89 research outputs found
On the stretch factor of the Theta-4 graph
In this paper we show that the \theta-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 \theta-graph for which it was not known whether its
stretch factor was bounded
Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition
A geometric graph is angle-monotone if every pair of vertices has a path
between them that---after some rotation---is - and -monotone.
Angle-monotone graphs are -spanners and they are increasing-chord
graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in
2014 and proved that Gabriel triangulations are angle-monotone graphs. We give
a polynomial time algorithm to recognize angle-monotone geometric graphs. We
prove that every point set has a plane geometric graph that is generalized
angle-monotone---specifically, we prove that the half--graph is
generalized angle-monotone. We give a local routing algorithm for Gabriel
triangulations that finds a path from any vertex to any vertex whose
length is within times the Euclidean distance from to .
Finally, we prove some lower bounds and limits on local routing algorithms on
Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
The Price of Order
We present tight bounds on the spanning ratio of a large family of ordered
-graphs. A -graph partitions the plane around each vertex into
disjoint cones, each having aperture . An ordered
-graph is constructed by inserting the vertices one by one and
connecting each vertex to the closest previously-inserted vertex in each cone.
We show that for any integer , ordered -graphs with
cones have a tight spanning ratio of . We also show that for any integer , ordered
-graphs with cones have a tight spanning ratio of . We provide lower bounds for ordered -graphs with and cones. For ordered -graphs with and
cones these lower bounds are strictly greater than the worst case spanning
ratios of their unordered counterparts. These are the first results showing
that ordered -graphs have worse spanning ratios than unordered
-graphs. Finally, we show that, unlike their unordered counterparts,
the ordered -graphs with 4, 5, and 6 cones are not spanners
Balanced Schnyder woods for planar triangulations: an experimental study with applications to graph drawing and graph separators
In this work we consider balanced Schnyder woods for planar graphs, which are
Schnyder woods where the number of incoming edges of each color at each vertex
is balanced as much as possible. We provide a simple linear-time heuristic
leading to obtain well balanced Schnyder woods in practice. As test
applications we consider two important algorithmic problems: the computation of
Schnyder drawings and of small cycle separators. While not being able to
provide theoretical guarantees, our experimental results (on a wide collection
of planar graphs) suggest that the use of balanced Schnyder woods leads to an
improvement of the quality of the layout of Schnyder drawings, and provides an
efficient tool for computing short and balanced cycle separators.Comment: Appears in the Proceedings of the 27th International Symposium on
Graph Drawing and Network Visualization (GD 2019
An Infinite Class of Sparse-Yao Spanners
We show that, for any integer k > 5, the Sparse-Yao graph YY_{6k} (also known
as Yao-Yao) is a spanner with stretch factor 11.67. The stretch factor drops
down to 4.75 for k > 7.Comment: 17 pages, 12 figure
Comment résumer le plan
International audienceCet article concerne les graphes de recouvrement d'un ensemble fini de points du plan Euclidien. Un graphe de recouvrement est de facteur d'étirement pour un ensemble de points si, entre deux points quelconques de , le coût d'un plus court chemin dans est au plus fois leur distance Euclidenne. Les graphes de recouvrement d'étirement (ci-après nommés \emph{-spanneurs}) sont à la base de nombreux algorithmes de routage et de navigation dans le plan. Le graphe (ou triangulation) de Delaunay, le graphe de Gabriel, le graphe de Yao ou le Theta-graphe sont des exemples bien connus de -spanneurs. L'étirement et le degré maximum des spanneurs sont des paramètres important à minimiser pour l'optimisation des ressources. En même temps le caractère planaire des constructions se révèle essentiel dans les algorithmes de navigation. Nous présentons une série de résultats dans ce domaine, en particulier: \begin{itemize} \item Nous montrons que le graphe (le Theta-graphe où cônes d'angle par sommet sont utilisées) est l'union de deux spanneurs planaires d'étirement deux. En particulier, nous établissons que l'étirement maximum du graphe est deux, ce qui est optimal. Des bornes supérieures sur l'étirement du graphe n'étaient connues que lorsque . Pour , la meilleure borne connue est d'environ et pour il était ouvert de savoir si le graphe était un -spanneur pour une valeur constante de . \item Nous montrons que le graphe contient comme sous-graphe couvrant un -spanneur planaire de degré maximum au plus~. \item Finalement, en utilisant une variante du résultat précédant, nous montrons que le plan Euclidien possède un -spanneur planaire de degré maximum au plus~. \end{itemize} La dernière construction, non décrite ici par manque de place, améliore une longue série de résultats sur le problème largement ouvert de déterminer la plus petite valeur telle que tout ensemble du plan possède un spanneur planaire d'étirement constant et de degré maximum . Le meilleur résultat en date montrait que
- …