Largest Empty Circle Centered on a Query Line

TL;DR

This paper introduces a query-based approach to find the largest empty circle centered on a query line within a convex hull, providing efficient algorithms for special and general cases.

Contribution

It presents the first solutions for the query version of the Largest Empty Circle problem with efficient preprocessing and query times for various special cases and the general case.

Findings

Queries run in O(log n) time for all cases.

Special case with horizontal query line preprocesses in O(n α(n) log n) time.

General case preprocesses in O(n^3 log n) time.

Abstract

The Largest Empty Circle problem seeks the largest circle centered within the convex hull of a set $P$ of $n$ points in $\mathbb{R}^2$ and devoid of points from $P$. In this paper, we introduce a query version of this well-studied problem. In our query version, we are required to preprocess $P$ so that when given a query line $Q$, we can quickly compute the largest empty circle centered at some point on $Q$ and within the convex hull of $P$. We present solutions for two special cases and the general case; all our queries run in $O(\log n)$ time. We restrict the query line to be horizontal in the first special case, which we preprocess in $O(n \alpha(n) \log n)$ time and space, where $\alpha(n)$ is the slow growing inverse of the Ackermann's function. When the query line is restricted to pass through a fixed point, the second special case, our preprocessing takes $O(n \alpha(n)^{O(\alpha(n))} \log n)$ time and space. We use insights from the two special cases to solve the general version of the problem with preprocessing time and space in $O(n^3 \log n)$ and $O(n^3)$ respectively.