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Given a point set P and a class C of geometric objects, GC(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 (ie. equilateral triangles with one of their… (More)

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 n/2. 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… (More)

- Ahmad Biniaz
- 2009

Range searching is one of the classical problems in computational geometry. Let P be a set of n points given in d-dimensional real space, R d , and a family of subsets of R d called ranges is considered. The goal is to preprocess P into a data structure so that for a given query range Q, the points in P ∩ Q can be counted or reported efficiently. The… (More)

- Ahmad Biniaz
- CCCG
- 2009

This paper introduces a new measure of quality for terrain surface simplification aiming at preserving the slope of the surface, as well as a new simplification algorithm which satisfies this measure. Experimental results and comparisons with other simplification algorithms show the effectiveness of proposed algorithm according to some quality measures.

Let R and B be two disjoint sets of points in the plane such that |B| |R|, and no three points of R ∪ B are collinear. We show that the geometric complete bipartite graph K(R, B) contains a non-crossing spanning tree whose maximum degree is at most max 3, |R|−1 |B| + 1 ; this is the best possible upper bound on the maximum degree. This solves an open… (More)

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) ∈ 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… (More)

Given a set P of n points in the plane, the order-k Gabriel graph on P , denoted by k-GG, has an edge between two points p and q if and only if the closed disk with diameter pq contains at most k points of P , excluding p and q. We study matching problems in k-GG graphs. We show that a Euclidean bottleneck perfect matching of P is contained in 10-GG, but… (More)

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… (More)

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) log m) time, improving the previously best result… (More)

Given an edge-weighted graph G = (V, E) and a subset R of V , a Steiner tree of G is a tree which spans all the vertices in R. A full Steiner tree is a Steiner tree which has all the vertices of R as its leaves. The full Steiner tree problem is to find a full Steiner tree of G with minimum weight. In this paper we present a 20-approximation algorithm for… (More)