3,596 research outputs found
Competitive Local Routing with Constraints
Let be a set of vertices in the plane and a set of non-crossing
line segments between vertices in , called constraints. Two vertices are
visible if the straight line segment connecting them does not properly
intersect any constraints. The constrained -graph is constructed by
partitioning the plane around each vertex into disjoint cones, each with
aperture , and adding an edge to the `closest' visible vertex
in each cone. We consider how to route on the constrained -graph. We
first show that no deterministic 1-local routing algorithm is
-competitive on all pairs of vertices of the constrained
-graph. After that, we show how to route between any two visible
vertices of the constrained -graph using only 1-local information.
Our routing algorithm guarantees that the returned path is 2-competitive.
Additionally, we provide a 1-local 18-competitive routing algorithm for visible
vertices in the constrained half--graph, a subgraph of the
constrained -graph that is equivalent to the Delaunay graph where the
empty region is an equilateral triangle. To the best of our knowledge, these
are the first local routing algorithms in the constrained setting with
guarantees on the length of the returned path
Routing on the Visibility Graph
We consider the problem of routing on a network in the presence of line
segment constraints (i.e., obstacles that edges in our network are not allowed
to cross). Let be a set of points in the plane and let be a set of
non-crossing line segments whose endpoints are in . We present two
deterministic 1-local -memory routing algorithms that are guaranteed to
find a path of at most linear size between any pair of vertices of the
\emph{visibility graph} of with respect to a set of constraints (i.e.,
the algorithms never look beyond the direct neighbours of the current location
and store only a constant amount of additional information). Contrary to {\em
all} existing deterministic local routing algorithms, our routing algorithms do
not route on a plane subgraph of the visibility graph. Additionally, we provide
lower bounds on the routing ratio of any deterministic local routing algorithm
on the visibility graph.Comment: An extended abstract of this paper appeared in the proceedings of the
28th International Symposium on Algorithms and Computation (ISAAC 2017).
Final version appeared in the Journal of Computational Geometr
GraphMaps: Browsing Large Graphs as Interactive Maps
Algorithms for laying out large graphs have seen significant progress in the
past decade. However, browsing large graphs remains a challenge. Rendering
thousands of graphical elements at once often results in a cluttered image, and
navigating these elements naively can cause disorientation. To address this
challenge we propose a method called GraphMaps, mimicking the browsing
experience of online geographic maps.
GraphMaps creates a sequence of layers, where each layer refines the previous
one. During graph browsing, GraphMaps chooses the layer corresponding to the
zoom level, and renders only those entities of the layer that intersect the
current viewport. The result is that, regardless of the graph size, the number
of entities rendered at each view does not exceed a predefined threshold, yet
all graph elements can be explored by the standard zoom and pan operations.
GraphMaps preprocesses a graph in such a way that during browsing, the
geometry of the entities is stable, and the viewer is responsive. Our case
studies indicate that GraphMaps is useful in gaining an overview of a large
graph, and also in exploring a graph on a finer level of detail.Comment: submitted to GD 201
Edge Routing with Ordered Bundles
Edge bundling reduces the visual clutter in a drawing of a graph by uniting
the edges into bundles. We propose a method of edge bundling drawing each edge
of a bundle separately as in metro-maps and call our method ordered bundles. To
produce aesthetically looking edge routes it minimizes a cost function on the
edges. The cost function depends on the ink, required to draw the edges, the
edge lengths, widths and separations. The cost also penalizes for too many
edges passing through narrow channels by using the constrained Delaunay
triangulation. The method avoids unnecessary edge-node and edge-edge crossings.
To draw edges with the minimal number of crossings and separately within the
same bundle we develop an efficient algorithm solving a variant of the
metro-line crossing minimization problem. In general, the method creates clear
and smooth edge routes giving an overview of the global graph structure, while
still drawing each edge separately and thus enabling local analysis
Discrete analogue computing with rotor-routers
Rotor-routing is a procedure for routing tokens through a network that can
implement certain kinds of computation. These computations are inherently
asynchronous (the order in which tokens are routed makes no difference) and
distributed (information is spread throughout the system). It is also possible
to efficiently check that a computation has been carried out correctly in less
time than the computation itself required, provided one has a certificate that
can itself be computed by the rotor-router network. Rotor-router networks can
be viewed as both discrete analogues of continuous linear systems and
deterministic analogues of stochastic processes.Comment: To appear in Chaos Special Focus Issue on Intrinsic and Designed
Computatio
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