Improved running time for continuous witness sets

Improve the running time of the exact algorithm for the Continuous Witness Set Problem in weak visibility polygons beyond \(O(n\log n+\rho^{2}(n+\rho^{2}))\), potentially by using \(\rho\) ray-shooting queries per state and a two-dimensional range-minimum data structure.

Background

The exact continuous algorithm computes a maximum witness set in $O(n\log n+\rho^{2}(n+\rho^{2}))$ time, where $n$ is the number of polygon vertices and $\rho$ is the number of reflex vertices.

The paper notes a possible refinement replacing visibility-polygon computations and scans over static candidates, which appears to yield $O(n\log n+\rho^{3}\log^{2}n)$, but does not establish whether this or another improvement is achievable.