# Improved Network Decompositions using Small Messages with Applications on MIS, Neighborhood Covers, and Beyond

## TL;DR

This paper introduces improved deterministic and randomized distributed algorithms for network decompositions with small messages, leading to faster algorithms for MIS, spanners, set cover, and neighborhood covers in distributed graph settings.

## Contribution

It presents novel deterministic and randomized algorithms for network decompositions with small messages, improving efficiency and success probability over prior methods.

## Key findings

- Faster deterministic network decomposition algorithm.
- Randomized algorithm with exponentially small failure probability.
- Enhanced algorithms for MIS, spanners, and neighborhood covers.

## Abstract

Network decompositions, as introduced by Awerbuch, Luby, Goldberg, and Plotkin [FOCS'89], are one of the key algorithmic tools in distributed graph algorithms. We present an improved deterministic distributed algorithm for constructing network decompositions of power graphs using small messages, which improves upon the algorithm of Ghaffari and Kuhn [DISC'18]. In addition, we provide a randomized distributed network decomposition algorithm, based on our deterministic algorithm, with failure probability exponentially small in the input size that works with small messages as well. Compared to the previous algorithm of Elkin and Neiman [PODC'16], our algorithm achieves a better success probability at the expense of its round complexity, while giving a network decomposition of the same quality. As a consequence of the randomized algorithm for network decomposition, we get a faster randomized algorithm for computing a Maximal Independent Set, improving on a result of Ghaffari [SODA'19]. Other implications of our improved deterministic network decomposition algorithm are: a faster deterministic distributed algorithms for constructing spanners and approximations of distributed set cover, improving results of Ghaffari, and Kuhn [DISC'18] and Deurer, Kuhn, and Maus [PODC'19]; and faster a deterministic distributed algorithm for constructing neighborhood covers, resolving an open question of Elkin [SODA'04].