Robust Max Selection (2409.06014v1)

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.

• The paper presents a framework that outputs a set of at least min{n, 2k+1} elements to ensure the uncorrupted maximum is included.
• Deterministic methods require Θ(nk) comparisons while randomized algorithms achieve O(n + k polylog k) queries with high confidence.
• The study provides key insights for designing resilient algorithms in adversarial environments, impacting distributed systems and fault-tolerant computing.

Robust Max Selection: An Analytical Overview

The paper "Robust Max Selection" by Trung Dang and Zhiyi Huang presents a comprehensive paper on algorithm design in the context of unreliable information, specifically addressing the problem of finding the uncorrupted maximum element in an array containing corrupted elements. This problem is particularly relevant in distributed systems where input data may be controlled by adversarial actors.

Model and Problem Statement

The authors introduce a scenario where a list of $n$ elements includes $k$ corrupted elements. These corrupted elements exhibit arbitrary behaviors in comparison queries, potentially causing cycles in the ordering of elements. The challenge is to design algorithms for selecting the maximum element such that the selection process remains robust against this adversarial interference.

Key Constraints and Observations:

1. Output Set Size:
   • It is theoretically impossible to guarantee the uncorrupted maximum by outputting a single element. For correctness, the algorithms must output a set.
   • The minimal size of the output set for any algorithm ensuring the inclusion of the uncorrupted maximum is shown to be $\min\{n, 2k + 1\}$.
2. Comparison Queries:
   • Deterministic algorithms necessitate $\Theta(nk)$ comparison queries to ensure the inclusion of the uncorrupted maximum in the output set.
   • Randomized algorithms can achieve a more efficient query complexity of $O(n + k \text{polylog} k)$, nearly matching a lower bound of $\Omega(n)$ for a guaranteed success with high probability.

Algorithmic Contributions

Deterministic Algorithms

The deterministic approach in the paper ensures that the maximum element is contained in the output set through an iterative process, maintaining a set $S$ of size $2k+1$ throughout the procedure. The algorithm conducts pairwise comparisons systematically:

1. Adds each new element $x_i$ to the set $S$.
2. Removes an element from $S$ if the set exceeds $2k+1$ elements, ensuring the removed element is smaller than at least $k+1$ other elements in the set.

This method requires $(2 + o(1))nk$ queries, closely matching the lower bound of $(1 - o(1))nk$.

Randomized Algorithms

The proposed randomized algorithm capitalizes on probabilistic techniques to significantly reduce the number of queries while maintaining high confidence in the correctness:

1. Stage 1: Prunes the initial element set to a size of approximately $k^{1+c}$ using $O(n)$ queries.
2. Stage 2: Further refines the element selection by sampling and ranking the remaining elements. It selects a subset expected to contain the maximum with high probability, using $O(k^{1+3c} \log k)$ queries.

The entire process results in $O(n + k \text{polylog} k)$ queries, which is highly efficient given the complexity of the problem.

Implications and Future Directions

The paper has significant practical and theoretical implications for the design of resilient algorithms in adversarial environments. The robust max selection framework could be extended to broader application areas such as data integrity in distributed systems, fault-tolerant data structures, and secure multi-party computations.

Future Research Directions:

1. Exact Deterministic Bounds: There remains a constant factor gap in the deterministic approach, suggesting potential for further optimization.
2. Randomized Lower Bounds: While the upper bound for randomized algorithms is nearly optimal, formalizing a tighter lower bound around $k \text{polylog} k$ could provide deeper insights.
3. Extended Problem Domains: Exploring the proposed model for more complex tasks like sorting or building resilient data structures (e.g., $k$-d trees) can open new avenues in robust algorithm design.

The robust max selection framework advances the understanding of resilient algorithm design under adversarial input conditions, setting a foundation for subsequent innovations in this domain.