## Section: New Results

### Other results

We initiated a collaboration with O. Devillers (INRIA,
Sophia-Antipolis) and D. Attali (Gipsa, Grenoble) on *smoothed
complexity* analysis of geometric data structures. In many cases,
the worst-case complexity analysis poorly represents the practical
performances of algorithms or data structures. The smoothed
complexity, which aims at
supplementing this gap, is defined as the maximum over the inputs of the expected
complexity over small perturbations of that input. We obtained some preliminary
results on the smoothed number of extreme points of a convex point
set subject to random noise; we submitted these results to the *Symposium on Computational
Geometry* in December [30] , [26] .

We completed some research on farthest-site Voronoi
diagrams of polygonal sites of total complexity n . We proved that the combinatorial complexity of
such diagrams is O(n) , and we presented an O(nlog^{3}n) time algorithm to compute it.
These results were submitted to the journal *Transactions on Algorithms* [33] .

One of our earlier results on computational topology was accepted this year in the journal
*Computational Geometry, Theory and Applications* [12] .
We considered the Fréchet distance between two curves in the plane is the minimum
length of a leash that allows a dog and its owner to walk along their
respective curves, from one end to the other, without backtracking.
We proposed a natural extension of Fréchet distance to more general
metric spaces, which requires the leash itself to move continuously
over time. For example, for curves in the punctured plane, the leash
cannot pass through or jump over the obstacles.

Finally, two of our earlier results in graph drawing were also accepted or published this year in the journal of
*Discrete and Computational Geometry* [17] , [18] .
The first result shows, in particular, that any planar graph with n vertices can be point-set embedded
with at most one bend per edge on a given set of n points in the plane. An
implication of this result is that any number of planar graphs admit a simultaneous
embedding without mapping with at most one bend per edge.