Given a set of n segments and a query shape Q , the windowing length query asks for finding the sum of the lengths of the parts of the segments that lie inside Q . The popular places problem of a set of curves asks for the subset of the plane where each query shape centered at a point of that region intersects with at least f distinct curves. For square queries, an optimal $$O(n^2)$$ O ( n 2 ) time algorithm and a matching lower bound exist. We solve the length query problem for convex polygons and disks as query shapes, with $$O(\log n+k)$$ O ( log n + k ) query time and polynomial preprocessing time that depends on the complexity of the query shape. We define a new version of the problem of finding popular places in a set of trajectories where the center of a query is a popular place if the length of the curves inside that query is at least f and use our data structure to solve the original problem as well as this new version. Other than length queries, we solve reporting queries that return the set of intersected segments. For disk queries, we design a point-location data structure for congruent disks with $$O(\log n)$$ O ( log n ) query time and $$O(n^3\log n)$$ O ( n 3 log n ) preprocessing. We also give algorithms for computing the length query for c -packed curves, which are a class of curves for which the length of the curve inside a disk of radius r is upper-bounded by cr , where c is a constant. Also, we use length queries for polygons to approximate the minimum value c for which a curve is c -packed, if such a c exists. Our results extend to MRC and MPC models for MapReduce, where we address these problems on a set of x -monotone curves. The round complexities of our MapReduce algorithms are constant. In addition, we also implemented our popular places algorithms on trajectories on inputs as big as 15K points to evaluate the efficiency of our algorithms in practice.

中文翻译：

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使用 Minkowski sum 的窗口查询及其对 MapReduce 的扩展

给定一组 n 个片段和一个查询形状 Q ，窗口长度查询要求找到位于 Q 内的片段部分的长度总和。一组曲线的流行地方问题要求平面的子集，其中每个查询形状以该区域的一个点为中心，与至少 f 条不同的曲线相交。对于方形查询，存在最优 $$O(n^2)$$O ( n 2 ) 时间算法和匹配下界。我们解决了凸多边形和圆盘作为查询形状的长度查询问题，$$O(\log n+k)$$O ( log n + k ) 查询时间和多项式预处理时间取决于查询形状的复杂性. 我们定义了一个新版本的问题，即在一组轨迹中找到热门地点，如果查询的中心是热门地点，如果该查询内的曲线长度至少为 f，并使用我们的数据结构来解决原始问题以及这个新版本。除了长度查询，我们解决了返回相交段集的报告查询。对于磁盘查询，我们为具有 $$O(\log n)$$ O ( log n ) 查询时间和 $$O(n^3\log n)$$ O ( n 3 log n ) 预处理。我们还给出了计算 c 压缩曲线长度查询的算法，这是一类曲线，其半径为 r 的圆盘内的曲线长度上限为 cr ，其中 c 是常数。还，我们使用多边形的长度查询来近似曲线被 c 填充的最小值 c，如果这样的 ac 存在。我们的结果扩展到 MapReduce 的 MRC 和 MPC 模型，我们在一组 x 单调曲线上解决这些问题。我们的 MapReduce 算法的回合复杂度是恒定的。此外，我们还在高达 15K 点的输入轨迹上实施了我们的流行地点算法，以评估我们算法在实践中的效率。