# Fast Deterministic Constructions of Linear-Size Spanners and Skeletons

## TL;DR

This paper presents the first deterministic distributed algorithms in the CONGEST model for constructing linear-size skeletons and spanners with various stretch factors, achieving efficient time complexity and small message sizes.

## Contribution

It introduces novel deterministic algorithms for sparse skeletons and spanners in distributed networks, improving over prior randomized or message-heavy methods.

## Key findings

- Deterministic construction of linear-size skeletons in subexponential time.
- Efficient algorithms for spanners with polylogarithmic stretch and near-linear edges.
- Lightweight algorithms requiring no heavy computations.

## Abstract

In the distributed setting, the only existing constructions of \textit{sparse skeletons}, (i.e., subgraphs with $O(n)$ edges) either use randomization or large messages, or require $\Omega(D)$ time, where $D$ is the hop-diameter of the input graph $G$. We devise the first deterministic distributed algorithm in the CONGEST model (i.e., uses small messages) for constructing linear-size skeletons in time $2^{O(\sqrt{{\log n}\cdot{\log{\log n}}})}$. We can also compute a linear-size spanner with stretch $polylog(n)$ in low deterministic polynomial time, i.e., $O(n^\rho)$ for an arbitrarily small constant $\rho >0$, in the CONGEST model. Yet another algorithm that we devise runs in $O({\log n})^{\kappa-1}$ time, for a parameter $\kappa=1,2,\dots,$ and constructs an $O({\log n})^{\kappa-1}$ spanner with $O(n^{1+1/\kappa})$ edges. All our distributed algorithms are lightweight from the computational perspective, i.e., none of them employs any heavy computations.