Ahmad Biniaz

Fixed-Orientation Equilateral Triangle Matching of Point Sets

Given a point set P and a class \(\mathcal{C}\) of geometric objects, \(G_\mathcal{C}(P)\) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some \(C \in \mathcal{C}\) containing both p and q but no other points from P. We study G ∇ (P) graphs where ∇ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-Θ6 graphs and TD-Delaunay graphs.

An Optimal Algorithm for Plane Matchings in Multipartite Geometric Graphs

Let P be a set of n points in general position in the plane which is partitioned into color classes. P is said to be color-balanced if the number of points of each color is at most \(\lfloor n/2\rfloor \). Given a color-balanced point set P, a balanced cut is a line which partitions P into two color-balanced point sets, each of size at most \(2n/3 + 1\). A colored matching of P is a perfect matching in which every edge connects two points of distinct colors by a straight line segment. A plane colored matching is a colored matching which is non-crossing. In this paper, we present an algorithm which computes a balanced cut for P in linear time. Consequently, we present an algorithm which computes a plane colored matching of P optimally in \(\Theta (n\log n)\) time.

Approximating the bottleneck plane perfect matching of a point set

A bottleneck plane perfect matching of a set of n points in 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 bottleneck. The problem of computing a bottleneck plane perfect matching has been proved to be NP-hard. We present an algorithm that computes a bottleneck plane matching of size at least n 5 in O ( n log 2 ? n ) -time. Then we extend our idea toward an O ( n log ? n ) -time approximation algorithm which computes a plane matching of size at least 2 n 5 whose edges have length at most 2 + 3 times the bottleneck.

Plane geodesic spanning trees, Hamiltonian cycles, and perfect matchings in a simple polygon

Let S be a finite set of points in the interior of a simple polygon P. A geodesic graph, G P ( S , E ) , is a graph with vertex set S and edge set E such that each edge ( a , b ) ź E is the shortest geodesic path between a and b inside P. G P is said to be plane if the edges in E do not cross. If the points in S are colored, then G P is said to be properly colored provided that, for each edge ( a , b ) ź E , a and b have different colors. In this paper we consider the problem of computing (properly colored) plane geodesic perfect matchings, Hamiltonian cycles, and spanning trees of maximum degree three.

Fixed-orientation equilateral triangle matching of point sets

Given a point set P and a class C of geometric objects, G C ( P ) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C ? C containing both p and q but no other points from P. We study G ? ( P ) graphs where ? is the class of downward equilateral triangles (i.e., equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half- ? 6 graphs and TD-Delaunay graphs.The main result in our paper is that for point sets P in general position, G ? ( P ) always contains a matching of size at least ? | P | - 1 3 ? and this bound is tight. We also give some structural properties of G ? ( P ) graphs, where ? is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of G ? ( P ) is simply a path. Through the equivalence of G ? ( P ) graphs with ? 6 graphs, we also derive that any ? 6 graph can have at most 5 n - 11 edges, for point sets in general position.

Approximation algorithms for the unit disk cover problem in 2D and 3D

Given a set P of n points in the plane, we consider the problem of covering P with a minimum number of unit disks. This problem is known to be NP-hard. We present a simple 4-approximation algorithm for this problem which runs in O ( n log ź n ) -time. We also show how to extend this algorithm to other metrics, and to three dimensions.

A Plane 1.88-Spanner for Points in Convex Position

Let S be a set of n points in the plane that is in convex position. For a real number t>1, we say that a point p in S is t-good if for every point q of S, the shortest-path distance between p and q along the boundary of the convex hull of S is at most t times the Euclidean distance between p and q. We prove that any point that is part of (an approximation to) the diameter of S is 1.88-good. Using this, we show how to compute a plane 1.88-spanner of S in O(n) time, assuming that the points of S are given in sorted order along their convex hull. Previously, the best known stretch factor for plane spanners was 1.998 (which, in fact, holds for any point set, i.e., even if it is not in convex position).

Plane Geodesic Spanning Trees, Hamiltonian Cycles, and Perfect Matchings in a Simple Polygon

Let S be a finite set of points in the interior of a simple polygon P. A geodesic graph, \(G_P(S,E)\), is a graph with vertex set S and edge set E such that each edge \((a,b)\in E\) is the shortest path between a and b inside P. \(G_P\) is said to be plane if the edges in E do not cross. If the points in S are colored, then \(G_P\) is said to be properly colored provided that, for each edge \((a,b)\in E\), a and b have different colors. In this paper we consider the problem of computing (properly colored) plane geodesic perfect matchings, Hamiltonian cycles, and spanning trees of maximum degree three.

Higher-order triangular-distance Delaunay graphs

We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set P of points in general position in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle ?, and there is an edge between two points in P if and only if there is an empty homothet of ? having the two points on its boundary. We consider higher-order triangular-distance Delaunay graphs, namely k-TD, which contains an edge between two points if the interior of the smallest homothet of ? having the two points on its boundary contains at most k points of P. We consider the connectivity, Hamiltonicity and perfect-matching admissibility of k-TD. Finally we consider the problem of blocking the edges of k-TD.

An optimal algorithm for the Euclidean bottleneck full Steiner tree problem

Let P and S be two disjoint sets of n and m points in the plane, respectively. We consider the problem of computing a Steiner tree whose Steiner vertices belong to S, in which each point of P is a leaf, and whose longest edge length is minimum. We present an algorithm that computes such a tree in O((n+m)logm) time, improving the previously best result by a logarithmic factor. We also prove a matching lower bound in the algebraic computation tree model.