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Preprocessing Imprecise Points for the Pareto Front
arXiv - CS - Computational Geometry Pub Date : 2021-01-15 , DOI: arxiv-2101.06079
Ivor van der Hoog, Irina Kostitsyna, Maarten Löffler, Bettina Speckmann

In the preprocessing model for uncertain data we are given a set of regions R which model the uncertainty associated with an unknown set of points P. In this model there are two phases: a preprocessing phase, in which we have access only to R, followed by a reconstruction phase, in which we have access to points in P at a certain retrieval cost C per point. We study the following algorithmic question: how fast can we construct the pareto front of P in the preprocessing model? We show that if R is a set of pairwise-disjoint axis-aligned rectangles, then we can preprocess R to reconstruct the Pareto front of P efficiently. To refine our algorithmic analysis, we introduce a new notion of algorithmic optimality which relates to the entropy of the uncertainty regions. Our proposed uncertainty-region optimality falls on the spectrum between worst-case optimality and instance optimality. We prove that instance optimality is unobtainable in the preprocessing model, whenever the classic algorithmic problem reduces to sorting. Our results are worst-case optimal in the preprocessing phase; in the reconstruction phase, our results are uncertainty-region optimal with respect to real RAM instructions, and instance optimal with respect to point retrievals.

中文翻译:

预处理帕累托阵线的不精确点

在不确定性数据的预处理模型中,我们给了一组区域R,该区域对与一组未知点P相关的不确定性进行建模。在此模型中,有两个阶段:预处理阶段,在该阶段我们只能访问R,通过重建阶段,在该阶段我们可以以每点一定的检索成本C访问P中的点。我们研究以下算法问题:在预处理模型中,构造P的pareto front的速度有多快?我们证明,如果R是一组成对不相交的轴对齐矩形,则可以预处理R以有效地重建P的Pareto前沿。为了完善我们的算法分析,我们引入了一种新的算法最优性概念,该概念与不确定性区域的熵有关。我们提出的不确定性区域最优性落在最坏情况最优性和实例最优性之间的频谱上。我们证明,每当经典算法问题归结为排序时,实例最优性就无法在预处理模型中获得。在预处理阶段,我们的结果是最坏的情况。在重建阶段,我们的结果是相对于实际RAM指令而言是不确定区域最优,而对于点检索而言是实例最优。
更新日期:2021-01-18
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