Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of detail. We present an $O(n^{3}m)$-time algorithm that takes a polygonal curve of n vertices and produces a set of consistent simplifications for m scales while minimizing the cumulative simplification complexity. This algorithm is compatible with distance measures such as the Hausdorff, the Fréchet and area-based distances, and enables simplification for continuous scaling in $O(n^5)$ time. To speed up this algorithm in practice, a technique is presented for efficiently constructing many so-called shortcut graphs under the Hausdorff distance, as well as a representation of the shortcut graph that enables us to find shortest paths in anticipated $O(n\log n)$ time on spatial data, improving over $O(n^2)$ time using existing algorithms. Experimental evaluation of these techniques on geospatial data reveals a significant improvement of using shortcut graphs for progressive and non-progressive curve simplification, both in terms of running time and memory usage.

在许多应用程序中，简化不同细节级别的多边形曲线是一个重要问题。现有的几何优化算法仅能使单个细节级别的简化曲线的复杂性最小化。我们提出一个$Ø（{ñ}^3米）$时间算法，该算法采用n个顶点的多边形曲线并为m个比例尺生成一组一致的简化，同时将累积的简化复杂性降至最低。该算法与诸如Hausdorff，Fréchet和基于区域的距离之类的距离度量兼容，并且可以简化连续缩放的过程。$Ø（{ñ}^5）$时间。为了在实践中加快该算法的速度，提出了一种在Hausdorff距离下有效构造许多所谓的捷径图的技术，以及表示该捷径图的一种表示形式，它使我们能够找到预期路径中的最短路径。$Ø（ñ\mathrm{日志}ñ）$ 时间上的空间数据，改善 $Ø（{ñ}^2）$时间使用现有算法。对这些技术在地理空间数据上进行的实验评估表明，无论是在运行时间还是在内存使用方面，都可以将快捷方式图用于渐进式和非渐进式曲线的简化。