Given a curve P with points in \(\mathbb {R}^d \) in a streaming fashion, and parameters ε > 0 and k, we construct a distance oracle that uses \(O(\frac{1}{\varepsilon })^{kd}\log \varepsilon ^{-1} \) space, and given a query curve Q with k points in \(\mathbb {R}^d \), returns in \(\tilde{O}(kd) \) time a 1 + ε approximation of the discrete Fréchet distance between Q and P. In addition, we construct simplifications in the streaming model, oracle for distance queries to a sub-curve (in the static setting), and introduce the zoom-in problem. Our algorithms work in any dimension d, and therefore we generalize some useful tools and algorithms for curves under the discrete Fréchet distance to work efficiently in high dimensions.

以流式方式给定一条在 \(\mathbb {R}^d \) 中具有点的曲线P ，以及参数 ε > 0 和k，我们构造一个使用 \(O(\frac{1}{\varepsilon })^{kd}\log \varepsilon ^{-1} \) 空间的距离预言机，并给定在 \(\mathbb {R}^d \) 中具有 k 个点的查询曲线 Q，返回 \(\tilde{O}(kd) \) 时间 Q 和 P 之间离散Fréchet距离的1 + ε近似值。此外，我们在流模型中构建了简化，用于对子曲线的距离查询（在静态设置中），并引入了放大问题。我们的算法适用于任何维度d，因此我们概括了一些有用的工具和算法，用于离散 Fréchet 距离下的曲线，以便在高维度下高效工作。