the problem of finding the uncorrupted maximum element of a list containing $n$ elements, among which are $k$ corrupted elements. Under our model, algorithms can perform black-box comparison queries between any pair of elements. However, queries regarding corrupted elements may have arbitrary output. In particular, corrupted elements do not need to behave as any consistent values, and may introduce cycles in the elements' ordering. This imposes new challenges for designing correct algorithms under this setting. For example, one cannot simply output a single element, as it is impossible to distinguish elements of a list containing one corrupted and one uncorrupted element. To ensure correctness, algorithms under this setting must output a set to make sure the uncorrupted maximum element is included. We first show that any algorithm must output a set of size at least $\min\{n, 2k + 1\}$ to ensure that the uncorrupted maximum is contained in the output set. Restricted to algorithms whose output size is exactly $\min\{n, 2k + 1\}$, for deterministic algorithms, we show matching upper and lower bounds of $\Theta(nk)$ comparison queries to produce a set of elements that contains the uncorrupted maximum. On the randomized side, we propose a 2-stage algorithm that, with high probability, uses $O(n + k \operatorname{polylog} k)$ comparison queries to find such a set, almost matching the $\Omega(n)$ queries necessary for any randomized algorithm to obtain a constant probability of being correct.