Section: New Results
Compact Routing
Compact routing with forbidden-set in planar graphs
Participant : Cyril Gavoille.
In [20] , we consider fully dynamic distance oracles and forbidden-set labeling schemes for planar graphs. For a given -vertex planar graph with edge weights drawn from and parameter , our forbidden-set labeling scheme uses labels of length . Given the labels of two vertices and and of a set of faulty vertices/edges, our scheme approximates the distance between and in with stretch , in time.
We then present a general method to transform forbidden-set labeling schemas into a fully dynamic distance oracle. Our fully dynamic distance oracle is of size and has query and update time, both the query and the update time are worst case. This improves on the best previously known dynamic distance oracle for planar graphs, which has worst case query time and amortized update time of .
Our forbidden-set labeling scheme can also be extended into a forbidden-set labeled routing scheme with stretch .
Planar Spanner of geometric graphs
Participants : Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse.
In [26] , we determine the stretch factor of -Delaunay and -Delaunay triangulations, and we show that this stretch is . Between any two points of such triangulations, we construct a path whose length is no more than times the Euclidean distance between and , and this bound is best possible. This definitively improves the 25-year old bound of by Chew (SoCG '86).
To the best of our knowledge, this is the first time the stretch factor of the well-studied -Delaunay triangulations, for any real , is determined exactly.