# Faster and Simpler Distributed Algorithms for Testing and Correcting Graph Properties in the CONGEST-Model

## TL;DR

This paper introduces faster distributed algorithms in the CONGEST model for testing and correcting various graph properties, including bipartiteness, cycle-freeness, and $H$-freeness, with improved round complexities over prior work.

## Contribution

It presents new, more efficient distributed testing algorithms for multiple graph properties and extends existing methods to broader classes of graphs and properties.

## Key findings

- Optimal bipartiteness tester with $O(rac{ ext{log} n}{ ext{epsilon}})$ rounds.
- Efficient algorithms for testing $H$-freeness for connected $H$ of size up to four.
- Generalization of $k$-path freeness testing to any tree of order $k$.

## Abstract

In this paper we present distributed testing algorithms of graph properties in the CONGEST-model [Censor-Hillel et al. 2016]. We present one-sided error testing algorithms in the general graph model. We first describe a general procedure for converting $\epsilon$-testers with a number of rounds $f(D)$, where $D$ denotes the diameter of the graph, to $O((\log n)/\epsilon)+f((\log n)/\epsilon)$ rounds, where $n$ is the number of processors of the network. We then apply this procedure to obtain an optimal tester, in terms of $n$, for testing bipartiteness, whose round complexity is $O(\epsilon^{-1}\log n)$, which improves over the $poly(\epsilon^{-1} \log n)$-round algorithm by Censor-Hillel et al. (DISC 2016). Moreover, for cycle-freeness, we obtain a \emph{corrector} of the graph that locally corrects the graph so that the corrected graph is acyclic. Note that, unlike a tester, a corrector needs to mend the graph in many places in the case that the graph is far from having the property. In the second part of the paper we design algorithms for testing whether the network is $H$-free for any connected $H$ of size up to four with round complexity of $O(\epsilon^{-1})$. This improves over the $O(\epsilon^{-2})$-round algorithms for testing triangle freeness by Censor-Hillel et al. (DISC 2016) and for testing excluded graphs of size $4$ by Fraigniaud et al. (DISC 2016). In the last part we generalize the global tester by Iwama and Yoshida (ITCS 2014) of testing $k$-path freeness to testing the exclusion of any tree of order $k$. We then show how to simulate this algorithm in the CONGEST-model in $O(k^{k^2+1}\cdot\epsilon^{-k})$ rounds.