COMPUTING GEOMETRIC MINIMUM-DILATION GRAPHS IS NP-HARD

The dilation coefficient of a graph representation is defined as the quotient of the longest and the shortest edge representation. The minimum of the dilation coefficients over all planar representations of a graph G is called the dilation coefficient of the graph G. The dilation coefficient of different planar representations of complete graphs is considered and upper and lower bounds for the dilation coefficients of complete graphs are given. Two iterative graph-drawing algorithms that try to minimize the dilation coefficient of a given graph are given. The calculated upper bounds for the dilation coefficients of complete graphs are compared to the values obtained by the graph-drawing algorithms.

2005

Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the length of a shortest path connecting p and q in G divided by their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers-Baumann, Grüne and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation δ ≥ π/2 ≈ 1.57. They conjectured that the lower bound is not tight. We use new ideas, a disk packing result and arguments from convex geometry, to prove this conjecture. The lower bound is improved to (1 + 10 −11 )π/2.

networks

Networks, 2007

The dilation-free graph of a planar point set S is a graph that spans S in such a way that the distance between two points in the graph is no longer than their planar distance. Metrically speaking, those graphs are equivalent to complete graphs; however they have far fewer edges when considering the Manhattan distance (we give here an upper bound on the number of saved edges). This article provides several theoretical, algorithmic, and complexity features of dilation-free graphs in the l 1-metric, giving several construction algorithms and proving some of their properties. Moreover, special attention is paid to the planar case due to its applications in the design of printed circuit boards.

Computational Geometry: Theory and Applications, 2007

Given two points on a closed planar curve, C, we can divide the length of a shortest connecting path in C by their Euclidean distance. The supremum of these ratios, taken over all pairs of points on the curve, is called the geometric dilation of C. We provide lower bounds for the dilation of closed curves in terms of their geometric properties, and prove that the circle is the only closed curve achieving a dilation of π/2, which is the smallest dilation possible. Our main tool is a new geometric transformation technique based on the perimeter halving pairs of C.

Algorithms and …, 2007

2005

Let S be a set of points in the plane. What is the minimum possible dilation of all plane graphs that contain S? Even for a set S as simple as five points evenly placed on the circle, this question seems hard to answer; it is not even clear if there exists a lower bound > 1. In this paper we provide the first upper and lower bounds for the embedding problem.

2020

The distance geometry problem consists in finding a realization of a weighed graph in a Euclidean space of given dimension, where the edges are realized as straight segments of length equal to the edge weight. We propose and test a new mathematical programming formulation based on the incidence between cycles and edges in the given graph.

Algorithm Engineering and Experimentation, 2005

We consider the problem of (repeatedly) computing single- source single-target shortest paths in large, sparse graphs. Previous investigations have shown the practical usefulness of geometric speed-up techniques that guarantee the correct- ness of the result for shortest-path computations. However, such speed-up techniques utilize a layout of the graph which typically comes from geographic information. This paper ex- amines the question

Journal of Heuristics, 2014

The complexity status of the Minimum Dilation Triangulation (MDT) problem for a general point set is unknown. Therefore, we focus on the development of approximated algorithms to find high quality triangulations of minimum dilation. For an initial approach, we design a greedy strategy able to obtain approximate solutions to the optimal ones in a simple way. We also propose an operator to generate the neighborhood which is used in different algorithms: Local Search, Iterated Local Search, and Simulated Annealing. Besides, we present an algorithm called Random Local Search where good and bad solutions are accepted using the previous mentioned operator. For the experimental study we have created a set of problem instances since no reference to benchmarks for these problems were found in the literature. We use the Sequential Parameter Optimization Toolbox for tuning the parameters of the SA algorithm. We compare our results with those obtained by the OV-MDT algorithm that uses the obstacle value to sort the edges in the constructive process. This is the only available algorithm found in the literature. Through the experimental evaluation and statistical analysis, we assess the performance of the proposed algorithms using this operator.