曲线间区间搜索的复杂性
On the complexity of range searching among curves
摘要：
现代跟踪技术已经使得大量密集采样的移动对象轨迹的收集广泛可用。我们认为一个基本的问题时遇到的分析这样的数据：给定$n$多边形曲线$S$在$\mathbb{R}^d$，预处理$S$到一个数据结构，回答查询曲线$q$和半径$ρ$的曲线$S$有\Frechet距离最多$ρ$到$q$。我们开始全面分析的空间/查询时间权衡这个数据结构化问题。我们的下限意味着，在指针模型模型中，任何达到$Q(n) + O(k)$查询时间的数据结构，其中$k$是输出大小，在最坏的情况下，必须使用大约$Ω\left((n/Q(n))^2\right)$空间，即使查询仅仅是点（对于离散的\Frechet距离）或线段（对于连续的\Frechet距离）。更重要的是，我们表明，更复杂的查询和输入曲线导致额外的对数因子的下限。粗略地说，添加的对数因子的数量与添加到查询和输入曲线复杂度的边的数量成线性关系。这意味着空间/查询时间的权衡是由输入和查询复杂性的指数因子决定的。这种行为解决了范围搜索文献中的一个悬而未决的问题：是否有可能避免在多级分区树的空间和查询时间中使用额外的对数因子。我们对这个问题的回答是否定的。在积极的一面，我们表明，我们可以建立数据结构的\Frechet距离使用半代数范围搜索。我们的离散\Frechet距离的解与下限一致，因为数据结构中的级别数是$O(t)$，其中$t$表示曲线的最大顶点数。对于连续的\Frechet距离，级别数增加到$O(t^2)$。
英文摘要：
Modern tracking technology has made the collection of large numbers of densely sampled trajectories of moving objects widely available. We consider a fundamental problem encountered when analysing such data: Given $n$ polygonal curves $S$ in $\mathbb{R}^d$, preprocess $S$ into a data structure that answers queries with a query curve $q$ and radius $ρ$ for the curves of $S$ that have \Frechet distance at most $ρ$ to $q$. We initiate a comprehensive analysis of the space/query-time trade-off for this data structuring problem. Our lower bounds imply that any data structure in the pointer model model that achieves $Q(n) + O(k)$ query time, where $k$ is the output size, has to use roughly $Ω\left((n/Q(n))^2\right)$ space in the worst case, even if queries are mere points (for the discrete \Frechet distance) or line segments (for the continuous \Frechet distance). More importantly, we show that more complex queries and input curves lead to additional logarithmic factors in the lower bound. Roughly speaking, the number of logarithmic factors added is linear in the number of edges added to the query and input curve complexity. This means that the space/query time trade-off worsens by an exponential factor of input and query complexity. This behaviour addresses an open question in the range searching literature: whether it is possible to avoid the additional logarithmic factors in the space and query time of a multilevel partition tree. We answer this question negatively. On the positive side, we show we can build data structures for the \Frechet distance by using semialgebraic range searching. Our solution for the discrete \Frechet distance is in line with the lower bound, as the number of levels in the data structure is $O(t)$, where $t$ denotes the maximal number of vertices of a curve. For the continuous \Frechet distance, the number of levels increases to $O(t^2)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
