# Local Computation Algorithms for Spanners

## TL;DR

This paper introduces local computation algorithms for graph spanners that efficiently determine edge inclusion in sparse spanners without fully constructing them, enabling scalable processing of massive graphs.

## Contribution

It presents the first LCAs for graph spanners with optimal size-stretch trade-offs and sublinear probe complexity, extending previous work and establishing lower bounds.

## Key findings

- Existence of LCAs for (2r-1)-spanners with optimal size and probe complexity for r=2,3.
- LCAs for k-spanners with improved probe complexity for graphs with bounded degree.
- Polynomial lower bounds on probe complexity for spanner LCAs and related problems.

## Abstract

A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known existential size-stretch trade-off efficiently. Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but also the output spanner might be too large for a single processor to store. To tackle this challenge, we initiate the study of local computation algorithms (LCAs) for graph spanners in general graphs, where the algorithm should locally decide whether a given edge $(u,v) \in E$ belongs to the output spanner. Such LCAs give the user the `illusion' that a specific sparse spanner for the graph is maintained, without ever fully computing it. We present the following results: -For general $n$-vertex graphs and $r \in \{2,3\}$, there exists an LCA for $(2r-1)$-spanners with $\widetilde{O}(n^{1+1/r})$ edges and sublinear probe complexity of $\widetilde{O}(n^{1-1/2r})$. These size/stretch tradeoffs are best possible (up to polylogarithmic factors). -For every $k \geq 1$ and $n$-vertex graph with maximum degree $\Delta$, there exists an LCA for $O(k^2)$ spanners with $\widetilde{O}(n^{1+1/k})$ edges, probe complexity of $\widetilde{O}(\Delta^4 n^{2/3})$, and random seed of size $\mathrm{polylog}(n)$. This improves upon, and extends the work of [Lenzen-Levi, 2018]. We also complement our results by providing a polynomial lower bound on the probe complexity of LCAs for graph spanners that holds even for the simpler task of computing a sparse connected subgraph with $o(m)$ edges.