Consider the natural question of how to measure the similarity of curves in the plane by a quantity that is invariant under translations of the curves. Such a measure is justified whenever we aim to quantify the similarity of the curves' shapes rather than their positioning in the plane, e.g., to compare the similarity of handwritten characters. Perhaps the most natural such notion is the (discrete) Fr\'echet distance under translation. Unfortunately, the algorithmic literature on this problem yields a very pessimistic view: On polygonal curves with $n$ vertices, the fastest algorithm runs in time $O(n^{4.667})$ and cannot be improved below $n^{4-o(1)}$ unless the Strong Exponential Time Hypothesis fails. Can we still obtain an implementation that is efficient on realistic datasets? Spurred by the surprising performance of recent implementations for the Fr\'echet distance, we perform algorithm engineering for the Fr\'echet distance under translation. Our solution combines fast, but inexact tools from continuous optimization (specifically, branch-and-bound algorithms for global Lipschitz optimization) with exact, but expensive algorithms from computational geometry (specifically, problem-specific algorithms based on an arrangement construction). We combine these two ingredients to obtain an exact decision algorithm for the Fr\'echet distance under translation. For the related task of computing the distance value up to a desired precision, we engineer and compare different methods. On a benchmark set involving handwritten characters and route trajectories, our implementation answers a typical query for either task in the range of a few milliseconds up to a second on standard desktop hardware. We believe that our implementation will enable the use of the Fr\'echet distance under translation in applications, whereas previous approaches would have been computationally infeasible.

中文翻译：

---

当利普希茨遛狗时：翻译下离散 Fr\'echet 距离的算法工程

考虑一个自然问题，即如何通过曲线平移不变的量来测量平面中曲线的相似性。每当我们的目标是量化曲线形状的相似性而不是它们在平面中的位置时，这种测量是合理的，例如，比较手写字符的相似性。也许最自然的这种概念是翻译下的（离散）Fr\'echet 距离。不幸的是，关于这个问题的算法文献产生了一个非常悲观的观点：在具有 $n$ 个顶点的多边形曲线上，最快的算法在 $O(n^{4.667})$ 时间内运行并且不能在 $n^{4- o(1)}$ 除非强指数时间假设失败。我们还能获得在现实数据集上有效的实现吗？受到 Fr\'echet 距离最近实现的惊人性能的刺激，我们对翻译下的 Fr\'echet 距离执行算法工程。我们的解决方案将来自连续优化的快速但不精确的工具（特别是用于全局 Lipschitz 优化的分支定界算法）与来自计算几何的精确但昂贵的算法（特别是基于排列构造的特定问题算法）相结合。我们将这两种成分结合起来，以获得平移下 Fr\'echet 距离的精确决策算法。对于计算距离值达到所需精度的相关任务，我们设计并比较了不同的方法。在涉及手写字符和路线轨迹的基准集上，我们的实现在标准桌面硬件上在几毫秒到一秒的范围内回答任一任务的典型查询。我们相信我们的实现将能够在应用程序中使用翻译下的 Fr\'echet 距离，而以前的方法在计算上是不可行的。