Antoine Vigneron

Motorcycle graphs and straight skeletons

We present a new algorithm to compute motorcycle graphs. It runs in

An algorithm for finding a K -median in a directed tree

O(n \sqrt{n}\log n)

Approximate shortest paths in anisotropic regions

time when n is the number of motorcycles. We give a new characterization of the straight skeleton of a nondegenerate polygon. For a polygon with n vertices and h holes, we show that it yields a randomized algorithm that reduces the straight skeleton computation to a motorcycle graph computation in expected

The Voronoi Diagram of Curved Objects

O(n\sqrt{h+1}\log^2 n)

Computing the Gromov hyperbolicity of a discrete metric space

time. Combining these results, we can compute the straight skeleton of a nondegenerate polygon with h holes and with n vertices, among which r are reflex vertices, in

Fitting a Step Function to a Point Set

O(n\sqrt{h+1}\log^2 n+r \sqrt{r} \log r)

Computing farthest neighbors on a convex polytope

expected time. In particular, we cancompute the straight skeleton of a nondegenerate polygon with n vertices in

Polynomial time algorithms for three-label point labeling

O(n\sqrt{n}\log^2n)

COMPUTING THE DISCRETE FRÉCHET DISTANCE WITH IMPRECISE INPUT

expected time.

Reachability by paths of bounded curvature in convex polygons

Abstract We consider the problem of finding a k -median in a directed tree. We present an algorithm that computes a k -median in O (Pk 2 ) time where k is the number of resources to be placed and P is the path length of the tree. In the case of a balanced tree, this implies O (k 2 n log n) time, in a random tree O (k 2 n 3/2 ), while in the worst case O (k 2 n 2 ) . Our method employs dynamic programming and uses O (nk) space, while the best known algorithms for undirected trees require O (n 2 k) space.