Preprocessing Uncertain Data into Supersequences for Sorting and Gaps

2026-01-27

AI Summary

This work addresses the challenge of efficiently reconstructing orderings or extreme gaps for interval-uncertain data once actual values are revealed. To this end, the paper proposes a novel preprocessing framework based on supersequences: during preprocessing, a single supersequence is constructed that embeds all possible realizations as subsequences, enabling rapid extraction of the ordering or gap information for any specific realization. This approach introduces supersequences as a general-purpose auxiliary structure in uncertain data processing—an idea not previously explored—and effectively decouples preprocessing from reconstruction, thereby simplifying algorithm design and facilitating modular composition. The proposed method achieves efficient reconstruction for sorting, minimum gap, and maximum gap problems, and further motivates a key open problem of independent theoretical interest.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Data transparency and provenance

Abstract

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.

Problem

• uncertain data
• preprocessing
• supersequence
• sorting
• gaps

Innovation

• supersequence
• uncertain data
• preprocessing
• sorting
• gap computation