In the imprecise geometry model, the input is an imprecise point set, which is a family of regions $F = (R_1, \ldots,R_n)$, where for each $R_i$ one may retrieve the true point $p_i \in R_i$. By preprocessing $F$, we can construct the output, in our case the Pareto front, on $P$ faster. We efficiently construct the Pareto front of an imprecise point set in the plane. Efficiency is interpreted in two ways: minimizing (i) the number of retrievals, and (ii) the computation time used to determine the set of regions that must be retrieved and to construct the Pareto front. We present an algorithm to construct the Pareto front for possibly overlapping rectangles that is \emph{instance-optimal} with respect to the number of retrievals, meaning that for every fixed input $(F, P)$, there is no algorithm that retrieves asymptotically fewer regions to compute the output. This is a strong algorithmic quality, as it means that our algorithm is competitive even to clairvoyant algorithms which know a correct guess of the output and only have to verify its correctness. In terms of algorithmic running time, instance-optimality is provably unobtainable. We instead present an algorithm which is within a $\log n$-factor of instance-optimality. This generalizes earlier results to overlapping input regions, at only a minor cost in running time. For unit squares, we present an algorithm that is not only instance-optimal in the number of retrievals, but also \emph{universally} optimal in terms of running time, meaning that for any fixed set of regions $F$, no algorithm has a better worst-case running time for all possible point sets $P$. This is the first universally optimal algorithm for overlapping planar input. Compared to previous work, this result improves the degree of overlap, the preprocessing time, the number of retrievals, and the running time.

翻译：在不精确几何模型中，输入是一族不精确点集，即区域族 $F = (R_1, \ldots,R_n)$，其中每个区域 $R_i$ 可返回真实点 $p_i \in R_i$。通过预处理 $F$，我们可以更快地构造 $P$ 上的输出——此处为帕累托前沿。本文高效构造平面上不精确点集的帕累托前沿，效率体现在两个指标上：最小化(i)检索次数，以及(ii)确定需检索的区域集并构造帕累托前沿所需的计算时间。我们提出了一种针对可能重叠矩形的帕累托前沿构造算法，该算法在检索次数上具有\emph{实例最优性}，即对任意固定输入$(F, P)$，不存在任何算法能以渐近更少的区域检索次数计算输出。这一强算法质量意味着我们的算法甚至能与预知输出正确假设的“先知”算法相竞争——后者仅需验证假设正确性即可。就算法运行时间而言，实例最优性被证明无法实现。我们转而提出一种算法，其运行时间与实例最优仅差一个$\log n$因子。这将以微小运行时间成本，将先前结果推广至重叠输入区域场景。针对单位正方形，我们提出的算法不仅在检索次数上达到实例最优，运行时间更实现\emph{全局最优}，即对任意固定区域集$F$，不存在任何算法能在所有可能点集$P$上取得更优最坏情况运行时间。这是首个面向平面重叠输入的全局最优算法。与现有工作相比，本结果在重叠程度、预处理时间、检索次数及运行时间上均实现改进。