arXiv subjects

Bradley McCoy

Publications and source records attributed to Bradley McCoy.

6 recordsLinked to original sources

arXiv2026

Optimistic Imprecise Shortest Watchtower in 1.5D and 2.5D

A 1.5D imprecise terrain is an $x$-monotone polyline with fixed $x$-coordinates, the $y$-coordinate of each vertex is not fixed but is constrained to be in a given vertical interval. A 2.5D imprecise terrain is a triangulation with fixed $x$ and $y$-coordinates, but the $z$-coordinate of each vertex is constrained to a given vertical interval. Given an imprecise terrain with $n$ intervals, the optimistic shortest watchtower problem asks for a terrain $T$ realized by a precise point in each vertical interval such that the height of the shortest vertical line segment whose lower endpoint lies on $T$ and upper endpoint sees the entire terrain is minimized. In this paper, we present a linear time algorithm to solve the 1.5D optimistic shortest watchtower problem exactly. For the discrete version of the 2.5D case (where the watchtower must be placed on a vertex of $T$), and we give an additive approximation scheme running in $O(\log(\frac{OPT}{\varepsilon})n^3)$ time, achieving a solution within an additive error of $\varepsilon$ from the optimal solution value ${OPT}$.

cs.CG↗

arXiv2025

Trees maximizing the number of almost-perfect matchings

We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings missing only one or two leaves. We also determine the trees that minimize the number of maximal matchings. We apply these results to extremal problems on the weighted Hosoya index for several choices of vertex-degree-based weight function.

math.CO↗

arXiv2023

Bounding mean orders of sub-$k$-trees of $k$-trees

For a $k$-tree $T$, we prove that the maximum local mean order is attained in a $k$-clique of degree $1$ and that it is not more than twice the global mean order. We also bound the global mean order if $T$ has no $k$-cliques of degree $2$ and prove that for large order, the $k$-star attains the minimum global mean order. These results solve the remaining problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-$k$-trees of $k$-trees.

math.CO↗

arXiv2023

From Curves to Words and Back Again: Geometric Computation of Minimum-Area Homotopy

Let $γ$ be a generic closed curve in the plane. Samuel Blank, in his 1967 Ph.D. thesis, determined if $γ$ is self-overlapping by geometrically constructing a combinatorial word from $γ$. More recently, Zipei Nie, in an unpublished manuscript, computed the minimum homotopy area of $γ$ by constructing a combinatorial word algebraically. We provide a unified framework for working with both words and determine the settings under which Blank's word and Nie's word are equivalent. Using this equivalence, we give a new geometric proof for the correctness of Nie's algorithm. Unlike previous work, our proof is constructive which allows us to naturally compute the actual homotopy that realizes the minimum area. Furthermore, we contribute to the theory of self-overlapping curves by providing the first polynomial-time algorithm to compute a self-overlapping decomposition of any closed curve $γ$ with minimum area.

cs.CG↗

arXiv2022

Catching Polygons

Consider an arrangement of $k$ lines intersecting the unit square. There is some minimum scaling factor so that any placement of a rectangle with aspect ratio $1 \times p$ with $p\geq 1$ must non-transversely intersect some portion of the arrangement or unit square. Assuming the lines of the arrangement are axis-aligned, we show the optimal arrangement depends on the aspect ratio of the rectangle. In particular, the optimal arrangement is either evenly spaced parallel lines or an evenly spaced grid of lines. We present the precise aspect ratios of rectangles for which each of the two nets is optimal.

cs.CG↗

arXiv2021

If You Must Choose Among Your Children, Pick the Right One

Given a simplicial complex $K$ and an injective function $f$ from the vertices of $K$ to $\mathbb{R}$, we consider algorithms that extend $f$ to a discrete Morse function on $K$. We show that an algorithm of King, Knudson and Mramor can be described on the directed Hasse diagram of $K$. Our description has a faster runtime for high dimensional data with no increase in space.

cs.CG↗