On the dilation spectrum of paths, cycles, and trees
Computational Geometry: Theory and Applications
Computing the dilation of edge-augmented graphs in metric spaces
Computational Geometry: Theory and Applications
Accentuating focus maps via partial schematization
Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
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Given a Euclidean graph $G$ in $\mathbb{R}^d$ with $n$ vertices and $m$ edges, we consider the problem of adding an edge to $G$ such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a graph with positive edge weights runs in $\cal{O}$$(nm+n^2 \log n)$ time, resulting in a trivial $\cal{O}$$(n^3m+n^4 \log n)$-time algorithm for computing the optimal edge. First, we show that a simple modification yields the optimal solution in $\cal{O}$$(n^4)$ time using $\cal{O}$$(n^2)$ space. To reduce the running time we consider several approximation algorithms.