# Robust and adaptive search

## TL;DR

This paper investigates the query complexity of search algorithms that are robust to errors and adapt to array disorder, providing tight bounds for various parameters and error models.

## Contribution

It introduces new bounds for robust search algorithms that handle imprecise queries and array perturbations, extending classical binary search to more realistic scenarios.

## Key findings

- Query complexities of log n + ck, (1+psilon)log n + ck, and sqrt cnk + o(nk) are optimal under different conditions.
- The results demonstrate how robustness to errors affects search efficiency in disordered arrays.
- The study provides nearly tight bounds for various parameters quantifying errors and disorder.

## Abstract

Binary search finds a given element in a sorted array with an optimal number of $\log n$ queries. However, binary search fails even when the array is only slightly disordered or access to its elements is subject to errors. We study the worst-case query complexity of search algorithms that are robust to imprecise queries and that adapt to perturbations of the order of the elements. We give (almost) tight results for various parameters that quantify query errors and that measure array disorder. In particular, we exhibit settings where query complexities of $\log n + ck$, $(1+\varepsilon)\log n + ck$, and $\sqrt{cnk}+o(nk)$ are best-possible for parameter value $k$, any $\varepsilon>0$, and constant $c$.