Consistent Simplification of Polyline Tree Bundles

TL;DR

This paper investigates the complexity of simplifying polyline bundles, especially tree structures, showing NP-hardness persists in planar cases and providing efficient algorithms for tree bundles with practical heuristics.

Contribution

It proves NP-hardness of PBS for planar inputs, develops polynomial-time algorithms for tree bundles, and introduces heuristics for general polyline bundles.

Findings

NP-hardness remains for planar inputs

Polynomial algorithms for tree bundles under Fréchet and Hausdorff distances

Heuristic decomposition improves applicability to real data

Abstract

The Polyline Bundle Simplification (PBS) problem is a generalization of the classical polyline simplification problem. Given a set of polylines, which may share line segments and points, PBS asks for the smallest consistent simplification of these polylines with respect to a given distance threshold. Here, consistent means that each point is either kept in or discarded from all polylines containing it. In previous work, it was proven that PBS is NP-hard to approximate within a factor of $n^{\frac{1}{3}-\varepsilon}$ for any $\varepsilon > 0$ where $n$ denotes the number of points in the input. This hardness result holds even for two polylines. In this paper we first study the practically relevant setting of planar inputs. While for many combinatorial optimization problems the restriction to planar settings makes the problem substantially easier, we show that the inapproximability bound known for general inputs continues to hold even for planar inputs. We proceed with the interesting special case of PBS where the polylines form a rooted tree. Such tree bundles naturally arise in the context of movement data visualization. We prove that optimal simplifications of these tree bundles can be computed in $O(n^3)$ for the Fr\'echet distance and in $O(n^2)$ for the Hausdorff distance (which both match the computation time for single polylines). Furthermore, we present a greedy heuristic that allows to decompose polyline bundles into tree bundles in order to make our exact algorithm for trees useful on general inputs. The applicability of our approaches is demonstrated in an experimental evaluation on real-world data.