Automated drawing of metro maps

@inproceedings{Nllenburg2005AutomatedDO,
  title={Automated drawing of metro maps},
  author={Martin N{\"o}llenburg},
  year={2005}
}
This work investigates the problem of drawing metro maps which is defined as follows. Given a planar graph G of maximum degree 8 with its embedding and vertex locations (e.g. the physical location of the tracks and stations of a metro system) and a set L of paths or cycles in G (e.g. metro lines) such that each edge of G belongs to at least one element of L, draw G and L nicely. We first specify the niceness of a drawing by listing a number of hard and soft constraints. Then we show that it is… 
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References

SHOWING 1-10 OF 63 REFERENCES
On Embedding a Graph in the Grid with the Minimum Number of Bends
  • R. Tamassia
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 1987
TLDR
An algorithm is presented that computes a region preserving grid embedding with the minimum number of bends in edges with use of network flow techniques, and runs in time $O(n^2 \log n)$, where n is the number of vertices of the graph.
Combining Graph Labeling and Compaction
TLDR
This work presents a branch-and--cut algorithm which computes optimally labeled orthogonal drawings for given instances of the COLA problem, the first algorithm especially designed to solve graph labeling problems.
A New Minimum Cost Flow Algorithm with Applications to Graph Drawing
TLDR
This work shows how to compute a planar orthogonal drawing with the minimum number of bends for an n- vertex embedded planar graph in time O(n7/4√log n), which is the first subquadratic algorithm for bend minimization.
Approximation algorithms for aligning points
TLDR
It is shown that for planar graphs the problem is NP-hard, and inapproximability results for general graphs are provided, and approximation algorithms whose performance depends upon the shape of the given regions and the set of orientations are given.
Planar Formulae and Their Uses
TLDR
Using these results, it is able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness forPlanar generalized geography.
Planar Embeddings of Graphs with Specified Edge Lengths
TLDR
The problem of finding a planar embedding of a (planar) graph with a prescribed Euclidean length on every edge is considered and it is shown that the problem is tractable—indeed, solvable in linear time on a real RAM—forPlanar embeddings of planar 3-connected triangulations, even if the outer face is not a triangle.
The Metro Map Layout Problem
TLDR
A set of aesthetic criteria for good metro map layouts is defined and a method to produce such layouts automatically using a variation of the spring algorithm with a suitable preprocessing step is presented.
Schematization of road networks
TLDR
This work studies the problem of computing schematized versions of network maps, like railroad maps, and applies to several types of schematizations, and certain additional constraints can be added.
A new polynomial-time algorithm for linear programming
TLDR
It is proved that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property: the ratio of the radius of the smallest sphere with center a′, containingP′ to theradius of the largest sphere withCenter a′ contained inP′ isO(n).
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