Two Error Bounds of Imperfect Binary Search

TL;DR

This paper introduces two lemmas that quantify the error bounds of an imperfect binary search algorithm, which uses a predictive comparison instead of actual comparisons to speed up search in sorted tables.

Contribution

The paper provides the first theoretical bounds on the accuracy of binary search when using an approximate comparison method.

Findings

Two error bounds for imperfect binary search are established.

The lemmas quantify how much the search results can deviate from the true index.

Results help in designing faster search algorithms with predictable accuracy.

Abstract

Suppose we know that an object is in a sorted table and we want to determine the index of that object. To achieve this goal we could perform a binary search. However, suppose it is time-consuming to determine the relative position of that object to any other objects in the table. In this scenario, we might want to resort to an incomplete solution: we could device an algorithm that quickly predicts the result of comparing two objects, and replace the actual comparison with this algorithm during a binary search. The question then is how far away are the results yielded by the imperfect binary search from the correct answers. We present two quick lemmas that answer this question.