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Relational Approximations for Subspace Primitives

Authors Xiang Liu , Kasturi Varadarajan

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • relational algorithm
  • Euclidean distance
  • subspace approximation

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Abstract

We explore fundamental geometric computations on point sets that are given to the algorithm implicitly. In particular, we are given a database which is a collection of tables with numerical values, and the geometric computation is to be performed on the join of the tables. Explicitly computing this join takes time exponential in the size of the tables. We are therefore interested in geometric problems that can be solved by algorithms whose running time is a polynomial in the size of the tables. Such relational algorithms are typically not able to explicitly compute the join. 
To avoid the NP-completeness bottleneck, researchers assume that the tables have a tractable combinatorial structure, like being acyclic. Even with this assumption, simple geometric computations turn out to be non-trivial and sometimes intractable. In this article, we study the problem of computing the maximum distance of a point in the join to a given subspace, and develop approximation algorithms for this NP-hard problem.

Cite As Get BibTex

Xiang Liu and Kasturi Varadarajan. Relational Approximations for Subspace Primitives. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.12

Author Details

Xiang Liu
  • Department of Computer Science, University of Iowa, Iowa City, IA, USA
Kasturi Varadarajan
  • Department of Computer Science, University of Iowa, Iowa City, IA, USA

Acknowledgements

The authors thank Kirk Pruhs for discussions that led to the problems studied here, and the organizers of the 2023 Workshop on Fined-Grained Complexity, Logic, and Query Evaluation at the Simons Institute for the Theory of Computing for facilitating these discussions.

References

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