Section: New Results

Compact Routing

Compact routing with forbidden-set in planar graphs

Participant : Cyril Gavoille.

In [20] , we consider fully dynamic $(1+\epsilon )$ distance oracles and $(1+\epsilon )$ forbidden-set labeling schemes for planar graphs. For a given $n$-vertex planar graph $G$ with edge weights drawn from $[1,M]$ and parameter $\epsilon >0$, our forbidden-set labeling scheme uses labels of length $\lambda =O({\epsilon }^{-1}{log}^{2}nlog\left(nM\right)·maxlogn)$. Given the labels of two vertices $s$ and $t$ and of a set $F$ of faulty vertices/edges, our scheme approximates the distance between $s$ and $t$ in $G\setminus F$ with stretch $(1+\epsilon )$, in $O\left(\right|F{|}^2\lambda )$ time.

We then present a general method to transform $(1+\epsilon )$ forbidden-set labeling schemas into a fully dynamic $(1+\epsilon )$ distance oracle. Our fully dynamic $(1+\epsilon )$ distance oracle is of size $O(nlogn·maxlogn)$ and has $\tilde{O}\left(n^{1/2}\right)$ query and update time, both the query and the update time are worst case. This improves on the best previously known $(1+\epsilon )$ dynamic distance oracle for planar graphs, which has worst case query time $\tilde{O}\left(n^{2/3}\right)$ and amortized update time of $\tilde{O}\left(n^{2/3}\right)$.

Our $(1+\epsilon )$ forbidden-set labeling scheme can also be extended into a forbidden-set labeled routing scheme with stretch $(1+\epsilon )$.

Planar Spanner of geometric graphs

Participants : Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse.

In [26] , we determine the stretch factor of $L_1$-Delaunay and $L_{\infty }$-Delaunay triangulations, and we show that this stretch is $\sqrt{4+2\sqrt{2}}\approx 2.61$. Between any two points $x,y$ of such triangulations, we construct a path whose length is no more than $\sqrt{4+2\sqrt{2}}$ times the Euclidean distance between $x$ and $y$, and this bound is best possible. This definitively improves the 25-year old bound of $\sqrt{10}$ by Chew (SoCG '86).

To the best of our knowledge, this is the first time the stretch factor of the well-studied $L_p$-Delaunay triangulations, for any real $p\ge 1$, is determined exactly.