计算几何
Additive manufacturing (AM) enables enormous freedom for design of complex structures. However, the process-dependent limitations that result in discrepancies between as-designed and as-manufactured shapes are not fully understood. The…
An additive manufacturing (AM) process often produces a {\it near-net} shape that closely conforms to the intended design to be manufactured. It sometimes contains additional support structure (also called scaffolding), which has to be…
The statistical rank tests play important roles in univariate non-parametric data analysis. If one attempts to generalize the rank tests to a multivariate case, the problem of defining a multivariate order will occur. It is not clear how to…
A deflated polygon is a polygon with no visibility crossings. We answer a question posed by Devadoss et al. (2012) by presenting a polygon that cannot be deformed via continuous visibility-decreasing motion into a deflated polygon. We show…
Classical separability problem involving multi-color point sets is an important area of study in computational geometry. In this paper, we study different separability problems for bichromatic point set P=P_r\cup P_b on a plane, where $P_r$…
Let P be a set of n points and each of the points is colored with one of the k possible colors. We present efficient algorithms to pre-process P such that for a given query point q, we can quickly identify the smallest color spanning object…
Distance Geometry Problem (DGP) and Nonlinear Mapping (NLM) are two well established questions: Distance Geometry Problem is about finding a Euclidean realization of an incomplete set of distances in a Euclidean space, whereas Nonlinear…
In this paper, we study the problem of computing the diameter of a set of $n$ points in $d$-dimensional Euclidean space for a fixed dimension $d$, and propose a new $(1+\varepsilon)$-approximation algorithm with $O(n+ 1/\varepsilon^{d-1})$…
In 2006, Alberto Bressan suggested the following problem. Suppose a circular fire spreads in the Euclidean plane at unit speed. The task is to build, in real time, barrier curves to contain the fire. At each time $t$ the total length of all…
We study continuous analogues of "vitality" for discrete network flows/paths, and consider problems related to placing segment barriers that have highest impact on a flow/path in a polygonal domain. This extends the graph-theoretic notion…
Range closest-pair (RCP) search is a range-search variant of the classical closest-pair problem, which aims to store a given set $S$ of points into some space-efficient data structure such that when a query range $Q$ is specified, the…
We study an old geometric optimization problem in the plane. Given a perfect matching $M$ on a set of $n$ points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip…
Following the seminal work of Erlebach and van Leeuwen in SODA 2008, we introduce the minimum ply covering problem. Given a set $P$ of points and a set $S$ of geometric objects, both in the plane, our goal is to find a subset $S'$ of $S$…
The visibility graph of a simple polygon represents visibility relations between its vertices. Knowing the correct order of the vertices around the boundary of a polygon and its visibility graph, it is an open problem to locate the vertices…
We consider a bichromatic two-center problem for pairs of points. Given a set $S$ of $n$ pairs of points in the plane, for every pair, we want to assign a red color to one point and a blue color to the other, in such a way that the value…
This work is about graphs arising from Reuleaux polyhedra. Such graphs must necessarily be planar, $3$-connected and strongly self-dual. We study the question of when these conditions are sufficient. If $G$ is any such a graph with…
The problem of nearest-neighbor (NN) condensation aims to reduce the size of a training set of a nearest-neighbor classifier while maintaining its classification accuracy. Although many condensation techniques have been proposed, few bounds…
We study the problem of $k$-center clustering with outliers in arbitrary metrics and Euclidean space. Though a number of methods have been developed in the past decades, it is still quite challenging to design quality guaranteed algorithm…
We study two fundamental problems dealing with curves in the plane, namely, the nearest-neighbor problem and the center problem. Let $\mathcal{C}$ be a set of $n$ polygonal curves, each of size $m$. In the nearest-neighbor problem, the goal…
We investigate a variety of problems of finding tours and cycle covers with minimum turn cost. Questions of this type have been studied in the past, with complexity and approximation results as well as open problems dating back to work by…