The Fr\'echet distance is a popular similarity measure between curves. For some applications, it is desirable to match the curves under translation before computing the Fr\'echet distance between them. This variant is called the Translation Invariant Fr\'echet distance, and algorithms to compute it are well studied. The query version, however, is much less well understood. We study Translation Invariant Fr\'echet distance queries in a restricted setting of horizontal query segments. More specifically, we prepocess a trajectory in $O(n^2 \log^2 n)$ time and space, such that for any subtrajectory and any horizontal query segment we can compute their Translation Invariant Fr\'echet distance in $O(\text{polylog} \, n)$ time. We hope this will be a step towards answering Translation Invariant Fr\'echet queries between arbitrary trajectories.

中文翻译：

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平移不变弗雷谢特距离查询

Fr'echet距离是曲线之间流行的相似性度量。对于某些应用，希望在计算平移曲线之间的Fr'echet距离之前对其进行匹配。该变体称为平移不变Fr'echet距离，并且对其计算算法进行了深入研究。但是，查询版本了解得很少。我们在水平查询段的受限设置中研究平移不变Fr'echet距离查询。更具体地说，我们在$ O（n ^ 2 \ log ^ 2 n）$的时间和空间中放置一条轨迹，这样对于任何子轨迹和任何水平查询段，我们都可以计算它们的平移不变Fr'echet距离，在$ O（ \ text {polylog} \，n）$时间。我们希望这将是迈向翻译不变式Fr \'的一步