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On Constructing Spanners from Random Gaussian Projections

Authors Sepehr Assadi, Michael Kapralov, Huacheng Yu

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Sepehr Assadi
  • Department of Computer Science, Rutgers University, New Brunswick, NJ, USA
Michael Kapralov
  • School of Computer and Communication Sciences, EPFL, Lausanne, Switzerland
Huacheng Yu
  • Department of Computer Science, Princeton University, NJ, USA

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Sepehr Assadi, Michael Kapralov, and Huacheng Yu. On Constructing Spanners from Random Gaussian Projections. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 57:1-57:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.57

Abstract

Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA'12) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area. We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA'21), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Sparsification and spanners
Keywords
  • sketching algorithm
  • lower bound
  • graph spanner

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