计算几何

Let $G$ be a complete edge-weighted graph on $n$ vertices. To each subset of vertices of $G$ assign the cost of the minimum spanning tree of the subset as its weight. Suppose that $n$ is a multiple of some fixed positive integer $k$. The…

We define a new framework that unifies the filtration and mapper approaches from TDA, and present efficient algorithms to compute it. Termed the box filtration of a PCD, we grow boxes (hyperrectangles) that are not necessarily centered at…

An important task in terrain analysis is computing \emph{viewsheds}. A viewshed is the union of all the parts of the terrain that are visible from a given viewpoint or set of viewpoints. The complexity of a viewshed can vary significantly…

We consider algorithmic problems motivated by modular robotic reconfiguration in the sliding square model, in which we are given $n$ square-shaped modules in a (labeled or unlabeled) start configuration and need to find a schedule of…

Given in the plane a set $S$ of $n$ points and a set of disks centered at these points, the disk graph $G(S)$ induced by these disks has vertex set $S$ and an edge between two vertices if their disks intersect. Note that the disks may have…

We prove that, for any two polyhedral manifolds $\mathcal P, \mathcal Q$, there is a polyhedral manifold $\mathcal I$ such that $\mathcal P, \mathcal I$ share a common unfolding and $\mathcal I,\mathcal Q$ share a common unfolding. In other…

We introduce a scalable witness-based persistent homology pipeline for full-brain MRI volumes that couples density-aware landmark selection with a GPU-ready witness filtration. Candidates are scored by a hybrid metric that balances…

I present a regression algorithm that provides a continuous, piecewise-smooth function approximating scattered data. It is based on composing and blending linear functions over Voronoi cells, and it scales to high dimensions. The algorithm…

We provide a simple proof of the existence of a planar separator by showing that it is an easy consequence of the circle packing theorem. We also reprove other results on separators, including: (A) There is a simple cycle separator if the…

Given a convex polytope $P$ defined with $n$ vertices in $\mathbb{R}^3$, this paper presents an algorithm to preprocess $P$ to compute routing tables at every vertex of $P$ so that a data packet can be routed on $P$ from any vertex of $P$…

Addressing irregular cutting and packing (C&P) optimization problems poses two distinct challenges: the geometric challenge of determining whether or not an item can be placed feasibly at a certain position, and the optimization challenge…

This paper presents a novel topology-aware dimensionality reduction approach aiming at accurately visualizing the cyclic patterns present in high dimensional data. To that end, we build on the Topological Autoencoders (TopoAE) formulation.…

Avraham et al. [AFK+15] presented an alternative approach to parametric search, called \emph{bifurcation}, that performs faster under certain circumstances. Intuitively, when the underlying decider execution can be rolled back cheaply and…

Periodic point sets model all solid crystalline materials (crystals) whose atoms can be considered zero-sized points with or without atomic types. This paper addresses the fundamental problem of checking whether claimed crystals are novel,…

We study the problem of gradually representing a complex graph as a sequence of drawings of small subgraphs whose union is the complex graph. The sequence of drawings is called \emph{storyplan}, and each drawing in the sequence is called a…

Standard sweep algorithms require an order of discrete points in Euclidean space, and rely on the property that, at a given point, all points in the halfspace below come earlier in this order. We are motivated by the problem of…

The Mapper algorithm is a fundamental tool in exploratory topological data analysis for identifying connectivity and topological clustering in data. Derived from the nerve construction, Mapper graphs can contain additional information about…

The NP-complete mutual-visibility (MV) problem currently lacks empirical analysis on its practical behaviour despite theoretical studies. This paper addresses this gap by implementing and evaluating three distinct algorithms -- a direct…

Constant workspace algorithms use a constant number of words in addition to the read-only input to the algorithm. In this paper, we devise algorithms to efficiently compute relative hulls in the plane using a constant workspace.…

Given a pointed metric space $(X,\mathsf{dist}, w)$ on $n$ points, its Gromov's approximating tree is a 0-hyperbolic pseudo-metric space $(X,\mathsf{dist}_T)$ such that $\mathsf{dist}(x,w)=\mathsf{dist}_T(x,w)$ and $\mathsf{dist}(x, y)-2…