Constrained set-up of the tGAP structure for progressive vector data transfer

A promising approach to submit a vector map from a server to a mobile client is to send a coarse representation first, which then is incrementally refined. We consider the problem of defining a sequence of such increments for areas of different land-cover classes in a planar partition. In order to submit well-generalised datasets, we propose a method of two stages: First, we create a generalised representation from a detailed dataset, using an optimisation approach that satisfies certain cartographic constraints. Second, we define a sequence of basic merge and simplification operations that transforms the most detailed dataset gradually into the generalised dataset. The obtained sequence of gradual transformations is stored without geometrical redundancy in a structure that builds up on the previously developed tGAP (topological Generalised Area Partitioning) structure. This structure and the algorithm for intermediate levels of detail (LoD) have been implemented in an object-relational database and tested for land-cover data from the official German topographic dataset ATKIS at scale 1:50 000 to the target scale 1:250 000. Results of these tests allow us to conclude that the data at lowest LoD and at intermediate LoDs is well generalised. Applying specialised heuristics the applied optimisation method copes with large datasets; the tGAP structure allows users to efficiently query and retrieve a dataset at a specified LoD. Data are sent progressively from the server to the client: First a coarse representation is sent, which is refined until the requested LoD is reached

A Network Flow Model for the Analysis of Green Spaces in Urban Areas

Green spaces in urban areas offer great possibilities of recreation, provided that they are easily accessible. Therefore, an ideal city should offer large green spaces close to where its residents live. Although there are several measures for the assessment of urban green spaces, the existing measures usually focus either on the total size of green spaces or on their accessibility. Hence, in this paper, we present a new methodology for assessing green-space provision and accessibility in an integrated way. The core of our methodology is an algorithm based on linear programming that computes an optimal assignment between residential areas and green spaces. In a basic setting, it assigns a green space of a prescribed size exclusively to each resident such that the average distance between residents and assigned green spaces is minimized. We contribute a detailed presentation on how to engineer an assignment-based method such that it yields reasonable results (e.g., by considering distances in the road network) and becomes efficient enough for the analysis of large metropolitan areas (e.g., we were able to process an instance of Berlin with about 130000 polygons representing green spaces, 18000 polygons representing residential areas, and 6 million road segments). Furthermore, we show that the optimal assignments resulting from our method enable a subsequent analysis that reveals both interesting global properties of a city as well as spatial patterns. For example, our method allows us to identify neighborhoods with a shortage of green spaces, which will help spatial planners in their decision making

Bicriteria Aggregation of Polygons via Graph Cuts

We present a new method for the task of detecting groups of polygons in a given geographic data set and computing a representative polygon for each group. This task is relevant in map generalization where the aim is to derive a less detailed map from a given map. Following a classical approach, we define the output polygons by merging the input polygons with a set of triangles that we select from a constrained Delaunay triangulation of the input polygons\u27 exterior. The innovation of our method is to compute the selection of triangles by solving a bicriteria optimization problem. While on the one hand we aim at minimizing the total area of the outputs polygons, we aim on the other hand at minimizing their total perimeter. We combine these two objectives in a weighted sum and study two computational problems that naturally arise. In the first problem, the parameter that balances the two objectives is fixed and the aim is to compute a single optimal solution. In the second problem, the aim is to compute a set containing an optimal solution for every possible value of the parameter. We present efficient algorithms for these problems based on computing a minimum cut in an appropriately defined graph. Moreover, we show how the result set of the second problem can be approximated with few solutions. In an experimental evaluation, we finally show that the method is able to derive settlement areas from building footprints that are similar to reference solutions

Balanced Independent and Dominating Sets on Colored Interval Graphs

We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely \emph{$f$-Balanced Independent Set} ($f$-BIS) and \emph{$f$-Balanced Dominating Set} ($f$-BDS). Let $G=(V,E)$ be a vertex-colored interval graph with a $k$-coloring $\gamma \colon V \rightarrow \{1,\ldots,k\}$ for some $k \in \mathbb N$. A subset of vertices $S\subseteq V$ is called \emph{$f$-balanced} if $S$ contains $f$ vertices from each color class. In the $f$-BIS and $f$-BDS problems, the objective is to compute an independent set or a dominating set that is $f$-balanced. We show that both problems are \NP-complete even on proper interval graphs. For the BIS problem on interval graphs, we design two \FPT\ algorithms, one parameterized by $(f,k)$ and the other by the vertex cover number of $G$. Moreover, we present a 2-approximation algorithm for a slight variation of BIS on proper interval graphs

EulerMerge: Simplifying Euler Diagrams Through Set Merges

Euler diagrams are an intuitive and popular method to visualize set-based data. In a Euler diagram, each set is represented as a closed curve, and set intersections are shown by curve overlaps. However, Euler diagrams are not visually scalable and automatic layout techniques struggle to display real-world data sets in a comprehensible way. Prior state-of-the-art approaches can embed Euler diagrams by splitting a closed curve into multiple curves so that a set is represented by multiple disconnected enclosed areas. In addition, these methods typically result in multiple curve segments being drawn concurrently. Both of these features significantly impede understanding. In this paper, we present a new and scalable method for embedding Euler diagrams using set merges. Our approach simplifies the underlying data to ensure that each set is represented by a single, connected enclosed area and that the diagram is drawn without curve concurrency, leading to well formed and understandable Euler diagrams

Map Matching for Semi-Restricted Trajectories

We consider the problem of matching trajectories to a road map, giving particular consideration to trajectories that do not exclusively follow the underlying network. Such trajectories arise, for example, when a person walks through the inner part of a city, crossing market squares or parking lots. We call such trajectories semi-restricted. Sensible map matching of semi-restricted trajectories requires the ability to differentiate between restricted and unrestricted movement. We develop in this paper an approach that efficiently and reliably computes concise representations of such trajectories that maintain their semantic characteristics. Our approach utilizes OpenStreetMap data to not only extract the network but also areas that allow for free movement (as e.g. parks) as well as obstacles (as e.g. buildings). We discuss in detail how to incorporate this information in the map matching process, and demonstrate the applicability of our method in an experimental evaluation on real pedestrian and bicycle trajectories