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.
翻译:波利线 Bundle 简化( PBS) 问题是古典多线简化问题的概括性。 在一系列多线性( 可能共享线段和点) 中, PBS 要求将这些多线性简化到一个给定的距离阈值。 这里, 一致意味着每个点要么被保存在包含它的所有多线性( PBS) 中, 要么被从包含它的所有多线性( PPBS) 中丢弃。 在先前的工作中, 事实证明 PBS 很难在${ frac{ 1 ⁇ 3} -\ varepsilon} 范围内, 任何值为 $( Valepsilon) > 0$( $) 的值, 其中, 美元表示输入输入的值是输入的点数。 这种硬性结果甚至保存在两个多线性( $ $ ) 。 在本文中, 最精确的 O 里程中, 我们的硬性数据会显示的是, 我们的直径( O) 的直径直径( ) 直径) 直径( O) 的直径) 的直径( O) 直径) 直径) 直径( O) 直) 的直径) 。