In the preprocessing framework for dealing with uncertain data, one is given a set of regions that one is allowed to preprocess to create some auxiliary structure such that when a realization of these regions is given, consisting of one point per region, this auxiliary structure can be used to reconstruct some desired output structure more efficiently than would have been possible without preprocessing. The framework has been successfully applied to several, mostly geometric, computational problems. In this note, we propose using a supersequence of input items as the auxiliary structure, and explore its potential on the problems of sorting and computing the smallest or largest gap in a set of numbers. That is, our uncertainty regions are intervals on the real line, and in the preprocessing phase we output a supersequence of the intervals such that the sorted order / smallest gap / largest gap of any realization is a subsequence of this sequence. We argue that supersequences are simpler than specialized auxiliary structures developed in previous work. An advantage of using supersequences as the auxiliary structures is that it allows us to decouple the preprocessing phase from the reconstruction phase in a stronger sense than was possible in previous work, resulting in two separate algorithmic problems for which different solutions may be combined to obtain known and new results. We identify one key open problem which we believe is of independent interest.

翻译：在针对不确定数据的预处理框架中，给定一组允许进行预处理的区域，通过构建某种辅助结构，使得当这些区域被实现为每个区域包含一个点时，该辅助结构能够比未经预处理时更高效地重建所需的目标输出结构。该框架已成功应用于多个（主要是几何计算）问题。本文提出使用输入项的超序列作为辅助结构，并探讨其在数字排序及计算最小/最大间隙问题中的应用潜力。具体而言，我们的不确定区域是实轴上的区间，在预处理阶段输出这些区间的超序列，使得任意实现值的排序结果/最小间隙/最大间隙均为该序列的子序列。我们认为超序列比以往工作中开发的专用辅助结构更为简洁。使用超序列作为辅助结构的优势在于，其能以比以往工作更强的形式实现预处理阶段与重建阶段的解耦，从而形成两个独立的算法问题——通过组合不同解决方案可获得已知及新的结果。我们指出了一个具有独立研究价值的关键开放性问题。