Computing the dilation of edge-augmented graphs in metric spaces

Abstract

Let G = (V , E) be an undirected graph with n vertices embedded in a metric space. We consider the problem of adding a shortcut edge in G that minimizes the dilation of the resulting graph. The fastest algorithm to date for this problem has O (n 4 ) running time and uses O (n 2 ) space. We show how to improve the running time to O (n 3 log n) while maintaining quadratic space requirement. In fact, our algorithm not only determines the best shortcut but computes the dilation of G ∪ {(u, v)} for every pair of distinct vertices u and v.

Geographic Information Science

“…This was later improved by Wulff-Nilsen [20] to O(n 3 log n) time. Note that repeatedly applying this greedy choice does not yield an optimal result.…”

Improving the dilation of a metric graph by adding edges

“…The problem stated is a major open problem in the field [6,15,21]. It is also one of twelve open problems posed in the final chapter of Narasimhan and Smid's book [17].…”

Improving the Dilation of a Metric Graph by Adding Edges

“…In particular, given an initial metric graph, and a budget of k edges, which k edges do we add to produce a minimum-dilation graph? The problem stated is a major open problem in the ield [6,15,21]. It is also one of twelve open problems posed in the inal chapter of Narasimhan and Smid's book [17].…”

“…Farshi et al [6] provided an O (n 4 ) time exact algorithm and an O (mn + n 2 log n) time 3-approximation. Wulf-Nilsen [21] improved the running time of the exact algorithm to O (n 3 log n), and in a follow-up paper Luo and Wulf-Nilsen [15] provided an O ((n 4 log n)/ √ m) time exact algorithm that uses linear space. Several of the papers that study the k = 1 case mention the k > 1 case as one of the main open problems in the ield.…”