Map Simplification with Topology Constraints: Exactly and in Practice

Abstract

We consider the classical line simplification problem subject to a given error bound but with additional topology constraints as they arise for example in the map rendering domain. While theoretically inapproximability has been proven for these problem variants, we show that in practice one can solve medium sized instances optimally using an integer linear programming approach and larger instances using an heuristic approach which for medium-sized real-world instances yields close-to-optimal results. Our approaches are evaluated on data sets which are synthetically generated, stem from the OpenStreetMap project [1], and the recent GISCup competition [3].

Area-Preserving Subdivision Simplification with Topology Constraints: Exactly and in Practice

“…We extended the heuristic of [10] and enabled it to also handle area-preservation constraints. We further improved the algorithm by introducing ε-cones, which replaces the previously used upper-bound and leads to faster execution times and/or more removed vertices.…”

“…In [10] Funke et al describe an algorithm based on a Delaunay triangulation, which solves MSLOCTOPOP quickly. They also introduced the aforementioned notion of local topology consistency, which we adopt.…”

“…We extend the approach of [10] by constraining the displaced area of the simplification. We also modify their algorithm to allow for a more precise yet quickly to evaluate method to bound the distance between the intermediate simplification and the input.…”

“…v 9 v 0 and an outer face which also contains a topology constraint point p. For sufficiently large values of ε, the simplification shown in Figure 2 We believe that, in map rendering applications, it is more natural to demand that shortcuts locally do not make points switch sides. We therefore employ the local topology definition from [10]. This definition also matches what is implied by strong homotopy according to [1].…”

Finding Optimal Sequences for Area Aggregation—A ⋆ vs. Integer Linear Programming

“…Reference Oehrlein and Haunert [46] aggregated the departments of France according to unemployment rates based on integer linear programming; they used a cutting-plane method to speed up solving their ILP. Reference Funke et al [18] simplified administrative boundaries based on an ILP. Their aim was to minimize the number of edges while keeping the resultant boundaries close to the original ones and avoiding any intersection.…”