Jean-Lou De Carufel

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We revisit the problem of searching for a target at an unknown location on a line when given upper and lower bounds on the distance D that separates the initial position of the searcher from the target. Prior to this work, only asymptotic bounds were known for the optimal competitive ratio achievable by any search strategy in the worst case. We present the(More)
In this paper we show that the θ-graph with 4 cones has constant stretch factor, i.e., there is a path between any pair of vertices in this graph whose length is at most a constant times the Euclidean distance between that pair of vertices. This is the last θ-graph for which it was not known whether its stretch factor was bounded.
We present tight upper and lower bounds on the spanning ratio of a large family of θ-graphs. We show that θ-graphs with 4k+2 cones (k ≥ 1 and integer) have a spanning ratio of 1 + 2 sin(θ/2), where θ is 2π/(4k + 2). We also show that θ-graphs with 4k + 4 cones have spanning ratio at least 1 + 2 tan(θ/2) + 2 tan 2 (θ/2), where θ is 2π/(4k + 4). This is(More)
Let P be a closed simple polygon with n vertices. For any two points in P , the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P. In 1989, Pollack, Sharir and Rote [Disc.(More)
We seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance between any two points on the augmented network. Unlike in the discrete setting where a shortcut connects two vertices and the diameter is measured between vertices, we take all points along the edges of the(More)
We present improved upper and lower bounds on the spanning ratio of θ-graphs with at least six cones. Given a set of points in the plane, a θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m, and adds an edge to the 'closest' vertex in each cone. We show that for any integer k ≥ 1, θ-graphs with 4k + 2 cones(More)
The sequence of adjacent nodes (graph walk) visited by a routing algorithm on a graph G between given source and target nodes s and t is a c-competitive route if its length in G is at most c times the length of the shortest path from s to t in G. We present 21.766-, 17.982-and 15.479-competitive online routing algorithms on the Delaunay triangulation of an(More)