Finding the best shortcut in a geometric network

Given a Euclidean graph G in Rd with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn + n2 log n) time, resulting in a trivial O(mn3 + n4 log n) time algorithm for computing the optimal shortcut. First, we show that a simple modification yields the optimal solution in O(n4 ) time using O(n2 ) space. To reduce the running times we consider several approximation algorithms. Our main result is a (2 + ε)-approximation algorithm with running time O(nm + n2 (log n + 1/ε3d )) using O(n2 ) space.

• Type: Conference
• Year: 2005
• Where: COMPGEOM
• Authors: Mohammad Farshi, Panos Giannopoulos, Joachim Gudmundsson
• Keywords: Algorithm, COMPGEOM 2005, Discrete Geometry, Euclidean Graph, Stretch Factor