Ordering Metro Lines by Block Crossings

A problem that arises in drawings of transportation networks is to minimize the number of crossings between different transportation lines. While this can be done efficiently under specific constraints, not all solutions are visually equivalent. We suggest merging crossings into block crossings, that is, crossings of two neighboring groups of consecutive lines. Unfortunately, minimizing the total number of block crossings is NP-hard even for very simple graphs. We give approximation algorithms for special classes of graphs and an asymptotically worst-case optimal algorithm for block crossings on general graphs. That is, we bound the number of block crossings that our algorithm needs and construct worst-case instances on which the number of block crossings that is necessary in any solution is asymptotically the same as our bound

Ordering metro lines by block crossings

A problem that arises in drawings of transportation networks is to minimize the number of crossings between different transportation lines. While this can be done efficiently under specific constraints, not all solutions are visually equivalent. We suggest merging single crossings into block crossings, that is, crossings of two neighboring groups of consecutive lines. Unfortunately, minimizing the total number of block crossings is NP-hard even for very simple graphs. We give approximation algorithms for special classes of graphs and an asymptotically worst-case optimal algorithm for block crossings on general graphs. Furthermore, we show that the problem remains NP-hard on planar graphs even if both the maximum degree and the number of lines per edge are bounded by constants; on trees, this restricted version becomes tractable. © 2015, Brown University. All rights reserved

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

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 $O(\sqrt{\log |L|})$-approximation algorithm for MLCM-P

Block Crossings in Storyline Visualizations

Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just minimizing pairwise crossings of curves, we propose to count block crossings, that is, pairs of intersecting bundles of lines. Our main results are as follows. We show that minimizing the number of block crossings is NP-hard, and we develop, for meetings of bounded size, a constant-factor approximation. We also present two fixed-parameter algorithms and, for meetings of size 2, a greedy heuristic that we evaluate experimentally.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Route Packing: Geospatially-Accurate Visualization of Route Networks

We present route packing}, a novel (geo)visualization technique for displaying several routes simultaneously on a geographic map while preserving the geospatial layout, identity, directionality, and volume of individual routes. The technique collects variable-width route lines side by side while minimizing crossings, encodes them with categorical colors, and decorates them with glyphs to show their directions. Furthermore, nodes representing sources and sinks use glyphs to indicate whether routes stop at the node or merely pass through it. We conducted a crowd-sourced user study investigating route tracing performance with road networks visualized using our route packing technique. Our findings highlight the visual parameters under which the technique yields optimal performance

Drawing with SAT: four methods and A tool for producing railway infrastructure schematics

Schematic drawings showing railway tracks and equipment are commonly used to visualize railway operations and to communicate system specifications and construction blueprints. Recent advances in on-line collaboration and modeling tools have raised the expectations for quickly making changes to models, resulting in frequent changes to layouts, text, and/or symbols in schematic drawings. Automating the creation of high-quality schematic views from geographical and topological models can help engineers produce and update drawings efficiently. This paper introduces four methods for automatically producing schematic railway drawings with increasing level of quality and control over the result. The final method, implemented in the open-source tool that we have developed, can use any combination of the following optimization criteria, which can have different priorities in different use cases: width and height of the drawing, the diagonal line lengths, and the number of bends. We show how to encode schematic railway drawings as an optimization problem over Boolean and numerical domains, using combinations of unary number encoding, lazy difference constraints, and numerical optimization into an incremental SAT formulation. We compare drawings resulting from each of the four methods, applied to models of real-world engineering projects and existing railway infrastructure. We also show how to add symbols and labels to the track plan, which is important for the usefulness of the final outputs. Since the proposed tool is customizable and efficiently produces high-quality drawings from railML 2.x models, it can be used (as it is or extended) both as an integrated module in an industrial design tool like RailCOMPLETE, or by researchers for visualization purposes.publishedVersio

Bundled Crossings Revisited

An effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges that travel in parallel into \emph{bundles}. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a \emph{bundled crossing}. We consider the problem of bundled crossing minimization: A graph is given and the goal is to find a bundled drawing with at most $k$ bundled crossings. We show that the problem is NP-hard when we require a simple drawing. Our main result is an FPT algorithm (in $k$) when we require a simple circular layout. These results make use of the connection between bundled crossings and graph genus.Comment: Appears in the Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019

The bundled crossing number

We study the algorithmic aspect of edge bundling. A bundled crossing in a drawing of a graph is a group of crossings between two sets of parallel edges. The bundled crossing number is the minimum number of bundled crossings that group all crossings in a drawing of the graph. We show that the bundled crossing number is closely related to the orientable genus of the graph. If multiple crossings and self-intersections of edges are allowed, the two values are identical; otherwise, the bundled crossing number can be higher than the genus. We then investigate the problem of minimizing the number of bundled crossings. For circular graph layouts with a fixed order of vertices, we present a constant-factor approximation algorithm. When the circular order is not prescribed, we get a 6c/c - 2 -approximation for a graph with n vertices having at least cn edges for c > 2. For general graph layouts, we develop an algorithm with an approximation factor of 6c/c - 3 for graphs with at least cn edges for c > 3. © Springer International Publishing AG 2016