Dilation-Optimal Edge Deletion in Polygonal Cycles

International Journal of Computational Geometry and Applications, 2010

Consider a geometric graph G, drawn with straight lines in the plane. For every pair a, b of vertices of G, we compare the shortestpath distance between a and b in G (with Euclidean edge lengths) to their actual Euclidean distance in the plane. The worst-case ratio of these two values, for all pairs of vertices, is called the vertex-to-vertex dilation of G.

Let P be a simple polygon in 2 with n vertices. The detour of P between two points, p, q ∈ P , is the length of a shortest path contained in P and connecting p to q, divided by the distance of these points. The detour of the whole polygon is the maximum detour between any two points in P . We first analyze properties of pairs of points with maximum detour. Next, we use these properties to achieve a deterministic O(n 2 )-algorithm for computing the maximum Euclidean detour and a deterministic O(n log n)-algorithm which calculates a (1+ε)approximation. Finally, we consider the special case of monotone rectilinear polygons. Their L 1 -detour can be computed in time O(n).

Computational Geometry, 2013

A constant-workspace algorithm has read-only access to an input array and may use only O(1) additional words of O(log n) bits, where n is the size of the input. We assume that a simple n-gon is given by the ordered sequence of its vertices. We show that we can find a triangulation of a plane straight-line graph in O(n 2 ) time. We also consider preprocessing a simple polygon for shortest path queries when the space constraint is relaxed to allow s words of working space. After a preprocessing of O(n 2 ) time, we are able to solve shortest path queries between any two points inside the polygon in O(n 2 /s) time.

Computational Geometry: Theory and Applications, 2004

We provide an O(log n)-approximation algorithm for the following problem. Given a convex n-gon P , drawn on a convex piece of paper, cut P out of the piece of paper in the cheapest possible way. No polynomial-time approximation algorithm was known for this problem posed in 1985.

International Journal of Computational Geometry & Applications, 1993

We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We give some variants that have linear or O(n log n) algorithms approximating polygonal chains of n segments. We also show that approximating subdivisions and approximating with chains with. no self-intersections are NP-hard.

Journal of Combinatorial Mathematics and Combinatorial Computing

In this paper we present optimal algorithms to compute monotone stabbers for convex polygons. More precisely, given a set of m convex polygons with n vertices in total we want to stab the polygons with an x-monotone polygonal chain such that each polygon is entered at its leftmost point and departed at its rightmost point. Since such a stabber does not exist in general, we study two related problems. In the rst problem we want to compute a monotone stabber that stabs as many convex polygons as possible. The second problem is to compute the minimal number of x-monotone stabbers that are necessary to stab all given convex polygons.

arXiv (Cornell University), 2017

Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A straight-line point-set embedding is a point-set embedding with no edge bends or curves. The point-set embeddability problem is NP-complete, even when G is 2-connected and 2-outerplanar. It has been solved polynomially only for a few classes of planar graphs. Suppose that S is the set of vertices of a simple polygon. A straight-line polygon embedding of a graph is a straight-line point-set embedding of the graph onto the vertices of the polygon with no crossing between edges of graph and the edges of polygon. In this paper, we present O(n)-time algorithms for polygon embedding of path and cycle graphs in simple convex polygon and same time algorithms for polygon embedding of path and cycle graphs in a large type of simple polygons where n is the number of vertices of the polygon.

Cccg, 2010

The straight skeleton of a simple polygon is defined as the trace of the vertices when the initial polygon is shrunken in self-parallel manner . In this paper, we propose a simple algorithm for drawing the straight skeleton of a monotone polygon. The time and space complexities of our algorithm are O(nlogn) and O(n) respectively.

Discrete & Computational Geometry, 1997

We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires O(n +h 2 log n) time.

In this paper we present optimal algorithms to compute monotone stabbers for convex polygons. More precisely, given a set of m convex polygons with n vertices in total we want to stab the polygons with an x-monotone polygonal chain such that each polygon is entered at its leftmost point and departed at its rightmost point. Since such a stabber does not exist in general, we study two related problems. In the rst problem we want to compute a monotone stabber that stabs as many convex polygons as possible. The second problem is to compute the minimal number of x-monotone stabbers that are necessary to stab all given convex polygons.