On Constructing Spanners from Random Gaussian Projections

TL;DR

This paper establishes near-optimal lower bounds for graph sketching algorithms in constructing spanners, explaining the difficulty of approximating shortest path metrics and advancing understanding of the limitations of sketching methods.

Contribution

The paper proves a strong lower bound for a broad class of graph sketching algorithms for spanner construction, matching recent upper bounds and addressing a longstanding open problem.

Findings

Lower bounds match recent algorithmic bounds

Progress on a conjecture regarding sketching limitations

Clarifies the difficulty of graph spanner approximation via sketching

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.