A Mixed-Integer Program for Drawing High-Quality Metro Maps

Nöllenburg, Martin; Wolff, Alexander

doi:10.1007/11618058_29

Cited by 40 publications

(47 citation statements)

References 8 publications

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“…For example, Cabello et al [3] give an algorithm that schematizes a network using 2 or 3 links per path, if possible. Nöllenburg and Wolff [12] use a method based on mixed-integer programming to generate metro maps using one edge per path. Both methods do not incorporate obstacles and are restricted to a small constant number of links per path.…”

Section: Introductionmentioning

confidence: 99%

Homotopic Rectilinear Routing with Few Links and Thick Edges

Speckmann

Verbeek

2010

LATIN 2010: Theoretical Informatics

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We study the problem of finding non-crossing thick minimum-link rectilinear paths homotopic to a set of input paths in an environment with rectangular obstacles. This problem occurs in the context of map schematization under geometric embedding restrictions, for example, when schematizing a highway network for use as a thematic layer. We present a 2-approximation algorithm that runs in O(n 3 + k in log n + k out ) time, where n is the total number of input paths and obstacles and k in and k out are the total complexities of the input and output paths, respectively. Our algorithm not only approximates the minimum number of links, but also minimizes the total length of the paths. An approximation factor of 2 is optimal when using smallest paths as lower bound.

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Section: Introductionmentioning

confidence: 99%

Homotopic Rectilinear Routing with Few Links and Thick Edges

Speckmann

Verbeek

2010

LATIN 2010: Theoretical Informatics

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“…For example, metro maps show connectivity of metro lines while abstracting from geographic reality (correct location) [11], and cartograms may show countries by using size (area) to depict total population [15]. Cartograms come in different types: contiguous area [7,9], non-contiguous area [12], rectangular [10], rectilinear [4], circle [6], and linear cartograms [5].…”

Section: Introductionmentioning

confidence: 99%

Time-Space Maps from Triangulations

Bies

Kreveld

2013

Lecture Notes in Computer Science

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Abstract. Time-space maps show travel time as distances on a map. We discuss the case of time-space maps with a single center; here the travel times from a single source location to a number of destinations are shown by their distances. To accomplish this while maintaining recognizability, the input map must be deformed in a suitable manner. We present three different methods and analyze them experimentally.

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“…Closely related to the first step are the works of Hong et al [8], Merrick and Gudmundsson [11], Nöllenburg and Wolff [12] and Stott and Rodgers [14]. The map labeling problem has also attracted the interest of several researchers.…”

Section: Introductionmentioning

confidence: 99%

Two Polynomial Time Algorithms for the Metro-line Crossing Minimization Problem

et al. 2009

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Abstract. The metro-line crossing minimization (MLCM) problem was recently introduced in [5] as a response to the problem of drawing metro maps or public transportation networks, in general. According to this problem, we are given a planar, embedded graph G = (V, E) and a set L of simple paths on G, called lines. The main task is to place the lines on the embedding of G, so that the number of crossings among pairs of lines is minimized. Our main contribution is two polynomial time algorithms. The first solves the general case of the MLCM problem, where the lines that traverse a particular vertex of G are allowed to use any side of it to either "enter" or "exit", assuming that the endpoints of the lines are located at vertices of degree one. The second one -which is more efficient in terms of time complexity-solves the restricted case, where only the left and the right side of each vertex can be used. To the best of our knowledge, this is the first time where the general case of the MLCM problem is solved. Previous work in the graph drawing literature was devoted to the restricted case of the MLCM problem under the additional assumption that the endpoints of the lines are either the topmost or the bottommost in their corresponding vertices, i.e., they are either on top or below the lines that pass through the vertex. Even for this case, we improve a known result of Asquith et al. [3] from O(|E| 5/2 |L| 3 ) to O(|V |(|E| + |L|)).

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A Mixed-Integer Program for Drawing High-Quality Metro Maps

Cited by 40 publications

References 8 publications

Homotopic Rectilinear Routing with Few Links and Thick Edges

Homotopic Rectilinear Routing with Few Links and Thick Edges

Time-Space Maps from Triangulations

Two Polynomial Time Algorithms for the Metro-line Crossing Minimization Problem

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