13,982 research outputs found
Automatic Metro Map Layout Using Multicriteria Optimization
This paper describes an automatic mechanism for drawing metro maps. We apply multicriteria optimization to find effective placement of stations with a good line layout and to label the map unambiguously. A number of metrics are defined, which are used in a weighted sum to find a fitness value for a layout of the map. A hill climbing optimizer is used to reduce the fitness value, and find improved map layouts. To avoid local minima, we apply clustering techniques to the map the hill climber moves both stations and clusters when finding improved layouts. We show the method applied to a number of metro maps, and describe an empirical study that provides some quantitative evidence that automatically-drawn metro maps can help users to find routes more efficiently than either published maps or undistorted maps. Moreover, we found that, in these cases, study subjects indicate a preference for automatically-drawn maps over the alternatives
An Algorithmic Framework for Labeling Network Maps
Drawing network maps automatically comprises two challenging steps, namely
laying out the map and placing non-overlapping labels. In this paper we tackle
the problem of labeling an already existing network map considering the
application of metro maps. We present a flexible and versatile labeling model.
Despite its simplicity, we prove that it is NP-complete to label a single line
of the network. For a restricted variant of that model, we then introduce an
efficient algorithm that optimally labels a single line with respect to a given
weighting function. Based on that algorithm, we present a general and
sophisticated workflow for multiple metro lines, which is experimentally
evaluated on real-world metro maps.Comment: Full version of COCOON 2015 pape
Snapping Graph Drawings to the Grid Optimally
In geographic information systems and in the production of digital maps for
small devices with restricted computational resources one often wants to round
coordinates to a rougher grid. This removes unnecessary detail and reduces
space consumption as well as computation time. This process is called snapping
to the grid and has been investigated thoroughly from a computational-geometry
perspective. In this paper we investigate the same problem for given drawings
of planar graphs under the restriction that their combinatorial embedding must
be kept and edges are drawn straight-line. We show that the problem is NP-hard
for several objectives and provide an integer linear programming formulation.
Given a plane graph G and a positive integer w, our ILP can also be used to
draw G straight-line on a grid of width w and minimum height (if possible).Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Efficient Generation of Geographically Accurate Transit Maps
We present LOOM (Line-Ordering Optimized Maps), a fully automatic generator
of geographically accurate transit maps. The input to LOOM is data about the
lines of a given transit network, namely for each line, the sequence of
stations it serves and the geographical course the vehicles of this line take.
We parse this data from GTFS, the prevailing standard for public transit data.
LOOM proceeds in three stages: (1) construct a so-called line graph, where
edges correspond to segments of the network with the same set of lines
following the same course; (2) construct an ILP that yields a line ordering for
each edge which minimizes the total number of line crossings and line
separations; (3) based on the line graph and the ILP solution, draw the map. As
a naive ILP formulation is too demanding, we derive a new custom-tailored
formulation which requires significantly fewer constraints. Furthermore, we
present engineering techniques which use structural properties of the line
graph to further reduce the ILP size. For the subway network of New York, we
can reduce the number of constraints from 229,000 in the naive ILP formulation
to about 4,500 with our techniques, enabling solution times of less than a
second. Since our maps respect the geography of the transit network, they can
be used for tiles and overlays in typical map services. Previous research work
either did not take the geographical course of the lines into account, or was
concerned with schematic maps without optimizing line crossings or line
separations.Comment: 7 page
Planar Octilinear Drawings with One Bend Per Edge
In octilinear drawings of planar graphs, every edge is drawn as an
alternating sequence of horizontal, vertical and diagonal ()
line-segments. In this paper, we study octilinear drawings of low edge
complexity, i.e., with few bends per edge. A -planar graph is a planar graph
in which each vertex has degree less or equal to . In particular, we prove
that every 4-planar graph admits a planar octilinear drawing with at most one
bend per edge on an integer grid of size . For 5-planar
graphs, we prove that one bend per edge still suffices in order to construct
planar octilinear drawings, but in super-polynomial area. However, for 6-planar
graphs we give a class of graphs whose planar octilinear drawings require at
least two bends per edge
Exact and fixed-parameter algorithms for metro-line crossing minimization problems
A metro-line crossing minimization problem is to draw multiple lines on an
underlying graph that models stations and rail tracks so that the number of
crossings of lines becomes minimum. It has several variations by adding
restrictions on how lines are drawn. Among those, there is one with a
restriction that line terminals have to be drawn at a verge of a station, and
it is known to be NP-hard even when underlying graphs are paths. This paper
studies the problem in this setting, and propose new exact algorithms. We first
show that a problem to decide if lines can be drawn without crossings is solved
in polynomial time, and propose a fast exponential algorithm to solve a
crossing minimization problem. We then propose a fixed-parameter algorithm with
respect to the multiplicity of lines, which implies that the problem is FPT.Comment: 19 pages, 15 figure
Octilinear Force-Directed Layout with Mental Map Preservation for Schematic Diagrams
We present an algorithm for automatically laying out metro map style schematics using a force-directed approach, where we use a localized version of the standard spring embedder forces combined with an octilinear magnetic force. The two types of forces used during layout are naturally conflicting, and the existing method of simply combining these to generate a resultant force does not give satisfactory results. Hence we vary the forces, emphasizing the standard forces in the beginning to produce a well distributed graph, with the octilinear forces
becoming prevalent at the end of the layout, to ensure that the key requirement of line angles at intervals of 45? is obtained. Our method is considerably faster than the more commonly used search-based approaches, and we believe the results are superior to the previous force-directed approach. We have further developed this technique to address the issues of dynamic schematic layout. We use a Delaunay triangulation to construct a schematic “frame”, which is used to retain relative node positions and permits full control of the level of mental map preservation.
This technique is the first to combine mental map preservation techniques with the additional layout criteria of schematic diagrams. To conclude, we present the results of a study to investigate the relationship between the level of mental map preservation and the user response time and accuracy
Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases
Crossing minimization is one of the central problems in graph drawing.
Recently, there has been an increased interest in the problem of minimizing
crossings between paths in drawings of graphs. This is the metro-line crossing
minimization problem (MLCM): Given an embedded graph and a set L of simple
paths, called lines, order the lines on each edge so that the total number of
crossings is minimized. So far, the complexity of MLCM has been an open
problem. In contrast, the problem variant in which line ends must be placed in
outermost position on their edges (MLCM-P) is known to be NP-hard. Our main
results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We
give an -approximation algorithm for MLCM-P
On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
We study two variants of the well-known orthogonal drawing model: (i) the
smooth orthogonal, and (ii) the octilinear. Both models form an extension of
the orthogonal, by supporting one additional type of edge segments (circular
arcs and diagonal segments, respectively).
For planar graphs of max-degree 4, we analyze relationships between the graph
classes that can be drawn bendless in the two models and we also prove
NP-hardness for a restricted version of the bendless drawing problem for both
models. For planar graphs of higher degree, we present an algorithm that
produces bi-monotone smooth orthogonal drawings with at most two segments per
edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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