Related papers: Algorithms for orthogonal partitioning into four p…
Two planar sets are circularly separable if there exists a circle enclosing one of the sets and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A linear-time…
We study a variant of a polygon partition problem, introduced by Chung, Iwama, Liao, and Ahn [ISAAC'25]. Given orthogonal unit vectors $\mathbf{u},\mathbf{v}\in \mathbb{R}^2$ and a polygon $P$ with $n$ vertices, we partition $P$ into…
Given an orthogonal polygon $ P $ with $ n $ vertices, the goal of the watchman route problem is finding a path $ S $ of the minimum length in $ P $ such that every point of the polygon $ P $ is visible from at least one of the point of $ S…
In this paper, we characterize planar point sets that can be partitioned into disjoint polygons of arbitrarily specified sizes. We provide an algorithm to construct such a partition, if it exists, in polynomial time. We show that this…
Suppose that a polygon $P$ is given as an array containing the vertices in counterclockwise order. We analyze how many vertices (including the index of each of these vertices) we need to know before we can bound $P$, i.e., report a bounded…
Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $p\in S$, $q\in T$ and for each $r\in S\cup T$, $r$ appears in at least one such pair.…
Fast exact algorithms are known for Hamiltonian paths in undirected and directed bipartite graphs through elegant though involved algorithms that are quite different from each other. We devise algorithms that are simple and similar to each…
Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two lines. For an integral with respect to an appropriate weight function defined on any…
Given in the plane a set of points and a set of halfplanes, we consider the problem of computing a smallest subset of halfplanes whose union covers all points. In this paper, we present an $O(n^{4/3}\log^{5/3}n\log^{O(1)}\log n)$-time…
Many problems of interest for cyber-physical network systems can be formulated as Mixed Integer Linear Programs in which the constraints are distributed among the agents. In this paper we propose a distributed algorithm to solve this class…
We consider the problem of finding minimum-link rectilinear paths in rectilinear polygonal domains in the plane. A path or a polygon is rectilinear if all its edges are axis-parallel. Given a set $\mathcal{P}$ of $h$ pairwise-disjoint…
We first prove a new separating hyperplane theorem characterizing when a pair of compact convex subsets $K, K'$ of the Euclidean space intersect, and when they are disjoint. The theorem is distinct from classical separation theorems. It…
A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a $O(n\log^2{n})$-time algorithm for deciding if a…
Let $P$ be a set of $n$ points in real projective $d$-space, not all contained in a hyperplane, such that any $d$ points span a hyperplane. An ordinary hyperplane of $P$ is a hyperplane containing exactly $d$ points of $P$. We show that if…
A planar orthogonal drawing {\Gamma} of a connected planar graph G is a geometric representation of G such that the vertices are drawn as distinct points of the plane, the edges are drawn as chains of horizontal and vertical segments, and…
The partitioning of space by hyperplanes in the context of discrete classification problem is considered. We obtain some relations for the number of partitions and establish a recurrence relation for the maximal number of partitions of R^n…
In this paper we present a combinatorial generalization of the fact that the number of plane partitions that fit in a $2a\times b\times b$ box is equal to the number of such plane partitions that are symmetric, times the number of such…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
A bottleneck plane perfect matching of a set of $n$ points in $\mathbb{R}^2$ is defined to be a perfect non-crossing matching that minimizes the length of the longest edge; the length of this longest edge is known as {\em bottleneck}. The…
We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.