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…
Figures and Tables from this paper
figure 1.1 
figure 1.2 
figure 1.3 
figure 2.1 
figure 2.2 
figure 4.1 
figure 4.2 
figure 4.3 
figure 5.1 
figure 5.2 
figure 5.3 
figure 5.4 
figure 5.5 
figure 5.6 
figure 5.7 
figure 6.1 
table 6.1 
figure 6.2 
figure 6.3 
figure 6.4 
figure 6.5 
figure 7 
table 7.1 
figure 7.1 
figure 7.10 
table 7.10 
figure 7.11 
table 7.11 
figure 7.12 
table 7.12 
figure 7.13 
figure 7.14 
figure 7.16 
figure 7.17 
figure 7.18 
figure 7.19 
table 7.2 
figure 7.2 
figure 7.20 
figure 7.21 
figure 7.22 
figure 7.23 
figure 7.3 
table 7.3 
figure 7.4 
table 7.4 
figure 7.5 
table 7.5 
figure 7.6 
table 7.6 
figure 7.7 
table 7.7 
figure 7.8 
table 7.8 
figure 7.9 
table 7.9
23 Citations
An ILP for the metro-line crossing problem
- Mathematics, Computer ScienceCATS
- 2008
It is proved that the problem, which is known to be NP-hard, can be rewritten as an integer linear program that finds the optimal solution for the problem in polynomial time.
Drawing and Labeling High-Quality Metro Maps by Mixed-Integer Programming
- Computer ScienceIEEE Transactions on Visualization and Computer Graphics
- 2011
This paper identifies seven design rules used in most real-world metro maps and translates them into an MIP model, which finds a metro map that satisfies all hard constraints and minimizes a weighted sum of costs that correspond to the soft constraints.
Drawing Metro Maps on Concentric Circles
- Computer Science
- 2016
This thesis examines algorithms for the drawing of metro maps by adapting the Topology-ShapeMetrics framework by Roberto Tamassia to create a drawing in which the lines bend as little as possible and shows that achieving an optimal compaction is also NP-hard.
Snapping Graph Drawings to the Grid Optimally
- MathematicsGraph Drawing
- 2016
It is shown that the problem is NP-hard for several objectives and an integer linear programming formulation is provided that can be used to draw G straight-line on a grid of width w and minimum height (if possible).
On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
- MathematicsAlgorithmica
- 2018
An algorithm is presented 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.
A Survey on Automated Metro Map Layout Methods
- Computer Science
- 2012
This survey distinguishes three subtasks of generating metro map layouts (schematic network layout, label placement, crossing minimization) and gives a list of ten design rules that form the basis of the layout algorithms.
Automatic Metro Map Layout Using Multicriteria Optimization
- Computer ScienceIEEE Transactions on Visualization and Computer Graphics
- 2011
An empirical study is described 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.
A survey of direction-preserving layout strategies
- Computer ScienceSCCG
- 2014
Different layout algorithms that preserve relative directions in geo-referenced networks, an important criterion for many sensor networks such as the electric grid and other supply networks, are analyzed.
On d-Regular Schematization of Embedded Paths
- Mathematics, Computer ScienceSOFSEM
- 2011
It is shown that deciding whether a path can be d-schematized is NP-hard for any integer d and that this approach generates reasonable route sketches for real-world data.
References
SHOWING 1-10 OF 63 REFERENCES
On Embedding a Graph in the Grid with the Minimum Number of Bends
- Computer Science, MathematicsSIAM J. Comput.
- 1987
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
- Computer ScienceGraph Drawing
- 1999
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
- Computer ScienceGraph Drawing
- 1996
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
- MathematicsSCG '03
- 2003
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
- Mathematics, Computer ScienceSIAM J. Comput.
- 1982
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
- MathematicsJ. Graph Algorithms Appl.
- 2007
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
- Computer ScienceInVis.au
- 2004
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
- Computer Science, MathematicsSCG '01
- 2001
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
- Mathematics, Computer ScienceComb.
- 1984
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).