Nandana Ghosh | Search arXiv

arXiv2026

Largest-Area Convex Quadrilateral in a $1.5$D Terrain

A $1.5$D terrain is a simple polygon bounded by a horizontal base and an $x$-monotone upper chain. We study the problem of finding a largest-area convex quadrilateral contained in an $n$-vertex terrain. We maximize area over the closure of the feasible nondegenerate quadrilaterals, allowing a triangular boundary optimum when necessary. Assuming that no three terrain vertices are collinear, we give a deterministic exact algorithm running in $O(n^2\log n)$ time and using $O(n)$ working space in the algebraic real-RAM. Among all optimum solutions, the algorithm returns a nondegenerate quadrilateral whenever one exists; otherwise, it returns a maximum-area terrain triangle. We also prove that a maximum-area axis-parallel rectangle contained in the terrain yields a tight $\frac12$-approximation and can be computed in $O(n\log n)$ time.

cs.CG↗

arXiv2026

Endpoint Covering of Axis-Parallel Segments:Bichromatic and Monochromatic One-Center

We study exact one-center optimization for axis-parallel segments using axis-parallel squares under endpoint-based coverage, where a segment is \emph{$1$-covered} if the square contains at least one of its endpoints. In the monochromatic problem, we seek a minimum-side-length square that $1$-covers all $n$ input segments. We obtain $O(n\log n)$-time algorithms for both unrestricted and segment-constrained centers. The unrestricted bound matches the known bound implied by the two-representative color-spanning-square problem, whereas the segment-constrained result is new. We also prove matching $Ω(n\log n)$ lower bounds for both center models in the fixed-order algebraic decision-tree model. In the bichromatic problem, an admissible square must fully contain all $m$ blue segments, minimize the number of red segments with an endpoint in its interior, and, subject to this minimum, maximize its side length within a prescribed bounding box. We give deterministic $O(m+n\log^2 n)$-time and $O(m+mn\log n)$-time algorithms for unrestricted and blue-segment-constrained centers, respectively.

cs.CG↗