arXiv subjects

Reilly Browne

Publications and source records attributed to Reilly Browne.

7 recordsLinked to original sources

arXiv2026

Decomposing a Simple Polygon with Geodesic Unit-Balls

We consider covering and partitioning a simple polygon into pieces which either have unit geodesic radius or unit geodesic diameter, using the $\ell_2$-metric for distances. There is no known method for finding an exact solution to these problems, even when the input size is constant, and the problem is known to be NP-hard in the case of polygons with holes. With this in mind, we instead devote our attention to developing simple approximation algorithms that run in polynomial time. For the radius problem, we present the first known approximation algorithms for both covering and partitioning, achieving a factor of 9. For the diameter problem, we are only able to give a positive result for the partition version of the problem, where we improve upon a complicated 72-approximation from Abrahamsen and Rasmussen [SODA '25], achieving a simple 15-approximation.

cs.CG↗

arXiv2026

Single-Criteria Metric $r$-Dominating Set Problem via Minor-Preserving Support

Given an unweighted graph $G$, the *minimum $r$-dominating set problem* asks for the smallest-cardinality subset $S$ such that every vertex in $G$ is within radius $r$ of some vertex in $S$. While the $r$-dominating set problem on planar graphs admits a PTAS from Baker's shifting/layering technique when $r$ is constant, it becomes significantly harder when $r$ can depend on $n$. Under the Exponential-Time Hypothesis, Fox-Epstein et al. [SODA 2019] showed that no efficient PTAS exists for the unbounded $r$-dominating set problem on planar graphs. One may also consider the harder *vertex-weighted metric $r$-dominating set*, where edges have lengths, vertices have positive weights, and the goal is to find an $r$-dominating set of minimum total weight. This led to the development of *bicriteria* algorithms that allow radius-$(1+\varepsilon)r$ balls while achieving a $1+\varepsilon$ approximation to the optimal weight. We establish the first *single-criteria* polynomial-time $O(1)$-approximation algorithm for the vertex-weighted metric $r$-dominating set on planar graphs, where $r$ is part of the input and can be arbitrarily large. Our algorithm applies the quasi-uniformity sampling of Chan et al. [SODA 2012] by bounding the *shallow cell complexity* of the radius-$r$ ball system to be linear in $n$. Two technical innovations enable this: 1. Since discrete ball systems on planar graphs are neither pseudodisks nor amenable to standard union-complexity arguments, we construct a *support graph* for arbitrary distance ball systems as contractions of Voronoi cells, with sparseness as a byproduct. 2. We assign each depth-($\geq 3$) cell to a unique 3-tuple of ball centers, enabling Clarkson-Shor techniques to reduce counting to depth-*exactly*-3 cells, which we prove are $O(n)$ by a geometric argument on our Voronoi contraction support.

cs.DS↗

arXiv2026

Covering and Partitioning Complex Objects with Small Pieces

We study the problems of covering or partitioning a polygon $P$ (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to write $P$ as a union of small pieces, and in partitioning, we furthermore require the pieces to be pairwise interior-disjoint. We show that these problems are in fact equivalent: Optimum covers and partitions have the same number of pieces. For covering, a natural local search algorithm repeatedly attempts to replace $k$ pieces from a candidate cover with $k-1$ pieces. In two dimensions and for sufficiently large $k$, we show that when no such swap is possible, the cover is a $1+O(1/\sqrt k)$-approximation, hence obtaining the first PTAS for the problem. Prior to our work, the only known algorithm was a $13$-approximation that only works for polygons without holes [Abrahamsen and Rasmussen, SODA 2025]. In contrast, in the three dimensional version of the problem, for a polyhedron $P$ of complexity $n$, we show that it is NP-hard to approximate an optimal cover or partition to within a factor that is logarithmic in $n$, even if $P$ is simple, i.e., has genus $0$ and no holes.

cs.CG↗

arXiv2024

An Overview of Minimum Convex Cover and Maximum Hidden Set

We give a review of results on the minimum convex cover and maximum hidden set problems. In addition, we give some new results. First we show that it is NP-hard to determine whether a polygon has the same convex cover number as its hidden set number. We then give some important examples in which these quantities don't always coincide. Finally, We present some consequences of insights from Browne, Kasthurirangan, Mitchell and Polishchuk [FOCS, 2023] on other classes of simple polygons.

cs.CG↗

arXiv2023

Convex Cover and Hidden Set in Funnel Polygons

We present linear-time algorithms for both maximum hidden set and minimum convex cover in funnel polygons. These algorithms show that funnel polygons are "homestead" polygons, i.e. polygons for which the hidden set number and the convex cover number coincide. We extend the algorithm to apply to maximum hidden vertex set and use the result to give a 2-approximation for all three problems in pseudotriangles.

cs.CG↗

arXiv2023

Cache-Oblivious Parallel Convex Hull in the Binary Forking Model

We present two cache-oblivious sorting-based convex hull algorithms in the Binary Forking Model. The first is an algorithm for a presorted set of points which achieves $O(n)$ work, $O(\log n)$ span, and $O(n/B)$ serial cache complexity, where $B$ is the cache line size. These are all optimal worst-case bounds for cache-oblivious algorithms in the Binary Forking Model. The second adapts Cole and Ramachandran's cache-oblivious sorting algorithm, matching its properties including achieving $O(n \log n)$ work, $O(\log n \log \log n)$ span, and $O(n/B \log_M n)$ serial cache complexity. Here $M$ is the size of the private cache.

cs.DS↗

arXiv2023

Fast American Option Pricing using Nonlinear Stencils

We study the binomial, trinomial, and Black-Scholes-Merton models of option pricing. We present fast parallel discrete-time finite-difference algorithms for American call option pricing under the binomial and trinomial models and American put option pricing under the Black-Scholes-Merton model. For $T$-step finite differences, each algorithm runs in $O(\left(T\log^2{T}\right)/p + T)$ time under a greedy scheduler on $p$ processing cores, which is a significant improvement over the $\Theta({T^2}/{p}) + \Omega(T\log{T})$ time taken by the corresponding state-of-the-art parallel algorithm. Even when run on a single core, the $O(T\log^2{T})$ time taken by our algorithms is asymptotically much smaller than the $\Theta(T^2)$ running time of the fastest known serial algorithms. Implementations of our algorithms significantly outperform the fastest implementations of existing algorithms in practice, e.g., when run for $T \approx 1000$ steps on a 48-core machine, our algorithm for the binomial model runs at least $15\times$ faster than the fastest existing parallel program for the same model with the speed-up factor gradually reaching beyond $500\times$ for $T \approx 0.5 \times 10^6$. It saves more than 80\% energy when $T \approx 4000$, and more than 99\% energy for $T > 60,000$. Our option pricing algorithms can be viewed as solving a class of nonlinear 1D stencil (i.e., finite-difference) computation problems efficiently using the Fast Fourier Transform (FFT). To our knowledge, ours are the first algorithms to handle such stencils in $o(T^2)$ time. These contributions are of independent interest as stencil computations have a wide range of applications beyond quantitative finance.

cs.CE↗