Simplification of polyline bundles of graphs and trees

Abstract

Polyline simplification is a well-studied optimization problem, in which a givenpolyline shall be replaced by a polyline with fewer vertices which still represents the shapeof the original polyline faithfully. In this paper, we propose and study a generalization ofthe polyline simplification problem. Instead of a single polyline, we are given a set of ℓ polylines possibly sharing some line segments and vertices. We call such a set apolylinebundle. The task is tosimplifyeach polyline L of a given polyline bundle by keeping a subsetof its vertices such that (i) the Hausdorff or Fréchet distance betweenLand its simplifiedcounterpart does not exceed a given distance threshold δ, (ii) a shared vertex is either kept ordiscarded in all polylines of the polyline bundle (we refer to this requirement asconsistency) and (iii) the number of kept vertices in the polyline bundle is minimized. To justify this definition, we argue that consistency is crucial to get meaningful and aesthetically pleasing outputs.

Regarding the computational complexity of polyline bundle simplification, we prove that this problem is NP-hard to approximate within a factor of n1/3−ε for any ε >0, where n is the number of vertices in the polyline bundle. This inapproximability even applies to planar inputs and also to instances with only ℓ = 2 polylines. However, we identify thesensitivity of the solution to the choice of the distance threshold δ as a reason for this strong inapproximability. In particular, we prove that if we employ the Fréchet distance and allow δ to be exceeded by a factor of 2 in the solution, then we can find a simplified polyline bundlewith no more than O(log(ℓ+n)) · OPT vertices in polytime, providing us with an efficient bi-criteria approximation. In addition, we show that the polyline simplification problem issolvable in polytime in case the polylines form a rooted tree. We further present a greedy heuristic that decomposes general bundles into tree bundles, which then can be simplified individually and optimally. In our experimental study, we compare the performance of the bi-criteria approximation algorithm and the tree bundle decomposition algorithm on public transit networks and movement trajectories. We show that in case the polylines form grid-like structures, the bi-criteria approximation algorithm outputs smaller simplifications, but the tree bundle decomposition algorithm scales better and produces superior results on polyline bundles derived from paths in embedded road networks.